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Probability and Mathematical Physics A Volume in Honor of Stanislav Molchanov

Volume 42

C CRM 1z

M

PROCEEDINGS & LECTURE NOTES Centre de Recherches Mathematiques Universite de Montreal

Probability and Mathematical Physics A Volume in Honor of Stanislav Molchanov Donald A. Dawson Vojkan Jakgic Boris Vainberg Editors The Centre de Recherches Mathematiques (CRM) of the Universite de Montreal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Universite de Montreal, the Province of Quebec (FCAR), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the Institut des Sciences Mathematiques (ISM) of Montreal, whose constituent members are Concordia University, McGill University, the Universite de Montreal, the Universite du Quebec a Montreal, and the Ecole Polytechnique. The CRM may be reached on the Web at www.crm.umontreal.ca.

American Mathematical Society Providence, Rhode Island USA

The production of this volume was supported in part by the Fonds pour la Formation de Chercheurs et 1'Aide a la Recherche (Fonds FCAR) and the Natural Sciences and Engineering Research Council of Canada (NSERC). 2000 Mathematics Subject Classification. Primary 60-06, 35-06.

Library of Congress Cataloging-in-Publication Data Probability and mathematical physics : a volume in honor of Stanislav Molchanov / Donald A. Dawson, Vojkan Jaksic, Boris Vainberg, editors. p. cm. - (CRM proceedings & lecture notes, ISSN 1065-8580 ; v. 42) Includes bibliographical references. ISBN 978-0-8218-4089-4 (alk. paper)

1. Probabilities-Congresses. 2. Mathematical physics-Congresses. I. Molchanov, S. A. (Stanislav A.) II. Dawson, Donald Andrew, 1937- III. Jaksic, Vojkan, 1964- IV. Vainberg, B. R. (Boris Rufimovich) QC20.7.P7P77 2007 530.15-dc22

2007060768

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America.

Q The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This volume was submitted to the American Mathematical Society in camera ready form by the Centre de Recherches Mathematiques. Visit the AMS home page at http://www.ams.org/ 10987654321 121110090807

Contents vii

Preface

Biography of Stanislav Molchanov

ix

Transition Asymptotics for Reaction-Diffusion in Random Media Gerard Ben Arous, Stanislav Molchanov, and Alejandro F. Ramirez Extreme Value Theory for Random Exponentials Leonid V. Bogachev

1

41

Singular Continuous and Dense Point Spectrum for Sparse Trees with Finite Dimensions

Jonathan Breuer

65

Some New Estimates on the Spectral Shift Function Associated with Random Schrodinger Operators

Jean-Michel Combes, Peter D. Hislop, and Frederic Klopp

85

On Phase Transitions and Limit Theorems for Homopolymers M. Cranston and S. Molchanov

97

Asymptotics of the Poincare Functions Gregory Derfel, Peter J. Grabner, and Fritz Vogl

113

Hamiltonian Extension and Eigenfunctions for a Time Dispersive Dissipative String Alexander Figotin and Jeffrey Schenker

131

Localization at Low Energies for Attractive Poisson Random Schrodinger Operators Francois Germinet, Peter D. Hislop, and Abel Klein

153

On the Influence of Random Perturbations on the Propagation of Waves Described by a Periodic Schrodinger Operator Yuri A. Godin, Stanislav Molchanov, and Boris Vainberg

167

Spectral Theory of 1-D Schrodinger Operators with Unbounded Potentials A. Gordon, J. Holt, and S. Molchanov

181

Fermi-Dirac Generators and Tests for Randomness Alexander Gordon, Stanislav Molchanov, and Joseph Quinn

199

The Spectral Problem, Substitutions and Iterated Monodromy Rostislav Grigorchuk, Dmytro Savchuk, and Zoran Sunic

225

V

CONTENTS

vi

On Scattering of Solitons for Wave Equation Coupled to a Particle Valery Imaykin, Alexander Komech, and Boris Vainberg

249

Purely Absolutely Continuous Spectrum for some Random Jacobi Matrices Uri Kaluzhny and Yoram Last

273

The Parabolic Anderson Model and Its Universality Classes Wolfgang Konig

283

An Inverse Problem for Gibbs Fields Leonid Koralov

299

Hierarchical Anderson Model Evgenij Kritchevski

309

Integral Representations of Solutions of Periodic Elliptic Equations Peter Kuchment

323

Inverse Spectral Problems for Schrodinger Operators with Energy Depending Potentials A. Laptev, R. Shterenberg, and V. Sukhanov

341

Theory of Point Processes and Some Basic Notions in Energy Level Statistics Nariyuki Minami 353

On the Law of Addition of Random Matrices: Covariance and the Central Limit Theorem for Traces of Resolvent L. Pastur and V. Vasilchuk

399

Green's Functions of Generalized Laplacians Philippe Poulin

417

Orthogonal Polynomials with Exponentially Decaying Recursion Coefficients Barry Simon

453

Poisson Statistics for Eigenvalues: From Random Schrodinger Operators to Random CMV Matrices Mihai Stoiciu

465

Preface A conference celebrating Stanislav Molchanov's 65th birthday took place from

June 27 to July 1, 2005 at the CRM in Montreal. The theme of the conference, "Probability and Mathematical Physics," reflected the wide range of Molchanov's scientific interests and in a similar spirit the present proceedings cover an exceptionally broad range of fields. As conference organizers and editors of the volume, we wish to thank all authors for their contributions. We also wish to thank Jacques Hurtubise and Francois Lalonde for making the conference and this volume possible, and to the CRM staff for the superb conference organization. Don Dawson, Vojkan Jaksic, Boris Vainberg February 2007

vii

Stanislav Molchanov

Stanislav Molchanov (Stas) was born on December 21, 1940, in Snetinovo, a small village in the Ivanovo region of Russia. His parents were high school teachers. After finishing high school in 1958, he became a student of Moscow State University. Starting from his sophomore year, Stas was a regular participant (together with his

classmates N. Krylov, M. Malutov and some older students, such as M. Freidlin and A. Wentzel) in a student research seminar chaired by Professor E. Dynkin. Later Dynkin became his Ph.D. adviser. Stas defended his Ph.D. thesis, entitled Martin boundary for Markov processes, in 1966. He then became an Assistant Professor in the probability theory division at the Department of Mechanics and Mathematics of Moscow State University. Stas received the advanced degree of Doctor of Physical and Mathematical sciences in 1983. His advanced thesis was entitled Spectral theory of one-dimensional Schrodinger operators. He taught at Moscow State University for 25 years, first as an Assistant, then as an Associate and eventually Full Professor. Stas arrived in the U.S. in 1991. He worked for three semesters at the University of California (Irvine) and a year at the University of Southern California (Los Angeles). Since 1994 Stas has been Professor at the University of North Carolina (Charlotte). Moscow State University provided a unique opportunity for undergraduate and graduate students as well as young professors to learn not only from senior professors, but also from each other. Stas has very warm memories of scientific discussions

with M. Freidlin after which his interest turned toward mathematical physics and (stochastic) differential equations. Later, under the influence V. Arnol'd he started working in the field of spectral theory. Stas published a very important paper [1] in 1975 on exact asymptotic expansions (for small t) for fundamental solutions of parabolic problems on Riemannian manifolds. His work in localization theory was triggered by his collaboration with I. Lifshitz and L. Pastur. In 1980, he published (together with I. Goldsheid and L. Pastur) a well-known paper [2] containing the first results on localization for one-dimensional Schrodinger operators. Later he also proved the exponential decay of eigenfunctions and the Poisson statistics for the eigenvalues for these operators [3, 4]. In the 90s, together with M. Aizenmann, Stas extended these results to the multidimensional case [7]. Now their approach to prove the localization is called the Aizenman-Molchanov method, and it is one of the most commonly used methods in the field. At the end of the 80s, S. Molchanov, in collaboration with J. Zeldovich, A. Ruzmaikin, and D. Sokolov, developed the intermittency theory for non-stationary random media [5, 6]. Their works started at the physical level of rigor and had applications to magneto-hydrodynamic of stars and intermittency of the temperature ix

STANISLAV MOLCHANOV

x

field of the ocean. The mathematical theory of intermittency was developed later by S. Molchanov in collaboration with Jii. Gartner, R. Carmona, and others [8, 9]. We mentioned only some contributions made by S. Molchanov in different areas of mathematics. In fact, he published more than 250 papers on stochastic differential equations, spectral theory for deterministic and random operators, localization and intermittency, mathematical physics and optics, and other topics. He was the

Ph.D. adviser of more than 40 students, but a much wider circle of people has learned from him and appreciated the chance to work with him. Vojkan Jaksic, Boris Vainberg

References 1. Diffusion processes and Riemannian geometry, Soviet Uspekhi Math, Sci. 30 (1975), no. 1, 2-59.

2. A pure point spectrum for 1D Schrodinger operator, Punct. Anal. and Appl 11 (1977), 1-10 (with I Goldsheid and L. Pastur). 3. The structure of eigenfunctions for 1D disordered systems, Math. USSR Izvestia 12 (1978), 69-101.

4. Local structure of the spectrum of 1D random Schrodinger operator, Comm. Math. Phys. 78 (1981), 429-446.

5. Intermittency, diffusion and generation in nonstationary random media, Sov. Sci. Reviews, Section C Math. Phys. 7 (1988), 1-110 (with Ja. Zeldovich, A. Ruzmaikin, D. Sokolov). 6. Ideas in the theory of random media, Acta Appl. Math. 12 (1991), 139-182. 7. Localization at large disorder and extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), 245-278 (with M. Aizenman). 8. Lectures in random media, Lecture notes, vol. 1581, 1994, Springer (with D. Bakry, R. Grill). 9. Parabolic Anderson model and intermittency, Memoir AMS 158 (1995), (with R. Carmona).

Centre de Recherches CRM Proceedings and Lecture Notes Volume 42, 2007

Transition Asymptotics for Reaction - Diffusion in Random Media Gerard Ben Arous, Stanislav Molchanov, and Alejandro F. Ramirez ABSTRACT. We describe a universal transition mechanism between annealed and quenched regimes in the context of reaction -diffusion in random media.

We study the total population size for random walks which branch and annihilate on Zd, with time-independent random rates. The random walks are independent, continuous time, rate 2dn, simple, symmetric, with t£ > 0. A random walk at x E Zd, binary branches at rate v+(x), and annihilates at rate v_ (x). The random environment w has coordinates w(x) = (v_ (x), v+ (x)) which are i.i.d. We identify a natural way to describe the annealed-Gaussian transition mechanism under mild conditions on the rates. Indeed, we introduce the exponents F9(t) := (Hl ((1 + 6)t) - (1 + B)H1(t))/0, and assume that (F2B(t) - Fe(t))/(0log(lct + e)) -> oo for 101 > 0 small enough, where HI (t) := log(m(0, t)) and (m(0, t)) denotes the average of the expected value of the number of particles m(0, t, w) at time t and an environment of rates w, given that initially there was only one particle at 0. Then the empirical average of m(x, t, w) over a box of side L(t) has different behaviors: if L(t) > eFE(t)/d for some e > 0 and large enough t, a law of large numbers is satisfied; if L(t) > eFE(2t)/d for some e > 0 and large enough t, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative E. As corollaries, we obtain more explicit statements under regularity conditions on the tails of the random rates, including examples in the four universality classes defined in [14]: potentials which are unbounded

of Weibull type, of double exponential type, almost bounded, and bounded of Frechet type. For them we also derive sharper results in the nonannealed regime.

2000 Mathematics Subject Classification. Primary 82B41, 82B44; Secondary 60J45, 60J65, 82C22.

Key words and phrases. parabolic Anderson model, random walk, branching processes, Feynman - Kac formula, principal eigenvalue. This work was partially supported by Fondo National de Desarrollo Cientifico y Tecnoldgico grant 7020686. The work of the last author is partially supported by Fondo Nacional de Desarrollo Cientifico y Tecnoldgico grant 1020686. This is the final form of the paper. ©2007 American Mathematical Society 1

2

G. BEN AROUS ET AL.

1. Introduction When studying the long time behavior of Markovian dynamics in random media, one is faced with an important distinction: quenched vs. averaged estimates. Should one work in the so-called quenched regime, where the randomness of the medium is frozen, i.e., where the dynamics are studied in one fixed random realization of the medium/environment? Or should one work in the averaged regime where both the randomness of the dynamics and of the medium are considered, i.e., when one studies the dynamics in a given realization but then also averages over the randomness of the medium?' The true significance of this distinction "quenched vs. averaged" is important when these two regimes give different answers, which is the case in many situations where the extreme values of the random environment might play an important role. A good generic class of examples where this distinction is significant is given by models of reaction-diffusion in random media ([12,13,17]). There are two opposing views about this distinction. The first approach is to think that the relevant and important asymptotic long-time estimates are the quenched ones. But, recognizing the obvious fact that the quenched estimates are the most intractable ones mathematically, the averaged/ annealed results are seen as a welcome first approximation. The second and opposing view is that, in many applications, the averaged asymptotic estimates are the only relevant ones. The quenched results, though mathematically more challenging, are not seen as useful or relevant. This second view is naturally based on the idea that some mechanism must be at work, which allows for averaging in the medium randomness. Some years ago, the authors of this paper held the two opposing opinions expressed above, based on their former collaborations with different domains of applications (specifically physics of pollution by underground waste storage on the one hand and chemical kinetics on the other). In the recent years we have built a common answer, which we believe provides not only a natural resolution of the scientific debate, but also introduces the idea that there exists in fact a very rich universal transition between the averaged and quenched results, which goes far beyond the reaction-diffusion context. This transition also explains, in our view,the true relevance of these regimes in various applications. The key idea is the following: one should work with a fixed realization of the

medium but introduce a new parameter, say L, which will be the scale of the spatial extent of the initial distribution of the dynamics. Then, depending on the respective sizes of the time scale t and the space scale L (when both t and L diverge),

one should see the transition we mentioned between the quenched and averaged asymptotic results. More precisely one should expect the following transition: for time scales t(L) short enough the averaged asymptotic results should be valid, whereas for very long time scales t(L) the quenched results should hold, and our new intermediate asymptotics should emerge in between. The mechanism of this transition is the following: for short time scales t(L), or equivalently for large space scales L(t) for the initial distribution, the spatial ergodic theorem should ensure the needed averaging mechanism in order to enforce the validity of averaged/ annealed results. For very large time scales no averaging is possible and the extreme values 'This regime is often called the annealed regime in the mathematical literature. This is an entrenched but misleading vocabulary since it is not the usual convention in physics

TRANSITION ASYMPTOTICS FOR RANDOM WALKS

3

of the environment play the prominent role. The transition regimes are then easily understood, they consist in regions of parameters t and L where one sees a gradual emergence of the extreme values versus the average. This scheme has been first established in the simplest possible context, i.e., i.i.d. samples [4]. More precisely sums of exponential of i.i.d. random variables are shown to exhibit this transition. In the context of reaction diffusion this could simply be seen as reaction with no diffusion! In this simple context a full transition is given from the Gaussian asymptotics to extreme value theory: one sees in [4] the gradual emergence of the importance of extreme values of the i.i.d. sample which gradually destroys the validity of the Central Limit Theorem and then of the Law of Large Numbers by enforcing a-stable fluctuations where the exponent a decreases through the whole possible range, i.e., from 2 to 0. This mechanism is analogous to the phase transition description of mean-field spin-glass equilibrium models such as the Random Energy Model ([8,9]). We then proceeded ([5]) to one important case of reaction-diffusion, i.e., annihilation (or absorption) for random walks in a random environment, more precisely random walks killed on random obstacles, building on the work of [17]. There, we studied the same transition mechanism for the natural quantity, which is the probability of survival. Our picture was less precise than in the i.i.d. context, in the sense that even though we get the proper scales for the intermediate regimes we cannot establish the stable nature of the fluctuations, due to a lack of precise enough understanding of the edge of the spectrum (for the generator of the dynamics which is the discrete Dirichlet Laplacian on a random domain of Zd). In this paper we address a rather general case of reaction-diffusion in random environments, i.e., of a system of noninteracting continuous-time Random Walks on the lattice Zd branching and annihilating with stationary random rates ([12,13]). Let us first describe the random environment {w(x) : x E Zd}, with w(x) := (v_(x),v+(x)) where v+ {v+(x) : x E Zd} represents the branching rates with v+(x) E [0, oc), while v_ {v_(x) : x E Zd} represents the annihilation rates, with v_ (x) E [0, oo] so that we admit the possibility of hard core obstacles. We assume that the random variables {w(x) : x E Zd} are i.i.d. and call their distribution P. We consider the following dynamics in a fixed random realization of this envi-

ronment. We start with one particle at each site of the box AL of side L in Zd. Each random walk moves independently of the others according to a continuous time simple symmetric rate 2d,c dynamics for some K > 0 (we admit the possibility that ic 0, i.e., no diffusion at all, as in [4]). A random walk at a site x E Zd, branches at rate v+(x), disappearing and producing two new independent offsprings, and annihilates at rate v-(x) (note that v-(x) = oc means the annihilation is instantaneous and certain as in [5]). We study the asymptotic behavior, when both t and L go to infinity, of the following observable of our system of random walkers in random environment: 1

ML(0,t,W) :_ AL

/.

xEAL

Note that mL (0, t, w) is simply the (normalized) size of the total population. Our main result, Theorem 2.3, describes sharp conditions on the scales oft and L for the validity of annealed asymptotics of rnL (0, t, w) and for these annealed asymptotics to cease to be true. It also describes the scales for t and L where the

4

G. BEN AROUS ET AL.

fluctuations about these annealed asymptotics are Gaussian and when they cease

to be so. We intend to give a full and complete picture of the expected stable fluctuations, all the way to quenched asymptotics, for a wide class of branching rates in a forthcoming work. The results we obtain here are general, in the sense that they are valid for a large class of product distributions for the random environment, under mild conditions on the branching and annihilation rates. They include examples in the four universality classes recently introduced by van den Hofstad et al. [14] describing all the cases of the random environment. For example, potentials which are unbounded of Weibull type, of double exponential type, almost bounded and bounded of Frechet type. Furthermore, our results include and generalize both[4, Theorems 2.1 and 2.2] and [5, Theorem 1(ii), (iii) and Theorem 2(i)]. Let us now be more precise about our assumptions about the distribution of the environment. In the recent paper [14], it is shown that under regularity assumptions on the tail of the law of the effective potential v(0) := v+(0) - v_(0), exactly four universality classes of environments can occur. Their assumptions are formulated in terms of the cumvlant generating function, (1.1)

H(t) = log(ev(O)t)

t > 0,

of the law of the effective potential v(0), where for any function of the environment

f , we

define (f) := f f dµ. Their basic assumption is that this function is defined and finite for every t > 0. Then, under two regularity assumptions on H they show that four universality classes can occur: (1) a first class where v is unbounded and has "heavy tails" at infinity, and which includes Weibull-type tails; (2) a second class of unbounded potentials with "lighter" tails which includes the double exponential law; (3) a class containing bounded and unbounded potentials; (4) a class of bounded potentials including those which have Frechet-type tails near their essential supremum, and the degenerate case of random walks on hard core random obstacles with µ[v(0) -oc] p < 1. In this paper we generalize [5, Theorem 1(ii) and (iii)] describing the passage to an annealed and Gaussian regime, to the previously described system of random walks on the random environment w, under mild conditions which include cases in these four universality classes. Throughout this article the following will be assumed. Assumption E. The law of the effective potential is such that µ[v(0) 1 and (ev(°)t) < oc, t > 0.

-oc]
0 whenever 0 < CBI < 1. We will see that Condition MI is satisfied whenever the following happens.

Assumption SI. For all 0 > 0 small enough, lim t- oc

Gee (t) - Go (t) = 00. Bt

Condition SI includes the first universality class of [14] and can be viewed as a strong intermittency requirement. It implies H(t)/t >> 1. Hence, using the bounds ea(t)-2dkt < (m(0, t)) < eH(t) (see, e.g., Theorem 3.1 of Gartner and Molchanov [12]), we see that if Assumption SI is satisfied, we have the asymptotics (1.5)

log(m(0, t)) - H(t),

which is much faster than the a.s. one (see [13]). As already mentioned the interest of the results of the present paper are their generality. Namely, their are valid only under the Assumptions E and MI. Part (i)

of Theorem 2.3, states that if for some e > 0 we have L(t) > eFE(t)/d eventually in t, the law of large numbers mL/(m) - 1 is satisfied in probability: hence we have the annealed behavior logmL(0,t) _ log(m(0,t)). On the other hand, if for some e > 0 we have L(t) < eF(t)/d eventually in t, in probability mL/(m) 0 we have L(t) > eFE(2t)/d eventually in t, then (mL - (2L + 1)d(m))/ Vary, mL converges in distribution to a centered normal law of unit variance A[(0, 1), where Vary denotes the variance. Also, if for some e > 0 we have L(t) < eF-E(2t)/d eventually in t, in probability (ML (2L + 1)d(m))/ Vary ML FE(2t)

Non-Gaussian behavior

dlogL(t) < FE(2t)

M /(m)_1

d log L (t) > FE (t)

mL/(rn) > 1, called the growth exponent, describing more explicitly the transition of Theorem 2.3. Indeed, in this case, a law of large numbers is satisfied when d log L(t) > 'yJ(t) for some ry > ryi; the CLT when d log L(t) > ryJ(t) for some 'y > -y2. Furthermore, if d log L(t) < -yJ(t) for some 'y < -yl, the law of large numbers is not satisfied, while if d log L(t) < -yJ(t) for some -y < 'y2, the CLT is not satisfied. Propositions 2.7, 2.8, 2.9 and 2.10 give the explicit value of -y', rye and J in the case of unbounded potentials with Weibull-type tails, unbounded potentials with double exponential type tails, potentials in the third universality class of [14] and bounded potentials with Frechet-type tails including the pure hard core case. Table 2 below summarizes those results. Also, in Theorem 2.11, we obtain sharper upper bounds for the order of magnitude of the averaged first moments in the nonannealed regime for the examples treated in Propositions 2.7, 2.8, 2.9 and 2.10. This theorem generalizes Case 3 of [5, Theorem 2]. The special case in which Assumption SI is satisfied expressed as Corollary 2.4.

This includes the case i 0, corresponding to sums of i.i.d. random exponentials where Condition MI reduces to, Go (t) _ 00, lim Gee (t) t- oc 9

(1.6)

for IBS > 0 large enough, and Hi(t) = H(t). Corollary 2.4 is a result complementary to theorem of [4], generalizing [4, Theorems 2.1 and 2.2] where Weibull and Frechettype tails are assumed on v(0). One of the main ingredients of the proof of Theorem 2.3 is a coarse graining technique, necessary to reduce the asymptotics of the averaged first moments, to a sum of independent random exponentials. This technique, was introduced in [5], TABLE 2. Transition and growth exponents in the four universality classes. In the pure hard core case µ[v(0) oc] p and c2 is a constant depending on p and d

POTENTIAL Weib ull

Double exponential

Thi r d cl ass exampl e P ure h ar d core

- log µ[v(0) > x] XP ,

p >1

ex/p

J(t)

ryl

H(t)

1 1p

t

p

2p

1

2

%2

21

e xa 2

log t

c2 tdy(d+2)

d+2

21

ryl

1

/i


7

but here it faces the extra difficulty that the terms of the sum defining the averaged probabilities are not uniformly bounded with respect to the time variable t or the scale L (whereas in [5], such a bound existed having the value 1). This requires more careful estimates on these quantities, which are performed, via the Feynman-Kac formula and spectral estimates. Once the reduction to a sum of i.i.d. exponentials is achieved, an analysis based on von Bahr-Esseen inequality finishes the proof (see also [4,5]). Besides this introduction, this paper has four other sections. The main results

are stated in Section 2. We first introduce in Section 2.1 the main definitions. In Section 2.2 we formulate Proposition 2.1, stating that a growth of the form limsupl,,_, v+(x)/(Ixl log xj) = 0 is enough to ensure well-defined first moments for the total number of particles. When this proposition is combined with [12, Proposition 2], one concludes that under the condition E, the reaction-diffusion process on the lattice Zd is such that the total number of particles at any given time has a finite first moment, for initial conditions with a finite total number of particles. In particular, there is no explosion, and the process is well defined. We then state Theorem 2.3 in Section 2.3. Corollary 2.4, under the Assumption SI is stated and proved next. The applications of Theorem 2.3 are given in Section 2.4. First, the regularity condition RI is introduced. This is applied to the case of unbounded effective potentials with Weibull-type tails, through Proposition 2.7, using the Kasahara exponential Tauberian theorem [6]. Next, Corollary 2.6 is applied to the case of unbounded potentials with double exponentially decaying type tails, through Proposition 2.8. Then, we treat the case of potentials falling in the third universality class (almost bounded) of [14] through Proposition 2.9. We end Section 2.4 considering the case of bounded potentials with tails near their upper-bound which are of the Frechet type (Proposition 2.10). In Section 2.5, we state Theorem 2.11, which improves the upper bounds describing the order of magnitude of the empirical average for the examples considered in Section 2.4. The proof of Proposition 2.1 is the content of the third section. In Section 4, the truncated first moments are introduced. These are the first moments of a reactiondiffusion process defined on a finite box, with Dirichlet boundary conditions. They are then used to approximate some important quantities related to the averaged first moments. Then, several important estimates for the moments and correlations of the first moments are derived. The proof of Theorems 2.3 and 2.11 are given in Section 5. In Section 5.1, the partition analysis method of [5] is recalled. This and together with the estimates of Section 4, and the von Bahr-Esseen inequality, is subsequently applied to prove Theorem 2.3. The paper finishes with Section 5.7, where Theorem 2.11 is proved.

2. Notation and results Here we will state the results of this paper. After introducing most of the notation and giving the main definitions in the first subsection, in Section 2.2 we state Proposition 2.2, which ensures that a.s. there is no explosion for the reactiondiffusion process under Assumption E. Then, the principal result of this paper, Theorem 2.3 is stated in Section 2.3, together with Corollary 2.4. In Section 2.4, we state Corollary 2.6, giving the form of Theorem 2.3, under certain regularity assumptions. Here we will consider applications of this results to several specific

G. BEN AROUS ET AL.

8

examples of distributions of the effective potential. We end the presentation of our results with Section 2.5, where we state Theorem 2.11.

2.1. Definition of the reaction -diffusion process. We begin defining a reaction-diffusion model corresponding to a set of random walks on the lattice Zd branching and annihilating at rates depending on the their position. Consider the set of natural numbers N endowed with the discrete topology. Define the set SZ : NZd representing the possible configuration of particles on the lattice. In this paper we will be interested only on the subset of configurations S2o C SZ characterized by the property that {x : 1)(x) > 0} has finite cardinality whenever rl E SZo. Let v+ {v+(x) : x E Zd} and v_ {v_(x) x E Zd}, where v+(x) E [0,oc) and v_ (x) E [0, oc]. Here v+(x) and v_ (x) represent the rate at which particles branch and annihilate at site x, respectively. Note that we admit the value oc for the annihilation rate: this represents a hard core obstacle, where particles are instantly annihilated. Call an ordered pair w : (v-, v+) E W, with coordinates ([0, oo) x [0, w(x) = (v_ (x), v+(x)), a field configuration, where W We will denote the set of hard core obstacle sites by c(w) {x E 7Gd : v_(x) = oo}. {y c Z : Ilx - yll < r} the Given r E [0, oc) and x E 7Gd we will call A(x, r) ball of radius r centered at x under the norm llxll sup1 0, ?7x,y is the configuration where a particle from site x has jumped to site y so that gx,y(z) = q(z) if z x, y, yx,y(x) = r1(x) - 1, and 71x,y(y) = 71(y) + 1; yx,+ is the configuration where there is an extra particle at site x and 71x- the configuration where one particle at site x has disappeared. It is a well known fact that it is possible to construct a strong Markov process, denoted by nU : {r1U(t) : t > 0}, corresponding to an infinitesimal generator of the form (2.1), and taking values on the Skorokhod space S := D([0, oc); SZo ). In fact, such a process falls in the category called I Ul-dimensional continuous time Markov branching process by Athreya and Ney (see Chapter V, Sections 7.1 - 7.2 of Athreya - Ney [3] ). Furthermore, it


9

can be shown that a.s. the expected value of each coordinate of such a process {qU (t) t > 0}, is finite, ensuring that there cannot be infinitely many particles produced in a finite time (see [3, Section 7.1]). Let us now call P(1') the set of :

probability measures defined on S2o endowed with the Borel a-algebra associated to the subspace topology of 1 as a subset of S2 with the product topology. Then, for each field configuration w E W and probability measure v E P(SZo) denote by Pv ,w the law of the process {rlU(t) t > 0} defined on S endowed with its Borel a-field B(S). We will call this process the reaction-diffusion process on U with field w and initial condition v. In the particular case in which U = An we will use the obvious notations Ln and PP 'w. Furthermore, we will call the process on :

A, the reaction - diffusion process at scale n. Now, note that using the natural coupling [15] and Kolmogorov's extension theorem, it is possible to define for each field configuration w E W and initial condition v E P(S2o) a probability measure Qv on the product space SN, endowed with its Borel a-algebra induced by the product topology, in such a way that if rln E S denotes the nth coordinate of an element rl E SN, rln(t) E 1 its value at time t > 0 and rln(t,x) E N the value at time t of the x-coordinate of rln(t), then, (i) for every A E B(S) and n > 1, QV [71n E A]

Pv,w [A].

In particular, for every B E B(SZo) we have that, Qv [rln(0) E B] (ii) for every n > 1, (t) > rln (t)]

Qv

v[B].

1.

(iii) for each n > 1 define the first exit time from the box An as Tn := inf{t > 0: sup rln(x, t) > 0}. XEA,

Then, for every n > 1, [rln(t) = r,n+l(t),Tn > t] = Qw [Tn > t]. Qw Let us now remark that due to property (ii), for every t > 0 and x C Zd the limit, 77 (t, x) := lim if (t, x), n-oc exists, possibly taking the value oo. Define q(t) {rl(t, x) : x E Zd}. We denote Zd

the stochastic process {q(t) : t > 0}, taking values on the space N , where N is the Alexandrov compactification of the natural numbers, and distributed according to the measure Q', the reaction - diffusion process with field w and initial law v. We will denote by Ev the corresponding expectation. In addition, for each t > 0, we define the total number of particles at time t by fi(t)

E rl(t,x) 2EZd

Also, whenever it is true that, Qv [rl(x, t) < oo, d x E Zd, t > 0] = 1,

we will say that with probability one there is no explosion. In the sequel we define for each x C Zd the probability measure Sx on (S2o , B) which assigns probability

1 to configurations with one particle at site x and none elsewhere. We will be interested in initial configurations where v = 8x. In such a case we will use the

G. BEN AROUS ET AL.

10

notation Px instead of P' and Ex for the corresponding expectation. In the case where x E C(w), we adopt the convention that Px is the probability measure which has a unique atom at the configuration rl 0 (rl(x) = 0 for every x E Zd). Let us denote by P(W) the set of probability measures defined on the space W = ([0, oo) x [0, oo])Zd endowed with its natural o--algebra. In the sequel we will take fields v+ and v- which are random, assigning a probability measure µ C P(W) in such a way that the field configuration {w(x) : x c Zd} has independent coor-

dinates with respect to p. Furthermore, we will use the notation () to denote expectation with respect to this law and variance. Now, let us define the quenched first moment on Zd of the total number of particles at time t starting from site x as, m(x, t, w) := Ex [((t)], and the annealed first moment on Zd of the total number of particles at time t starting from site x as, (m(x, t)) := f m(x, t, w) dµ. Furthermore, we call the sets {m(x, t, w) : x C Z'} and {(m(x, t)) : x C Zd}, the fields of quenched first moments and the field of annealed first moments, respectively. Depending on the context, we might write m or m(x, t) in place of m(x, t, w), dropping the dependence on the field configuration w, and (m) instead of (m(x, t)). The quantity which will give us a transition mechanism between of the quenched

first moments is the averaged first moment at scale L and time t defined for a reaction-diffusion process starting from site x as, 1

A(x L)

mL (x, t, w)

yeE

m(y, t, w) L)

2.2. Results giving conditions for no explosion. Here we will give a criteria on the field configuration w, stated as Proposition 2.1, which ensures that there is no explosion in the reaction-diffusion process with field w. Proposition 2.1. Consider the reaction diffusion process with field w and initial law v. Assume that, (2.2)

lim sup v+(x) xHoc x log x1

- 0.

Then for every t > 0 we have that

E [((t)] < oo. Hence, there is no explosion.

We state below with the name of Proposition 2.2, a result of Gartner-Molchanov [12, Lemma 2.5] giving a sufficient condition on the law p in order that the first moment of the total number of particles ((t) at time t, exists p-a.s., and hence that

there is no explosion. Given a field configuration w = (v+, v-), we now need to introduce the effective potential {v(x) : x E Zd}, defined by v(x) : v+(x) -v-(x). Furthermore, set log+ x log x if x > e and log+ x 1 otherwise, while define the positive part x+ := max(0, x).

Proposition 2.2. Consider the reaction diffusion process with field w and initial law v. Assume that the field configuration w is distributed according to a product probability measure p C P(W). Suppose that (2.3)

d) v+(0) < oo. log+ v(0) )


11

Then p-a.s. it is true that, lim sup

log x1 _" xV+(x)

0,

and therefore, p-a.s. there is no explosion for the reaction -diffusion process with field w and arbitrary initial law in P(S2o).

Note that the last statement of Proposition 2.2 follows from Proposition 2.1. Furthermore, Assumption E is enough for (2.3) to be satisfied, and hence to ensure no explosion.

2.3. The Gaussian-annealed transition results. Here we will state the main result of this paper, which shows how under different growth of scales, the averaged first moment has an asymptotic behavior where a law of large numbers is satisfied, and a central limit theorem can describe the fluctuations around this law of large numbers. We will assume Condition E, ensuring the existence of the annealed first moments. We will also need to consider the growth functions {Hi(t) = log(m(0, t)) : t > 0} already defined in equation (1.2) of the introduction and the intermittency exponents {Fo : 0 E R1, defined in equation (1.3). Let us now state the main result of this paper.

Theorem 2.3. Consider a reaction - diffusion process with initial law do and (v+,v_) distributed according to a product measure p E P(W). Consider the intermittency exponents {Fo : 0 E R} defined in equation (1.3) and the growth functions {Hi(t) : t > 0} defined in equation (1.2). Assume that Conditions E and MI are satisfied. Then the following statements are true, (i) Law of large numbers. Assume that there is an e > 0 such that eventually in t, L(t) > exp{FE(t)/d}. Then in p-probability we have field w

(2.4)

mL (0, t, w)

ti 1,

(m(0,t))

as t - oc. Furthermore, assume that there is an e > 0 such that eventually in t, L(t) < exp{F_E(t)/d}. Then in p-probability we have (2.5)

ML (0, t, w) (m(0, t) )

« 1,

ast -oo. (ii) Central limit theorem. Assume that there is an e > 0 such that eventually in t, L(t) > exp{FE(2t)/d}. Then, (2.6)

the mL(0, t, w) - (m(0, t)) Varµ (m(0, t))

=

where N(0, 1) is a centered normalized normal law and the convergence is in the sense of distributions. Furthermore, assume that there is an e > 0 such that eventually in t, L(t) < exp{F_E(2t)/d}. Then in p-probability we have that, (27)

mL(0't, w) - (m(0, t)) « 1. VarN, (m(0, t))

To provide a better insight on the meaning of Theorem 2.3, we will see the form that it takes under the stronger Assumption SI, which includes the first universality class of [14]. This will be formulated as a corollary, which in the case r, = 0

G. BEN AROUS ET AL.

12

generalizes [4, Theorems 2.1 and 2.2] to include distributions µ of the field, which not necessarily have regularly varying log-tails.

Corollary 2.4. Consider a reaction - diffusion process with initial law d0 and field w (v+,v_) distributed according to a product measure p C P(W). Consider the cumulant intermittency exponents {Go : 8 C RI defined in equation (1.4) and the cumulant generating function {H(t) : t > 0} defined in equation (1.1). Assume that Conditions E and SI are satisfied. Then the following statements are true, (i) Law of large numbers. Assume that there is an c > 0 such that eventually in t, L(t) > exp{GE(t)/d}. Then in p-probability we have ML(0,t,w)

1

(m(0, t))

oc. In particular, log mL(0, t, w)/H(t) 1. Furthermore, assume that there as t is an e > 0 such that eventually in t, L(t) < exp{G_E(t)/d}. Then in p-probability we have

ML(0't,w) (m(0, t))

«1,

ast- oc. (ii) Central limit theorem. Assume that there is an e > 0 such that eventually in t, L(t) > exp{GE(2t)/d}. Then, tlim ML(0, t, w)

(m(0, t))

Varµ (m,(0, t))

N(0,1),

where the convergence is in the sense of distributions. Furthermore, assume that there is an e > 0 such that eventually in t, L(t) < exp{G-E(2t)/d}. Then in p-probability we have that,

mL(0,t,w) - (m(0,t)) Var,, (m(0, t))

0 is a direct consequence of the fact that eH(t)-2dkt < (m(0,t)) < eH(t), stated in [12, Theorem 3.1], and the observation that Go (t) > 0 for 8 > 0 and Go (t) < 0 for 8 < 0, which follows from Jensen's inequality. Now, the following proposition, which will be proved in Section 4, shows

that the condition, (2.8)

slim tH11(t) = 00, 00

is sufficient for Assumption SI to be true. This condition implies a kind of domination of the branching over the annihilation.

Proposition 2.5. Consider the cumulant exponents {Go : 8 C R}. Assume that condition (2.8) is satisfied. Then, Condition SI is satisfied. Furthermore: (i) for every 8 0, there is a to > 0, such that the function Go(t) is monotone in t, for t > to; (ii) there is a t1 > 0, such that the function Go(t) is monotone in 8

fort>t1.

Condition (2.8) implies that the branching dominates the annihilation, in the

sense that H(t)/t - oc as t - oc, which implies that the essential supremum of the random variable v(0) is infinite (see [12]).


13

2.4. Regularity assumptions on the intermittency exponents. For the purpose of applications, it will be important to identify cases where the assumptions in Theorem 2.3 can be formulated in a more explicit way. As it will be shown, the

following assumption on the intermittency exponents, turns out to fall in one of these situations.

Assumption RI. The intermittency exponents {Fo (t)

:

0

RI satisfy the

mild intermittency condition MI. In addition, there exist two increasing functions fl, f2: R \ {0} R and a function J(t) : [0, oc) [0, oc), such that for 0 0 small enough,

(i) Fo(t) - f1(0)J(t), and Fo(2t) ,., f2(0)J(t) (ii) There exists two constants -y' and 'y2i such that lim0_0 f, (0) _ -y' and lim0+0 f2 (0) = 'y2

Throughout the sequel, the constants 'yl and 'y2 will be called transition exponents and the function J growth exponent. It will be shown that there exist several important cases of random fields which fall in this category. Furthermore, the following corollary of Theorem 2.3 shows the convenience of Assumption RI. Corollary 2.6. Consider a reaction - diffusion process with initial law 60 and field w (v+,v_) distributed according to a product measure p C P(W). Suppose that Assumptions E and RI are satisfied with transition exponents -yl and `y2 and growth function J. Then the following statements are true, (i) Law of large numbers. Assume that there is a ry > ryl such that eventually in t, log L(t) > 'yJ(t)/d. Then in p-probability we have m'(0, t, w) (m(0, t) )

as t - oc. Furthermore, assume that there is a 0 < 'y < 'yl such that eventually in t, log L(t) < 'yJ(t)/d. Then in p-probability we have ML (0, t, w)

(m(0, t))

«l

ast -oc. (ii) Central limit theorem. Assume that there is a 'y > `y2 such that eventually in t, logL(t) > -yJ(t)/d. Then, mL(0,t,w) (m(0,t)) lim V(0,1), t-'°° Var(m(m(0, t)) where the convergence is in the sense of distributions. Furthermore, assume that there is a 0 < 'y < ry2 such that eventually in t, logL(t) < -yJ(t)/d. Then in p-probability we have that, ML (0, t, w) - (m(0, t) )

c 1.

Var,, (m(0, t))

In the next subsection we will apply Corollary 2.6 to four situations each one falling in one of the universality classes described by van den Hofstad et al.in [14]. These classes encompass all possible situations under three conditions. The first is Condition E, ensuring the existence of the positive moments defining the cumulant generating functions (1.1). The second and third condition avoids different qualitative behaviors of the potential at different scales. Let us formulate next the second condition of [14].

G. BEN AROUS ET AL.

14

Assumption H. The function H(t)/t is in the de Haan class. A function f is said to be in the de Haan class if for some regularly varying function g : (0, oc) f8, we have that (f (At) f (t)) /g(t) converges to a limit different from 0 as t oc, for A > 0. Let us recall that a function h is regularly varying at infinity with index p, if for any a > 0 we have h(ax)/h(x) = aP.

This property will be stated as h c R. Whenever H is satisfied, then H(t) is regularly varying with index 'y > 0. Furthermore, in [14, Proposition 1.1], it is

proved that under Assumption H there exist a function H: (0, oo) - R and a continuous function k(t) : (0, oo) - (0, oc) such that, (2.9)

tli m

yH(t)

H(ty)

k (t)

00

= H(y) , 0,

for y G (0, 1) U (1, oc). It is also shown that k(t) is regularly varying of index ry. We can now recall the third assumption of [14].

Assumption K. The limit k* = limt_,,,, k(t)/t exists in [0, oc]. Under Assumptions E, H, and K, the four universality classes defined in [14] are:

(1) -y>1,or-y=landk*=oc. (2) -y = 1 and k* c (0, oc). (3) -y = 1 and k* = 0.

(4) ry x] = exp{-h(x)} for x > 0, with h G RP for some 1 < p < oc. In the terminology of [4] in the context of i.i.d. random exponentials (ic 0 in our situation), this is called Case B. The following proposition shows that Assumption RI is satisfied in this situation, and hence Corollary 2.6, which generalizes [4, Theorems 2.1 and 2.2] in Case B form , = 0 to r, > 0. Proposition 2.7. Suppose that the essential supremum of the effective potential v(0) is oc with Weibull-type tails µ[v(0) > x] = exp{-h(x)} for x > 0, and h c RP for some 1 < p < oc. Then Assumption RI is satisfied with transition exponents 1

`}'1 =

P

p-1

,

`}2 = 2 P-1

1

p1

,

and growth exponent,

J(t) = H(t). Let us now prove Proposition 2.7. Note that under the conditions on the tail of the law of v(0), the cumulant generating function H(t) of v(0) is well defined, smooth, nondecreasing and tends to infinity as t - oc. Furthermore, by the Kasahara exponential Tauberian theorem (see Bingham et al. [6, Theorem 4.12.7]), we


15

know that H E RP', where the index p' is defined by the equation, 1

1

+

P

1. P'

From this observation it is easy to check that Assumption SI of Corollary 2.4 is satisfied, and that for every c 0, the cumulant growth exponents GE(t) and GE(2t) satisfy

G,(t) ti (1 + E)P' (1 + E) H(t) E and

G,(2t) H (t)

,,

2p' (1 + )P - (1 + E) E

Now note that (1+E)) /E is increasing in c and converges to ry1 = 1/(p-1) as c 0, and similarly 2,0' ((1 + )P' (1 + E)) /E is increasing in c and converges to 'Y2 = 2PAP-1)'Yl as c

0.

2.4.2. Unbounded potentials with double exponential type tails. Here we consider the second universality class so that H is regularly varying with index -y = 1 and k* C (0, oc). As shown in [14, Proposition 1.1], this is equivalent to the existence of a constant p E (0, oc) such that

yH(t) py log y t for all y C (0, 1) U (1, oc) (this and 0 < p < oc is called Assumption H in [13]). Furthermore, under assumption (2.10) it is true that, lim

(2.10)

H(yt)

t -+

H(t)

= oo. t This second universality class includes the case of unbounded potentials which are lim t-.oc

double exponentially distributed with parameter p, 0 x] = exp{-ex/P}, for x c R. We then have the following interesting proposition.

Proposition 2.8. Suppose that assumption (2.10) is satisfied for some p E (0, oc). Then Assumption RI is satisfied with transition exponents 'Y1 = P,

'Y2 = 2P,

and growth exponent,

J(t) = t. To prove Proposition 2.8, we quote [13, Theorem 1.2], which shows that under assumption (2.10) we have that, (m(0, t))

exp{H(t)

2dkx ( P ) t + o(t) r,

for , > 0, where x(x) : z inft,EP(z) [S(p) + pI (p)] for x > 0, P(Z) is the space of probability measure on Z, S: P(Z) [0, oc) is the Donsker-Varadhan functional ) 2, while I: P(Z) [0, oo) is the defined by S(p) := xEZ ( p(x + 1) -

G. BEN AROUS ET AL.

16

entropy functional defined by I(p) := - >,,,E, p(x) log p(x). On the other hand, from the previous discussion we can conclude that, for every c 54 0,

Fi t)

c

p(1+c)log(l+c)

and

F,(2t) t

- 2P(l + c)

log(1 + c) c

From these limiting behaviors, we see that Assumption MI is satisfied. Furthermore, form the fact that (1 + c) log(1 + c) /c is increasing for c small enough and converges

to 1 as c - 0, we obtain the transition exponents at p and 2p of Proposition 2.8. 2.4.3. Almost bounded potentials. We now focus on the third universality class, 1 and k* 0. As shown in [14, Theorem 1.4], in this case it is true that,

where 'y (2.11)

log(m(0, t)) ,., H(tad d)

at

where at : [0, oc) - [0, oc) is the so called scaling function, which is implicitly defined for all t > 0 sufficiently large by the equation, (2.12)

k(tat d) tat d

1

a2 t

As shown in [14, Proposition 1.2], this function is unique up to asymptotic equivalence. Furthermore, for the third universality class, it is a slowly varying function (regularly varying of index 0).

Proposition 2.9. Suppose that H is regularly varying of index 1 and that (2.9) is satisfied for k(t) such that k(t) x] = exp{-ex2},

for x > 0. As it can be deduced from the discussion of [14, Example 1.4.3], in this case the scale function is given by at - 21/2(logt)1/4 and k(t) - t/(2 logt). Furthermore, p = 1 and the growth exponent can be chosen as J(t) - t/(2(log t)1/2) (see Table 2 of the introduction).


17

2.4.4. Bounded potentials with Frechet-type tails. We now continue with an example falling in the universality class (4) of [14]. We will consider the case in which the essential supremum of v(0) is 0 and the tails are of Frechet type: µ[v(0) > -x] = exp{-h(x-1} for x > 0, with h E R. for some 0 < p < oc. Using the terminology of [4] in the context of i.i.d. random exponentials, this is Case A. To state appropriately this result, we will need to recall the work of Biskup and Konig [7], who studied the asymptotics of the annealed and quenched first moments in the case of product environments p such that the cumulant generating function

H(t) is in the so called -y-class for some 'y E [0, 1). We say that H is in the -yclass if the essential supremum of v(0) is 0 and if there is a nondecreasing function

at E (0, oo) and a function H: [0, oo) - (-oc, 0], H j 0, such that (2.14)

slim

_

attd+2 H C ad

y) = H(t),

t

for y > 0, uniformly on compact sets in (0, oo). We will denote the function at the scale function. It is not difficult to show, using the de Bruijn exponential Tauberian theorem [6], that if v(0) is in Case A for some p > 0, then H is in the p'-class for p' = p/(p + 1), or 1/p' - 1/p = 1. We can now sate the following proposition.

Proposition 2.10. Suppose that the essential supremum of the effective potential v(0) is 0 with Frechet-type tails µ[v(0) > -x] = exp{-h(x-1)} for x > 0, and h C Rp for some 0 < p < oc. Then Assumption RI is satisfied with transition exponents 2

'Y2 = 21--"-yl,

'Y1 = (d+2+2p)' and growth exponent,

J(t) =

Xt

2

t

for some constant x C (0, oc). As a consequence of Proposition 2.10 we can now apply Corollary 2.6, generalizing Case A of [4, Theorems 2.1 and 2.2] from r, = 0 to r, > 0. In contrast to Proposition 2.7, where there is no change in the value of the transition exponents -yl and 'y2 from pc = 0 to ,c > 0, here there is. Let us now prove Proposition 2.10. In [7, Proposition 2.1], it is shown that whenever p is in the p'-class, then the scaling function at E R,,, for, V :=

1-p' d+2-dp'-

Furthermore, [7, Theorem 1.2] states that when p is in the p'-class, there exists a x c (0, oc) such that, log(m(0, t M ti

-x2ot pt

,

for every 0 E (0, oc). Then Condition MI of Theorem 2.3 is satisfied, and for every e y4 0, the growth exponents F,(t) and F,(2t) satisfy F, (t)

(1 + e) - (1 + e

G. BEN AROUS ET AL.

18

and

F,(2t)

,,,

t/at

21-"2 (1 +

E)1_"2

E) - (1 + e

As in the proof of Proposition 2.7, we can show that ((1 + E) - (1 + E)1-"2)/E is increasing in E and converges to yl = v2 as E --> 0, and similarly the expression 21-"2 ((1 + E) - (1 + E)1-"2)/E is increasing in e and converges to rye = 21-"2y1 as E -4 0.

2.5. The critical regime. For the examples discussed in the four previous sections, it is possible to obtain the following improvement of Theorem 2.3.

Theorem 2.11. Consider a reaction - diffusion process with initial law b0 and field w = (v+, v_) distributed according to a product measure .t E P(W). Then the following statements are satisfied.

(i) Weibull type. Assume that t has Weibull-type tails so that µ[v(0) > x] = exp{-h(x)} for x > 0, h E Rp for 1 0 in u-probability,

mL(0, t)

e(aw('y)+a)H(t) 0 in ,u-probability,

rL(0, t) « 1, exp{H((aD(y) + b)t)/(aD(y) + S)} where aD(-Y)

ye(y-n)/P.

(iii) Almost bounded potentials. Assume that p is such that H is regularly varying of index 1 and that (2.9) is satisfied for k(t) such that k(t) 0 in µ-probability, mL (0, t)

exp{H((aA(y) + 8)t)/(aD(y) + S)}

« 1,

where

aA(y) := ye(7-v)/n

(iv) Frechet type. Assume that µ is such that essupv(0) = 0 and is of Frechet type so that µ[v(0) > -x] = exp{-h(x-1)} for x > 0, and h E Rp for some p E (0, oo). Then if d log L(t) < yJ(t) eventually in t, and y < yl = v2, we have that for every b > 0 in p-probability, mL(0, t)

e-(aF(-Y)-6)J(t) 0} is a set of independent processes such that (x (s) 0 for s < 0, (x (s) dxk for s 0 while {(x',, (s) : s > 0} has the law Px 'w. Let us now define for each n C N the maximum value of the field v+ on the box An by vn : maxxEAn v+(x)

Lemma 3.1. Consider the reaction -diffusion process at scale n with field w and initial condition 80. Then,

G. BEN AROUS ET AL.

20

(i) For every t > 0 and n > 1, Eo [C"''(t)]
0 and n > 2kt, Eo [an (t)] < 4d exp (vn - 2r,)t - n log

n 2ert

JJJ

(iii) For every t > 0 and m > n > 1, Eo [(m(t)] < Eo [(n(t)] +Eo [(n(t)] Eo [(2n,+m(t)].

The following elementary lemma will be used to prove Lemma 3.1. We will need to define,

I(y) := sup{Ay - (cosh.

(3.2)

1)}

y

sink-1

y-

1 + y2 + 1,

aE

Note that I: [0, oc) - [0, oo) is one to one and that I(x) > 0 for x > 0.

Lemma 3.2. Let {Xt : t > 0} be a simple symmetric continuous time random walk on Z of total jump rate 2t > 0 starting from 0. For a nonnegative real x inf{t > 0 : IXtl > x} as the first exit time of this random walk from define Tx :

the interval Ax. Then, if P is its law, we have (3.3)

P[rx < t] < 4exp` - 201 (2t I } < 4exp{

xlog2t

the second inequality being satisfied only for x > 2rt. The proof of Lemma 3.2 is a simple large deviation estimate and will be omitted.

(i) Let us remark that for every bounded nondePROOF OF LEMMA 3.1. creasing function f : N - I18 which is eventually constant we have, Ln.f ((") < vn(n (f ((n + 1)

For a natural N > 1 fixed, choose f (m) we then conclude that,

f ((n))

m A N. Using the fact that (T'(0)

1,

t

(3.4)

('(t) A N

1 - vn J (n(s)B[O,N_1] ((n(s)) ds,

where for A C I18, BA is the indicator function of the set A, is a super-martingale. Hence, since the integrand of the integral in (3.4) is a positive function, by Fubini's theorem, t

Eo [Cn(t)8[O,N-1] ((n(t))] _< I+ vn f Eo [CC(s)B[o,N-1] (("(s))] ds.

Therefore, by Gronwall's lemma, EO [Cn(t)8[o,N_1](CC(t)] < exp{vnt}.

oc and using the monotone convergence theorem we Taking the limit when N conclude the proof of part (i) of the lemma.


21

(ii) Let us note the following identity, t], Eo [an(t)] < where rn is the first exit time of a simple symmetric continuous time random walk

(3.5)

of total jump rate 2ds;, starting from the origin 0, from the box An and P is its law. Now, from the second inequality of equation (3.3) applied to each of the d coordinates of such a random walk, we conclude that,

P[Tn N it is true that

vn < 6(nlog(n/(2ekt)) - 4nlog4)/t whenever n > N. By part (i) of the lemma we have that,

Ew [an(t)] < exp(6 nlog l

n ) - 4nlog4 I },

\ 2eKt

l

/ ))J

while by part (ii) we have,

Ew [(n(t)] < 4dexp{ -(1 - 6)nlogl n I - 4Snlog4 ),

\2ekt)

J

whenever n > N. Choosing m = 2n > N in part (iii) of the same lemma, it follows that, An + Bn Ew [(4n(t)],

Ew [C2n(t)]

4de-(1-6)nlog(n(2ent))-46nlog 4. Repeating where An := esnlog(n/(2ekt)) and Bn := the bound for 2n and m = 4n and substituting back we get,

Ex [(2n(t)] < An + BnA2n + BnB2n Ex [(8n(t)1

Now, by induction on m, we get that, 'rn-1

r2

(3.6)

Ex [Sn(t)]
1, H(at) aH(t) lim

t->DC

00

t

PROOF. (i) Let t1, t2 > 0. From Holder's inequality, we have that 0(t1)q(t2) < 0(t1 + t2). Hence, H(t1 + t2) > H(t1) + H(t2). By induction on

n we conclude the proof.

(ii) From the assumption, note that we can write H"(t)

f(t)/t, where

limt-DC f (t) = oc. Integrating the function H" from t to at, it follows that, H' (at)

(4.2)

H' (t) > inf f (s) - log a.

Integrating again we obtain for u > t that,

H(at) - aH(t) > (t - u) inf f (s) a log a + c(u), S>U

where c(u) H(au) aH(u). Dividing by t, taking the limit when t then the limit when u oc, we obtain (4.1).

oc and

Let us now prove Proposition 2.5(i) and (ii). Note that, aG

(1 + e)H' ((l + 8)t) - (1 + e)H'(t)

at

0

Then, inequality (4.2) implies part (i) of the lemma. On the other hand we have that, OtH'((1 + O)t) H((1 + 9)t) + H(t) aG 00

02


23

By the mean value theorem there exists a B such that 0 t1, where t1 is independent of o and o. This proves that Go (t) is monotone in o for t > t1. We now show that condition (2.8) ensures SI. We will without loss of generality assume that , > 0. Let us define real valued functions f, g by f (x) H(ot + xt) H(xt) and g(x) := 2x for real x. By the generalized mean value :

theorem applied in the interval [1,1 + o], there exists a o1 E (0, o), such that (f(1 + o) - f(1))/(g(1 + o) - g(1)) = f'(1 + o1)/g'(1 + 01). In other words, the expression G20(t) - Go(t) = (H((1 + 28)t) - 2H((1 + 8)t) + H(t))/(20), equals, 022H"((1+01+02)t), 2 (H'((l + 0 + 01)t) H'((l+o1)t)) = where in the last equality we have applied the mean value theorem and 02 E (0, 0). It therefore follows that there is a function o: [0, oc) (0, 20) such that, G20(t)

Go(t) =

t

2

H"((I + e(t))t).

Our hypothesis limt-,,, tH"(t) oc, shows that the expression above tends to oc as t - oo. The proof of the case in which o < 0 is similar and the details will be omitted. We now continue in Section 4.2, defining the truncated quenched first moments, and then describing the parabolic Anderson equation satisfied by the quenched first moments and the corresponding Feynman-Kac representations.

4.2. Truncated quenched first moments. In the sequel, given a real function f (x) defined on Zd, we will define the discrete Laplacian by, (4.3)

A f (x) :_ E (f (x + e)

f (X)).

eEZd:Iel=1

Let us now for each finite set U C Zd and environment w E W, define the field wU := (vU,v+) with vu(x) = v_(x) for x E U, while vu(x) = oc for x V U. We now define for x E Zd and t > 0,

mU(x,t,w) := m(x,t,wU). As it will be seen later, this expression satisfies the parabolic Anderson equation with Dirichlet boundary conditions. We will denote this quantity the truncated quenched first moment on U at time t for a reaction-diffusion process starting from x. Also, we will call the set {mu(x,t,w) : x E Zd}, the field of truncated first moments on U at time t. Now, in the particular case in which U = A(x, r) for some r > 0, we will use the notation rnr(x, t, w) instead of rnU(x, t, w). We will refer to this quantity as the truncated quenched first moment at scale r at time t for a reaction-diffusion process starting from site x. Furthermore, we will call the sets {mr(x, t, w) : x E Z'}, the field of truncated quenched first moments at scale r at time t.

G. BEN AROUS ET AL.

24

4.3. The parabolic Anderson equations. Here we will recall the moment equations satisfied by the field of quenched first moments {m(x, t, w)} and by the corresponding truncated fields. Following [12], we have the proposition.

Proposition 4.2. Let U C Z' be a finite set and w E W an environment. Consider the field of quenched first moments {m(x, t, w) : x E Zd} on Zd at time t and the field of truncated quenched first moments {rfu (x, t, w) : x E Z' } on U at time t. Then the following statements are true. (i) The field of truncated quenched first moments {mu(x, t, w) : x E Z'} of the total number of particles at time t on U, satisfies the equation, amU

at

='Amu + v(x)rnU, for x c U n c(w)°

rnU (x, 0, w) = 1,

for x E 7Gd

rnU(x, t, w) = 0,

for x V U nc(w)°, t > 0.

(ii) The field of quenched first moments {m(x, t, w) number of particles at time t on Zd, satisfies the equation,

at

KAm + v(x)m,

:

x E Zd} of the total

for x E C(w)° for x E Zd for x V C(w)°, t > 0.

m(x, 0, w) = 1, m(x, t, w) 0,

PROOF. Consider the family of functions {uz(x, t)

:

E' [zi(t)]}, parametrized

by complex z such that 0 < jzl < 1. It is easy to see that,

at =

t 1 uz + v+(x)uz - (v+ (x) + v- (x))uz + v- (x), uz(x,0) = z,

for x c g(w)°, while uz(x,t) 1 fort > 0 and x E 9(w). Differentiating the above equation with respect to z we obtain part (i). A similar proof can be carried out for part (ii).

4.4. Bounds on the quenched first moments. We will now obtain upper and lower bounds for the annealed moments of the quenched first moments. Let us first recall two elementary inequalities. For n natural, let a1, ... , an be arbitrary real numbers. Then, for r > 1, we have Jensen's inequality, r

n

(4.4)

E ai

r,

< nr-

i=1

ai

i=1

while for 0 < r < 1 we have, n

(4.5)

r

E ai i=1

We will also need to introduce for L > 0 the notation, ML :

max Iv(x)

xEAL

Let us recall the following lemma, contained in the statement of [12, Theorem 2.1].


25

Lemma 4.3. Consider a finite subset U C Zd and p E P(W). Assume that p satisfies Condition E. Then, p-a.s. for every x E Z° and t > 0, the quenched first moment m(x, t, w) on 7Ld at time t starting from site x admits the Feynman -Kac representation m(x, t, w) = Ex [ef-' v(Xs) dsl(T4(w) > t)]

(4.7)

and the truncated quenched first moment mU(x, t, w) on U at time t starting from x also, mU (x, t, w)

(4.8)

Ex e1° v (X,g) ds 1 (TU" uG(w) >

t)],

where in both (4.7) and (4.8), {Xt : t > 0} is a simple symmetric random walk of total jump rate 2dt starting from x, of law Px, Ex is the expectation related to this law, and for A C Zd we define TA := inf{t > 0 : Xt V Al.

We can now apply Lemma 4.3 to obtain the first estimates on the quenched first moments.

Proposition 4.4. Consider a finite subset U C Zd and p E P(W). Assume that p satisfies Condition E. Then, (i) For each x C Zd, t > 0 and /3 > 0 there exists a constant C such that, (4.9)

eH(Qt)-2drct < (m(x, t)Q) < C(I£ + t)deH(Qt),

(ii) For each x C U, t > 0 and /3 > 0, (4.10)

eH(Qt)-2dkt < (mu(x, t) /3) < C(k + t)deH(Ot),

where C is the constant of part (i). (iii) For each /3 > 0, 'y > 0 and a > 0 we have that, (4.11)

m(x t) - tn,y(kt)a (x, t) 11) < C('y(rt)a + l)de-2prtz(ti(-t)a-l/2)eH(/st)

for some constant C > 0, where I: [0, oc)

[0, oc) is defined in equation (3.2).

PROOF. (i) The first inequality of equation (4.9) can be obtained from the Feynman-Kac representation (4.7), taking only into account the contribution of the path XS which stays during the whole time interval [0, t] at x ([12, p. 637]). To prove the second inequality of (4.9), let us note that by translation invariance it is enough to prove the estimate for (m(0, t) ). On the other hand, 1(TG(w) > t)efo v(X.) ds
0} from the box AR., with Rn := R02n and Ro := max{ct,1}, while MR, := maxx:llxll 0, for each natural n we have the following inequality which will be used soon, (4.13)

(e*MR-) < (2(Rn + 1))d exp{H(,3t)}.

G. BEN AROUS ET AL.

26

In fact, (e'tMR-) _< ExE[-R.a,R.]d (eOtMR 1(v(x) = MR )). Let us now consider the case 0 < 0 < 1 Then, by an application of inequality (4.5) to estimate (4.12), we conclude that, 00

m(0 t F,

< e/3tMRo +

eatMR P[Tn 1 < t] n=1

Taking expectations on both sides of this inequality and applying the estimate (4.13) and the second inequality of equation (3.3) for each of the d coordinates of the underlying random walk, we obtain, -/3Ro2n-1 1og(Ro2n-1/(2ekt)) )

(m(0, t)/') < 2d(Ro + 1)deH(Qt) (I + 4d

J

n=1

e-13Ro2--11og(Ro2n-1/(2ekt)) < e-)32'-11og(2'-1/(2e)) and Ro + 1 < 2Ro, Now, since we see that there is a constant C such that inequality (4.9) is satisfied for 0 < /3 < 1.

Let us now consider the case /3 > 1. Let /3' > 1 be defined by 1//3 + 1//3' = 1. Then, if we represent the left-hand side of (4.12) as 00

E k=0

etMR "P[Tn_1 < t < Tn]1/O'P[Tn-1 < t < Tn]1/Q

by Holder's inequality we get that, 1//3

00

(etMP[Tn_i < t < Tn])

m(0, t, w) < n

A computation similar to the case 0 < 0 < 1 finishes the proof of part (i). (ii) The first inequality of equation (4.10) can be deduced by an argument analogous to the one leading to the first inequality of equation (4.9). Now, note from the representations (4.7) and (4.8) that my (x, t, w) < m(x, t, w). Hence, the second inequality of equation (4.10) is a corollary of the second inequality of equation (4.9). (iii) Let us remark from the Feynman-Kac representations (4.7) for m(0, t) and (4.8) for rn,y(kt)a (0, t) that,

m(0,t)-my(,t)a(0,t)=Ex

[ef°v(xs)asl(TA-7

iu(z, t, w). We also have the expansion,

mu(z,t,w) _ Euo

can see

that,

,t) I

,w l(U))2

GUI

> 1 eao(U,w)t >('t/)o,w)2(z)

V

toeao(U,w)t

zEU

where we have used in the second inequality the fact that Oo `w (x) > 0 and in the last inequality the normalization condition EzEU( U,W)2(z) = 1. Let us now prove the second inequality of (4.16). By Jensen's inequality (4.4), in the case 0 > 1, or inequality (4.5), in the case 0 < /3 < 1, applied to (4.18), note that for some constant c(a, d, 0), (m(x t)Q) < c((Ct)da,3/2 + 1) (e

Qtao(x,(Qrt)°',w))

+

4de-2pKtz((prt)a-1/2) (eptM(t)a ).

Now, by the first inequality of equation (4.9) and a computation similar to the one leading to (4.13), the second term of the right hand side of the above inequality, is upper bounded by, (4.21)

4d2d pKt)a + 1) d exp{

,t 20Kt1 C

On the other hand, by inequality (4.20) with U (eatA0(x,(/3rt)a,w))
1 the second term in the right-hand factor of the above inequality is negligible with respect to the first term.


29

Let us now prove the first inequality of equation (4.16). In the case 0 < ,(3 < 1, by inequality (4.5) and the bound (4.20), we have that, mQ(x, t, w) > eptao(U,w)

(4.22) xEU

But when 13 > 1, by Jensen's inequality we have that a

E mQ(x t w) > IUI-(Q-I)

(Y,

m(x t w))

and (4.22) is still satisfied. Choosing U = A(0, (/30)a), and using translation invariance we get, I)-d(e/3ta0(o,(Q-t)a,w)).

(m(O,t)°) > (2[(/Kt)a] + Using again the bound (4.18), neglecting the second term, we finish the proof in

thecase 0 0.

Corollary 4.6. Suppose that Assumptions (E) and MI are satisfied. Then, for every 0

0,

Fo(t) 00. tl oc 0log(#ct + e) = F20(t)

(4.23)

4.5. Correlation and variance estimates on the field of quenched first moments. In order to prove Theorem 2.3(ii), it will be important to have a control on the variance of the the quenched first moments. In the sequel of this paper, to avoid heavy notation, we will use ma instead of m(,ct)a. Given x, y C Zd and t > 0, let us define, (m(x, t)m(y, t)) - (m(x, t)) (m(y, t)), c(x, y, t) which we will call the correlation between sites x and y at time t of the field of quenched first moments. Similarly let us define for a > 0, ca(x,y,t) : (i a(x,t)m'a(y,t)) - (Fna(x,t))(ma(y,t)), the correlation between sites x and y at time t of the truncated field of quenched first moments at scale (r't)a. Let us begin with the following lemma. :

Lemma 4.7. Let t > 0. Consider the fields of quenched first moments {m(x, t, w) : x C Zd} and truncated quenched first moments at scale (kt)a, {ma(x, t, w) : x C Zd}. Then the following statements are true. (i) The sum of the correlations between site 0 and the other sites of the field oc like the sum of the of quenched first moments, behaves asymptotically as t correlations between site 0 and the other sites of the truncated field of quenched first moments at scale (,ct)a. In other words, E c(O,y,t) ^' E ca(0,y,t) yEZd

yEZd

G. BEN AROUS ET AL.

30

(ii) Let {Ut : t > 0} be a collection of subsets of the lattice Zd indexed by t > 0. Assume that I Ut Ut,(K,t)a I as t , oc, where Ut,T {x E Ut : dist(x, Ut) > 2r},

for r > 0. Then, (4.24)

m(x, t) ti Ut

Varµ

Var,

(4.25)

c(0, y, t), yEZd

xEAut

rna(x, t)

UtI Y, ca(0,y,t),

yEZd

xEAUt

and

Var,, ) , m(x, t) - Var1, )

(4.26)

, in-,, (x, t). xEAut

xEAvt

PROOF. From the Feynman-Kac representation (4.7) of Lemma 4.3, note that it is possible to write, m(x, t, w) = Ex

leE=Ezd v(z)L(z,t)1(A)I

,

t}, £(z, t) := fo 6z(X,,) ds is the local time at the point z of the random walk {Xt t > 0} starting from x, and dz : Zd {0, 1} is the indicator function of the set {z}. From here, using Fubini's theorem it follows that we have the following representation for the correlations between site x and y of where A

:

the quenched field of first moments. (4.27)

c(x, y, t)

v(L+Z)1(A n A)) - eE,Ezd vfl(A))(eE-Ezd vz l(A))] Ex y = where G(z, t) := fo 6z (X8) ds is the local time at the point z of the random walk {Xt : t > 0}, independent of {Xt : t > 0}, starting from y, with law Py, and A is an identical copy of A, but defined in terms of the random walk {Xt : t > 0}. Furthermore, Ex,y := Ex 0 Ey, denotes the expectation with respect to the law of the independent random walks {Xt} and {Xt}. Now, the expression (4.27) for the

correlations can be written in terms of the cumulant generating function defined in equation (1.1), (4.28)

c(x, y, t) = Ex,y

H(f+Z)

- ed H(L)eE,EZd

H(IE)

where we have used the independence of the coordinates of the effective potential {v(x) : x E Zd} under p. Note that the super-additivity of H (Lemma 4.1(i)),

implies that this expression is nonnegative. On the other hand, a reasoning similar to the one leading to the representation (4.28), this time based on the FeynmanKac representation (4.8) of Lemma 4.3, enables us to deduce that, (4.29)

ca(x, y, t)

= Ex,y [1(Ta > t)1(Ta > t) where Ta

inf{t > 0 : Xt (4.30)

H(c+Z)

- eE=Ezd H(L)eE,Ezd H(Z)

7n(x,(,t)a) = inf{t > 0 : Xt V A(x, (rt)a)} and Ta := 7A(x,(Kt)a) A(y, (r t)a)}. From (4.28) and (4.29) it follows that,

E c(0, y, t) > E ca (0, y, t). yEZ'

yEZd


31

But note that in reality, due to the independence between any pair of truncated quenched first moments at time t at two points at a distance larger than 2(ict)a, we have ca(0, y, t) = yEA2(wt)a Ca (0, y, t). Furthermore, yEZd (4.31)

E c(0, y, t) = E c(0, y, t) + yEA2(K.t)a

yE7Gd

c(0, y, t). YVA2(wt)a

Now, an application of the first inequality of equation (4.9) and Proposition 4.4(ii), Ca(0, y, t). shows that since a > 0, it is true that yEAz(,t)a c(0, y, t) -

And a second application of equation (4.9) and the first inequality in equation (3.3), shows that EyVA2(,t)a c(0, y, t) q. For our purposes, the relevant fact is that L' < Li < L' + 1. In the sequel, for any given pair of real numbers a, b we will use the notation [a, b] 1 for [a, b] n Z. We now will subdivide the box [-L, L]1 in intervals according to equation (5.1). Thus, we define I1 := [-L, -L+L' -1]l and for 1 < i < q we let Ii [-L+r_j=1 L', -L+Ei=1 L'- 1]1. Note that Iq = [L-L'q+1, L]1 and IIil = L. Next, we introduce a second parameter r which is a natural number smaller than or equal to L'. We will call r the fine scale. Then, for each Ii we define an interval Ji such that Ji C Ii, I Ji I = L' - 2r and the endpoints of Ji are at a distance larger than r to the endpoints of Ii. To do so, first let ri r + Bq(i). Then define J1 : [-L + r, -L + L' 1 ri]1 and for 1 1.

Consider the field of quenched first moments

{m(x, t, w) : x E Zd} and of truncated quenched first moments {ma(x, t, w) : x EZd}

at scale (,t)a. Let L(t) : [0, oo) -+ N be such that (ct)a fx±},

the precise meaning of (1.1) is then furnished by the assumption that the function

h is regularly varying at infinity with index o E (1, oo) (case B) or o E (0, oo) (case A). For example, a normal distribution fits in case B with o = 2. Systematic study of this class of random exponentials has been initiated by Ben Arous et al. [4] (case B only), later on extended to both cases B and A in [5] (see also an earlier preprint [3]). (Let us also mention that the "limiting" value o = 0 in case B, leading to random variables of the form Xz , has been recently considered by Bogachev [6].) Motivation comes from various areas in theoretical and applied probability, including the problem of a unified treatment of limit laws for sums and extreme values (see Schlather [22]), analysis of the free energy and its fluctuations in the Random Energy Model (REM) of spin glass (see Bovier et al. [7] and further references therein and in [5]), branching random walks in random environments (see [3,5]), and risk theory (see [3,5] and references therein).

One can expect that if the number of terms N in the sum SN(t) grows fast enough relative to the parameter t, then the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT) should hold true in a conventional form. Ben Arous et al. [5] have shown that an appropriate growth scale of N = N(t) is of the form (1.3)

N ,., e Ho(t) (t -' oo),

where the rate function Ho(t) is a certain asymptotic version of the cumulant generating function (1.4)

H(t)

log E[etX]

By a Tauberian theorem of Kasahara-de Bruijn (see Section 2 below), the function H is regularly varying with index o' such that (1.5)

f

1 0

-+

1 0'

= 1.

The link between h(x) and Ha(t) H(t) can be characterized explicitly, and in particular H0 (t) can be found (for all t large enough) as the unique solution of the equation (1.6)

Ho = oh((o'Ho/t)±)

EXTREME VALUE THEORY FOR RANDOM EXPONENTIALS

43

For example, if h(x) = xel o (x > 0) then the solution Ho is given by Ho(t) = &'1 0'. As shown in [5], the critical values of the parameter A are given by (1.7)

Al =

P

r

r ,

9

A2 = 29'0) 9

so that the LLN and CLT break down if A < Al and A < A2, respectively. Moreover,

for 0 < A < A2 one can prove (under a slightly more restrictive condition of norrnalized regular variation of h, see Section 2 below) that the distribution of SN(t), properly centered and normalized, converges to a stable law with characteristic exponent /A (1.8)

aa(o,A):

-

(note that the critical values (1.7) correspond to the canonical values al = 1,

a2 = 2). Let us point out that the centering constant vanishes for 0 < A < A1, while the normalization constant is given for all 0 < A < A2 by B(t) = ett71(t> , where ,q, is the (unique) root of the equation h(rli) = AHo(t) (note that the righthand side of this equation is asymptotically equivalent to log N, due to the scaling condition (1.3)). A more precise statement of these results will be given below in Section 3.

In the present paper, we complement this analysis by studying the asymptotic behavior of the upper extreme values of the exponential sample {etXi}'y1. Surprisingly enough, the limiting picture here replicates the classical results in the i.i.d. extreme value theory, known in the case of attraction to the Frechet distribution. In particular, the maximal term M1,N(t) normalized by B(t) = e±t71(t) (for all A > 0) converges in distribution to the Frechet law 4),, with the distribution function (1.9)

fa(x) = exp(-x-'), x > 0

(Theorem 4.3). The proof was first sketched in [4] (in case B) and later on extended

to both cases B and A in [3]. This result should be contrasted with the limit distribution of the maximum of "plain" exponentials, {exi }N 1, which appears to be the Gumbel (double exponential) law A with the distribution function (1.10)

A(x) = exp(-e-'),

x E IR

(Theorem 4.1). Limit theorems for the (joint) distribution of individual order statistics confirm the Frechet-type nature of exponential extremes (Theorems 4.8, 4.9). Of particular interest is Theorem 4.10, which reveals a Poisson limiting structure of the empirical extremal measure (in the sense of convergence of finite-dimensional distributions), with a "stable" intensity measure d(x-a). We also explore the role of the maximum in the sum. To this end, we use some representations of order statistics via uniform and exponential random variables, which allows us to work with a.s.-convergence. Using this approach, we obtain series representations for the joint limit distribution of SN(t) and M1,N(t). The rest of the paper is laid out as follows. In Section 2 we specify our regularity assumptions and derive some consequences. Section 3 summarizes results about

the limit laws for the sums SN(t) proved earlier. Section 4 deals with the limit (joint) laws for upper order statistics. In Section 5 we work out certain useful representations of order statistics, in order to obtain certain explicit characterizations

L. V. BOGACHEV

44

of the limit laws of the sums SN(t). Finally, the Appendix contains a few direct proofs of a formula for the expected value of the limiting ratio SN(t)/M1,N(t) in

the case 0 0, set

(2.10)

(2.11)

rlx = ix (t)

flog x

77i(t)

t Combining equations (1.6) and (2.10) as t ---j oo and using that h E Re, one can find

lim trll (t) t-»oo Ho(t)

(2.12)

-o

_ a

.

The next lemma plays the crucial role.

Lemma 2.2. Uniformly in x on each interval [x0, x1] C (0, 00),

tlI m[h(rlx(t)') -

alogx.

PROOF. Note that Kx (t) :=

Wj) _ (1 ± t (t))

-41

(t ---. 00),

uniformly in x E [xo, x1] C (0, oo). Therefore, for all large enough t the function icx (t) is uniformly bounded, 0 < no < ,c, (t) < /c1 < oo. Applying Lemma 2.1 and using (2.10), in the limit as t --f oo we obtain, uniformly in x, h(771(t)±) - h(r11(t)±)(r-x(t)e - 1) _ AHo(t)

- AHo(t) according to (2.12).

log x) ((i ± trll (t) ologx tr7i(t)

fe

alogx,

-1

L. V. BOGACHEV

46

3. Limit theorems for the sums The main results of the paper [5] can be summarized in the following two theorems. We assume throughout that lim Ne-AH°(t) = 1, (3.1) t-oo where A > 0 is a parameter (see (1.3)). Recall that a is given by equation (1.8).

Theorem 3.1. Set (3.2)

A(t)

A > Al A = Al

0,

0 0 lim B(t) = 0. t- 00 A(t) Therefore, Theorem 3.1 implies the following LLN.

Theorem 3.2. If A > Al then (3.6)

A((t) +

1

(t , oo),

where A(t) is given by (3.2).

Remark 3.3. In fact, in the regions A > A, and A > A2 the LLN and CLT, respectively, hold under the plain regularity condition h E Re, without any extra assumptions. Moreover, the function H(t) can be used here instead of Ho(t), and the scaling condition (3.1) can be relaxed to log N >> A,H(t) and log N >> A2H(t), respectively (see [3,5]). 1For 1 < a < 2, we use an analytic continuation of the gamma function in (3.5), given by

P(1 - a) = 1'(2 - a)/(1 - a). 2See [12, #8.367].


47

Theorem 3.1 can be proved using the known asymptotic methods for sums of independent random variables (see, e.g., [17]). The proofs are technically quite involved since the condition of normalized regular variation guarantees only the limited smoothness of the distribution of X (i.e., ultimate monotonicity and a.e.differentiability). However, this condition (in particular, Lemma 2.2), combined with the scaling condition (3.1) provide enough analytical control. For instance, in view of (2.1) the tail probability for etX is asymptotically given by (3.7)

N-leh(n1 )-h(,7x )

P{etX > xB(t)} = P{X > ±ijx(t)} = e

- N-le-alogx = N-lx-a where B(t) = e+t'71(t) (see (3.3)). Therefore, the Levy-Khinchin spectral function, being the main ingredient of the limiting infinitely divisible law [17, pp. 81-82], is given for x > 0 by (3.8)

L(x) :_ - tli m N P{etX > xB(t)}

= -x-a,

which indicates that a is indeed the characteristic exponent of the limit (stable) law (see [15, Theorem 2.2.1]). Using (3.1), note that for A < A2 we have

B(t) = ef"1(t)

,., N±t71(t)/()Ho(t))

(t --1 cc),

and (2.12) implies that in case B, N is being raised to the power tijl (t) AHo(t)

0

1

a >a

This should be compared to classical results in the i.i.d. case (see, e.g., [15, Theorem 2.1.1]), where the normalization is essentially of the form N1/a. As we see, in case B the sums SN (t) have a stable limit by virtue of a nonclassical (heavier) normalization. As for case A, we have B(t) - N-t71(t)/()Ho(t)) _, 0, which has no analogies in the classical theory. However, another look at the tail probability reveals the mechanism of settling down to a stable law, analogous to that in the i.i.d. situation and acting in both cases A and B. Indeed, in order that i.i.d. random variables {Y } belong to the

domain of attraction of a stable law with characteristic exponent a > 0, it is sufficient that for each x > 0 (3.9)

P{Y > Nl/ax} -

N-lx-a

(N -+ oo)

(see [15, Theorem 2.6.1]). If we set Y2 := etXi/B(t)N-l/a then, according to (3.7),

P{Y > Nl/ax} = P{etX > xB(t)} -

N-lx-a

(t --> oo),

which mimics the condition (3.9). Thus, in the normalizing function represented in the form B(t) = B(t)N-l/a N'/a, the factor B(t)N-11' is responsible for the correct behavior of the distribution tail, whereas the conventional power Nl/a performs averaging towards a stable law with characteristic exponent a. This observation helps explain heuristically the many similarities between the limit behavior of random exponentials etXj and that of the usual i.i.d. random variables-from convergence to a stable law (Theorem 3.1) to the asymptotic properties of extreme values (Sections 4, 5 below).

L. V. BOGACHEV

48

We conclude this section with the following LLN for log SN (t): as t --* oo, a(A ± 1) A> Al, log SN (t) 1,

(3.10)

t?Jl (t)

1fl,

0 I as t --4 oc. If A > A1i then from (3.1) and (3.2)

logA(t) = log N f H(t) ". (A ± 1)Ho (t),

and so (3.10) follows by (2.12). If A _ Al = o'/g (i.e., a = al = 1), then A(t) < Ne+H(t) (see (3.2)), so from (3.11) we get, according to (2.12) and (3.1), logSN(t) < logN fH(t) (A1 f 1) (3.12) 0(t) till (t) till (t) Al e

- fl

(cf. (1.5)). On the other hand, SN(t) > M1,N(t) := max{etXi, i = I,-, N}, hence logSN(t) > logM1,N(t)

fl

(3.13)

(t --> co),

till (t) ti71(t) as we will show below (see (4.14)). Now, combination of (3.12) and (3.13) yields (3.10).

Remark 3.4. The limit (3.10) has the meaning of the free energy in the REM (see [3, Section 9; 5; 7] and further references therein). It is easy to check that (3.10), as a function of A, is continuous and continuously differentiable everywhere including the critical point A _ A1, but its second derivative has a jump at this point. This corresponds to a "third order" phase transition [7].

4. Asymptotic behavior of extreme values 4.1. The plain exponential maximum. Let us first obtain the distribution of the maximum for plain exponentials, eXi (i.e., without the large scaling parameter t in the exponent). Namely, we will show that under our conditions on Xi, the random variables eXi belong to the domain of attraction of the Gumbel (double exponential). distribution A (see (1.10)).

Theorem 4.1. Denote Y := eXi, Y1,N := max{Yl,... , YN}, and set (4.1)

aN

exp(±h'-(log N)t),

bN

faNlogaN ologN

Then

aN

(

(4.2)

lim P(

N-oo

l

Y1' bN

ll

< x1 = exp(-e-x),

x E JR.

PROOF. It is not difficult to verify available sufficient conditions for convergence

of the maximum's distribution to A (see, e.g., [11, Theorem 2.1.3]). However, it appears even simpler to prove (4.2) directly. Using (2.1), we can write

(4.3) P{Y1,N < aN +xbN} = (P{X < log(aN + xbN)})N = (1 where

LN(X) :=+log(aN+xbN).

e-h(LN(X)}))N,


49

Note that, according to (4.1),

LN(0)l = ±(logaN)± = h'-(logN),

(4.4)

and since h E NRe, equation (4.4) can be inverted for all large enough N to yield

h(LN(0)±) = logN.

(4,5)

Observe that, as N -- oo, bN (4.6)

±

aN log aN

0

1

o log N

and, moreover, bN

(4.7)

aN

h-(logN)f logaN = = f ologN ologN

since h'-(x) e Rl/e and h'-(x)±/x E R±l/p_1 with ±1/o-1 = -1/0' < 0. In view of (4.6) and (4.7), we obtain (4.8)

rN(X) :_

LN(x)±

LN(0)=

1+

log(1 + xbN/aN)1 log aN

J -j 1 (N --> oo)

and hence LN(x)± - LN(0)± --4+00 (see (4.4)). Using (4.3) and (4.5), it follows that the limit (4.2) is reduced to showing that

lim [h(LN(x)±) - h(LN(0)±)] = x.

N-oo

To this end, we apply Lemma 2.1 and use (4.5), (4.8) to get

h(LN(0)t)

h(LN(0)) )(tN(x)e - 1) tog N

(1 +

log(1 + xbN/aN))+e - 1

\\

tog N

log aN

J

(+o)xbN aN log aN = x'

0

according to (4.6).

Passing to logarithms, it is easy to show that the background random variables

Xi = log e'i also belong to the domain of attraction of the Gumbel distribution (see also [6, Proposition 10.1]).

Corollary 4.2. Let X1,N := max{Xl, ... , XN} and set (4.9)

aN := fh (log N)fbN :_

h (log N o log N

Then

(4.10)

lim P

N-.oo

X1,N - aN bN

<x = ex p(-e-'), I

x E R.

Since bNliiN -4 0, the limit (4.10) readily implies the following LLN: (4.11)

h-(logN)!

±1 (N -+ oo).

L. V. BOGACHEV

50

4.2. Limit distribution of the exponential maximum. We shall now see that when a nonlinear (power) scaling is switched on, e X c-, etx the limit law for the maximum changes dramatically. Denote by Mk,N - Mk,N(t) := etxk,N the nonincreasing order statistics of the sample In particular, Mk,N(t) _ i = 1, ... , N}. For all A E (0, oo) denote B(t) := eft?jl(t)

(4.12)

where i11(t) is defined by equation (2.10) (cf. (3.3)). In this section, we obtain the limit distribution of M1,N(t).

Theorem 4.3. For all A > 0, as t -> oo, M1,N(t)/B(t) converges in distribution to the Frechet law 4)a (see (1.9)). PROOF. Using (2.1), (2.11), and (4.12), for x > 0 we have (4.13)

P{M1,N(t) < xB(t)} = P{X1,N
0 and each k E N, (4.17)

( Mk,N(t)

lim P{

l

B(t)

-03

e-

<x

j!

)))

PROOF. We have P{Mk,N < xB(t)} = P{Xk,N -a log xi (see Lemma 2.2), we arrive at (4.21).

El

5. Sums of random exponentials via order statistics 5.1. Some representations of extreme values. In this section, we record a few (basically well known) representations in distribution for extreme values Mk,N

and hence for the sum SN(t). These representations are expressed in terms of auxiliary sequences of i.i.d. random variables with either exponential or uniform distribution. In particular, they will be used to study the asymptotic behavior of SN (t) in the case 0 < A < A. The advantage of such an approach (usually called the "method of common probability space") is that the random variables of interest will have a limit with probability one, rather than just in distribution. Consider the random variable 'E := ±h'-(fi)t, where has the unit exponential distribution, that is, P{ > x} = e-x (x > 0). A key observation is that E a X. Indeed, since h is right-continuous, we can use the property (2.4) to obtain

P{E < x} =

x} =

P{ < h(±xt)} = 1 - e-h(tx}) = P{X < x}, according to (2.1). Furthermore, if 1, ... , Z;N are i.i.d. random variables with unit are also indeexponential distribution, then the random variables _i := pendent and hence the random vector (°i)N1 has the same distribution as (Xi)f'1.

In particular, the joint distribution of the order statistics E1,N > > EEN,N coincides with that of the order statistics X1,N > ... > XN,N. We also note that since the function ±h'-(x)± is nondecreasing in its domain, the order statistics Sk,N cantt be represented through the underlying exponential order statistics ... SN,N as k,N = ±h`-(Sk,N)t (k = 1, ... , N). S1,N Furthermore, let us recall the following representation in distribution of the exponential order statistics (see [9, Section 2.7]).

L. V. BOGACHEV

54

Lemma 5.1. For each k = 1, . . , N, the distribution of the kth order statistic ek,N coincides with the distribution of the random variable .

Tk,N

(5.1)

where (c,) is an auxiliary sequence of i.i.d. random variables, each having the unit

t

exponential distribution. Moreover, (Z;1,N,

,

SN,N)

a

(Ti,N, ... , TN,N).

As a result, the joint distribution of the order statistics Mk,N = etXk,N coincides e+th-(TT,N)t. Hence, the sum with the joint distribution of the random variables 1 etXi = X:N 1 etXk,N has the same distribution as the sum SN(t) = EN i= k=

N

E efth'-(Tk,N)f k=1

given by (5.1). It is convenient to rewrite Tk,N as N /

Tk,N =

I

\

z

log Z

i=k

i

1

I+ log N

1

k

Lemma 5.2. 00

(ZZ -log a

(5.4)

converges with probability one.

PROOF. Let us represent the series (5.4) as (5.5)

>Cii -log i + °O

1

°O ( - 1

°O / 1

log

i+ i

1

and show that both series on the right are convergent. Recalling that (z are exponentially distributed with mean 1, we have E[(( - 1)/i] = 0 and

ccVar(ti1 )

00

i=1

i=1

Var

00

oo),

hence the last series in (5.5) converges.

Remark 5.3. A similar approach was used by Hall [14] to obtain a canonical representation for limiting extreme values in the i.i.d. scheme, following the idea suggested by Renyi [18] (see also [19, Chapter VIII, Section 9]]). LePage et al. [16] have used the same approach to study convergence to a stable law in the classical situation of i.i.d. random variables.


55

Lemma 5.2 implies that for each k > 1 the sum of the series

Zk :_

log i

i

is finite with probability one. It is not difficult to find the (joint) distribution of

4k Lemma 5.4. Let us set Tk :=

ke-zk (k = 1, 2.... ). Then ('rk) d (o-k), where

Ak := i+ - +(% and ((z) is a sequence of independent exponential random variables with mean 1. In particular, 'rk has the gamma distribution with mean k.

Remark 5.5. The random variables times of a Poisson process with unit rate.

are distributed as a sequence of arrival

PROOF OF LEMMA 5.4. According to the definitions (5.6) and (5.1), we have N

Zk =N-+oo Urn 1: ( (.Zi

log

i+1

lim (Tk,N - log N) + log k.

N-+oo

2

i=k

Hence, Tk = e-zk +logk = N

(5.7)

Recall that by Lemma 5.1, the random variable Tk,N has the same distribution as the kth (decreasing) order statistic Sk,N of N independent exponential random variables (Si)N1 (with mean 1). From this, one could derive the distribution of Tk using the known limit results for the exponential order statistics (see [11, Example 1.3.1 for k = 1 and Theorem 2.8.1 for k > 1]). However, a more neat proof is possible that requires almost no calculations and simultaneously allows one to establish independence of the successive differences (i = 1,.. . , N) Tk+i - Tk. Namely, observe that the random variables Ui := d are independent and uniformly distributed in [0, 1]. Therefore, (e-fk,N)k 1 < UN,N are the order statistics of (Ui)N 1. In turn, it (Uk,N)N_1, where U1,N < is well known (see [10, Chapter III, §3]) that (5.8)

(UI,N, ... , UN,N)

d

ol1

, ... ,

QN+1

ON

QN+ 1

where Uk are described in the lemma. Returning to (5.7), the distribution of the vector (r1, ... , Tk) for each k > 1 can be computed as the weak limit NUk C QN+1

NQk ,

UN+/ 1 d

(Or1, ... , 9k)

(N - oc),

where we used that, due to the Law of Large Numbers, o-N+1 IN -+ 1.

5.2. Preparatory estimates. The main goal of this section is to establish a suitable uniform upper bound (with probability one) for the terms of the sum (5.2)) (see Lemma 5.7 below), which will allow us to pass to the limit in (5.2) as t --+oo.

Lemma 5.6. For each fixed k > 1, with probability one, exp{±th'-(Tk,N)'} = k-1/aezk/a, lira (5.9) t--.oo B(t)

L. V. BOGACHEV

56

where B(t) is given by (3.3) and Zk is defined in (5.6). PROOF. Note that a number sequence (an) has a limit a c ll if and only if for each c a,

lim 1 oo),

while on the right, using Lemma 2.2 and the scaling relation (3.1), we have

h(rl"(t)')

-

AHo(t) - log N --> alogc (t---> oo).

Therefore, in the limit t -i oo, N -p oo, inequality (5.13) takes the form Zk -log k < alogc, or equivalently, k-1/ctieZk/a < c. Comparing this inequality with (5.11) and applying (5.10), we obtain (5.9).

Let us note that if Ei ai is a convergent series, then its partial sums E k ai are uniformly bounded. Indeed, set sn ai, so := 0, then s* := supra sn < +oo,

s* :=infrasn> -ooand IE kaiI = Isn-sk-1I <s*-s* < oo for all n and k xN(t) -..

1T

I

aP/ > \1 T

> 0, since a > a. Hence, we can apply Lemma 2.1 and, using the elementary inequality

xn-1>p(x-1) (p>1orp ±gh(rli)(xk - 1) = oh(rli) atrll

(Z*+c-

alogk\J NZ*+e-alogk a

a

uniformly in k < N. We also note that by (2.10) and (3.1), (5.21)

log(N + 1) = h(rll) + o(1) (t -+ oo).

Estimates (5.20), (5.21) imply that inequality (5.18) will be proved once we check

that

Z* -logk+o(1) < (z*

+6- a log k

(1+0(1)),

where all 0(1) are uniform in k < N. Rearranging, this is reduced to the inequality

0 < e+ (1 -

-

\

a

+0(1) ) logk+o(1),

which holds as t --* oo, since s > 0, 1 - a/a > 0 and log k > 0.

5.3. Convergence of the representations of sums. We are now in a position to prove the following result.

Theorem 5.8. Assume that 0 < a < 1. Then, as t --* oo, N (5.22)

B(t)k=1 >

efth

o0

(TI N )f

V. := E

k-1/aezk/«

k=1

where the last series converges with probability one.

for k < N(t) and Vk(t) := 0 otherwise. By Lemma 5.6, Vk(t) - k-1/ae2k/" as t -3 00. Let us pick some s > 0 and for a given a E (0, 1) choose a number a such that a < a < 1. Then, PROOF. Denote Vk(t) :_

(1/B(t))e+th-(Tk,,)}

according to Lemma 5.7 (see (5.17)), with probability one for all t large enough we

L. V. BOGACHEV

58

k-1/ae(Z.+E)/a,

k = 1, 2, ... Since 1/6 > 1, the series Ek vk* have Vk(t) < v* := convergent, so the Lebesgue dominated convergence theorem yields 1

B(t)

N k=1

00

e±th-(Tk,N)}

is

co

-4

vk(t) as.

k-'/"e Zka . Va (t k=1

k=1

also implying a.s.-convergence of the limiting series.

Remark 5.9. The representation of the limit (5.22) can be rewritten as 00

(5.23)

Va =

Tk

,

k=1

where (Tk) are defined in Lemma 5.4. Note that convergence of the series (5.23) is k_11a < 00. obvious, since a.s. Tk - k as k -+ oo (by the strong LLN) and >k

Recalling that SN(t) has the same distribution as the sum (5.2), from Theorem 5.8 it follows that the distribution of SN(t)/B(t) weakly converges, as t ---p oc, to the distribution of the random variable Va. Comparing this result with Theorem 3.1, we arrive at the following assertion.

Theorem 5.10. For 0 < a < 1, the random variable Va defined in (5.22) has the stable distribution Fa with characteristic function 0a given by formula (3.5). This theorem can be viewed as a series representation of the stable distribution

.Fa (with 0 < a < 1 and a = 1). However, being rewritten in the form (5.23) this representation amounts to one of the known formulas (cf. [21, Theorem 1.4.5]).

Theorem 5.11. For a E (0, 1), the ratio SN(t)/M1,N(t) has a proper limit distribution, which can be represented via the random variable k

00

(5.24)

Wa := e-Z1/a Va

-1

exp k_1

a 2-1 i

where ((z) is a sequence of independent exponential random variables with mean 1 involved in the representation (5.6).

PROOF. As stated after Lemma 5.1, the joint distribution of M1,N(t) = etxl,N and SN(t) = Tk 1 etxk N coincides with that of the pair N

exp{±th'-(T1,N)±},

Eexp{±th'-(Tk,N)±}. k=1

In particular, (5.25)

SN(t) d M1,N(t)

EN exp{+th-(Tk,N)'} 1 exp{+th'-(T1,N):'}

Dividing both the numerator and denominator by the function B(t) defined in (3.3) and applying Lemma 5.6 (with k = 1) and Theorem 5.8, we deduce that the right-hand side of (5.25) with probability one converges to (5.26)

Vae-Z1/a

= 1+

0"

k=2

k-1/ae-(Z1-Zk)/a.


59

so (">.6) it follows that for k > 2 k-1

Z1 - Zk =

i _ log i + 1

k-I

i=1 flat

z

log k.

i=1

e, (5.26) is reduced to the expression

/

00

1 k-I

a

k=2

i=1

2/

which is the same as the right-hand part of (5.24).

Remark 5.12. The random variable (5.24) can be represented as (cf. (5.23)) 00 I/a

Wa = > CTk f

(6.28)

,

where (Tk) are defined in Lemma 5.4.

Theorem 5.13. The random variable Wa defined in (5.24) has the characteristic function given by eiU (5.29)

f. (u) =

1 - a f0 (eiux - 1) dx/xa+1

Remark 5.14. Remembering that Wa has emerged in Theorem 5.11 in relation to the limit of SN(t)/M1,N(t), it is worthwhile to compare Theorem 5.13 with the analogous result by Darling [8, Theorem 5.1]; see also Arov and Bobrov [1, Corollary 4], stating that if (Y) is a sequence of i.i.d. random variables with distribution belonging to the domain of attraction of a stable law with exponent 0 < a < 1, then for SN := Yl + + YN and M1,N := max{Yl,... , YN} the ratio SN/M1,N has the limit distribution with characteristic function (5.29). PROOF OF THEOREM 5.13. Using the observation in Remark 5.14, we will prove the theorem's statement indirectly, via establishing a representation analogous to (5.24) for the limit of the ratio SNIMI,N. Clearly, it suffices to do this with a suitable choice of random variables Yi. Let us set Y = ecu/a, where is an i.i.d. sequence of exponentially distributed random variables with mean 1. Note that Yj > 1 and

a log x} = e-a log x = x-a

P{Yi > x} =

(x > 1).

Hence (see [15, Theorem 2.6.1]), the distribution of Y is in the domain of attraction

of a stable law .''a (with,3 = 1) and SN

Nl/a =

N11

a

Passing to the order statistics 6,N > we represent the sum SN as N

SN = k=1

-d To, (N--> oo). > eN,N, in a way similar to the above d CND k=1=1

eTk,N/a,

L. V. BOGACHEV

60

where Tk,N are given by (5.1). Analogously to (5.13) and (5.14), we obtain N 00 sN d -1/a eZk/a = Va (N -> 00), = e (Ti. N-1ogN)/a a.s. k (5.30)

Nl/a

-

k=1

k=1

where a term-by-term passing to the limit can be justified as before, using (5.16). Similarly, one shows that M1,N d (T1 N-log N)/a ± ezl/a (5.31) (N -* oo). N1/a Hence, dividing (5.30) by (5.31) we obtain (cf. (5.24))

-e -Z /a Va=_Wa

SN d M1,N

1

Comparing this with the result by Darling [8] mentioned above, we conclude that Wa has distribution with the characteristic function (5.29).

Remark 5.15. It would be interesting to derive formula (5.29), or otherwise characterize the distribution of Wa directly from representation (5.24) (or (5.28)). The following result, being an immediate consequence of Theorem 5.13, is of interest due to the striking simplicity of the answer. Corollary 5.16. For 0 < a < 1, the expected value of Wa is given by E[Wa]

(5.32)

=

1

a

1

PROOF. Differentiate formula (5.29) at u = 0.

Remark 5.17. In the Appendix, we will give three alternative proofs of identity (5.32) based on representations (5.24) or (5.28).

Remark 5.18. Taking the expectation of Wa using (5.24) (see (A.1) below) and comparing with (5.32), we arrive at the following curious identity: 00

k

Eri (1+ia) k=1 i=1

a

1-a

(0 0+. To find the distribution of Z1, note (I = Z1 + log Wa cording to the definition of 'rk in Lemma 5.4, we have Z1 = - log Ti, where the exponential distribution with mean 1. Hence, P{Z1 < x} = P{-F, > =- exp(-e-x).

l .emark 5.21. The limit theorem (5.33) can be used to explore the limiting -+ 0+ of Shlather's conjecture concerning random norms (see [3, §9; 6; A general result of this kind was proved purely analytically by Zolotarev Theorem 5; 25, Theorem 2.9.1].

We conclude this section by stating a series representation theorem for the limit t lie case Al < A < A2, analogous to Theorem 5.8.

Theorem 5.22. N

B(t)

(a) For Al < A < A2, 00

(e±th-(Tk,N)}

)E

-E[e±th_(Tk,N)}])

k=1

eZk/a fi,

k=1

El rzk/a] ae

(b) ForA_A,, N I T(e±trt (TkN)*

B(t)

k=1

k=1

Here the limiting series converge with probability one.

This theorem can be proved along the same lines as in the case 0 < A < Al above. However, the proof is technically more involved and will be presented elsewhere.

Appendix A. Direct proofs of Corollary 5.16 A.I. A proof using representation (5.24). From (5.24) it follows 00

(A.1)

k

] =1+a+i. 00

E[Wa ] =1+E fl E[e- Ci l(ia) k=1i=1

k

i

k=1i=1

Note that the right-hand side of (A.1) coincides with the hypergeometric function [12, #9.100] 00

F( a, b ;c ;z ) ;

:

=1+ k=1

n

(a + i)(b + i)

ti=0 (c+i)(l+i)

taken at the values a = 1, b = 1, c = 1 + a-1, z = 1. Furthermore, it is known [12, #9.122(1)] that for z = 1 one has

r(c)r(c-a-b) (c> a+b).

F(c - a)F(c - b) For the above specific values of the parameters this yields I'(1 + a-1)F(a-1 - 1) (A.2) F(1 1; 1 + a-; 1) _ F(a -1 ) 2

L. V. BOGACHEV

62

Using that F(1 + x) = xF(x), we obtain F(1 +

F(a

- 1) = a-iF(a-1)

a-1)F(a-1

a

F(a) -1 1-a

2

=

1)

so substituting this into (A.2) we get (5.32).

A.2. A proof using representation (5.28). The proof above may not seem quite satisfactory, as it relies heavily on the èxternal' analytic aid from tables of formulas. Here we give a simpler, self-contained proof based on another representation of W,,, given by equation (5.28). Namely, using Lemma 5.4 and relation (5.8) W" a we deduce from (5.28) that 1 + Ek1 U11, where Ui,k is the minimum of independent random variables U1, . . . , Uk with uniform distribution on [0, 1]. Therefore, CO

E[WE]

(A.3)

k=1

Note that P{Ul,k > x} = (1 - x)k (0 < x < 1), and hence E

[U] =

f x1k(1 - x)1 dx.

Substituting this expression into (A.3), we obtain 00

E[WE] =1+ Jlxl/"Ek(1-x)k-1dx=1+ Jxl/a-2dx= 0

k =1

0

la

and (5.32) follows.

A.3. Yet another proof. Finally, following an elegant idea of S. A. Molchanov (private communication), we give an elementary proof of identity (5.32) proceeding directly from representation (A. 1). For 0 < a < 1, set 1

al

a2

a-1 - 1'

_

i

a-1 + i - 1

(i > 2),

k

(k > 1).

ft ai

Ak

i=1

It is easy to check that for all k > 1, k

i=1

k

i

a-1 + i = i=1

It follows that the series on the right-hand side of (A.1) is reduced to 00

1 + >(Ak - Ak+1) = 1 + lim (A1 - An) = 1 + A1, n-*co

k=1

since An -* 0 as n -* oo. But

1+A1=1+a1 and formula (5.32) is proved.

1

1-a'


63

Iternark A.I. The different proofs of relation (5.32) given above, being of h*t ' vst, in their own right, may be useful for a direct proof of Theorem 5.13 (cf. frinnrk 5.14). Acknowledgments. Financial support by the conference organizers and the C:Ì(?v1 (Montreal) is gratefully acknowledged. This paper, as well as my talk at the Euflference, is based in part on joint work with G. Ben Arous and S. A. Molchanov. My special thanks are due to Stanislav Alekseevich Molchanov, my former supervisor and subsequently a colleague, who has had a profound impact on my vision of Mathematics. For more than thirty years, I have been privileged - alongside the

uncountably many other lucky ones-to witness his awesome creativity, benefit from his insightful ideas and enjoy his art as a champion story-teller. I am now very happy to add my voice to the choir of sincere congratulations on the occasion of his 65th birthday. Cheers!

References I.

D. Z. Arov and A. A. Bobrov, The extreme terms of a sample and their role in the sum of independent random variables, Theory Probab. Appl. 5 (1960), no. 4, 377-396.

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia Math. Appl., vol. 27, Cambridge Univ. Press, Cambridge, 1989. 3. G. Ben Arous, L. V. Bogachev, and S. A. Molchanov, Limit theorems for sums of random exponentials, Technical Report N103078-IGS, Isaac Newton Institute for Mathematical Sciences, 2.

4.

Cambridge, 2003, http://www.newton.cam.ac.uk/preprints/NI03078.pdf. , Limit theorems for sums of random exponentials, Recent Developments in Stochastic

Analysis and Related Topics (Beijing, 2002) (S. Albeverio, Z.-M. Ma, and M. Roeckner, eds.), World Sci., Singapore, 2004, pp. 45-65. , Limit theorems for sums of random exponentials, Probab. Theory Related Fields 132 5. (2005), no. 4, 579-612. 6. L. V. Bogachev, Limit laws for norms of iid samples with Weibull tails, J. Theoret. Probab. 19 (849-873), no. 4, 2006. 7.

8.

A. Bovier, I. Kurkova, and M. Lowe, Fluctuations of the free energy in the REM and the p-spin SK models, Ann. Probab. 30 (2002), no. 2, 605-651. D. A. Darling, The influence of the maximal term in the addition of independent random

variables, Trans. Amer. Math. Soc. 73 (1952), 95-107. H. A. David, Order statistics, 2nd ed., Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1981. 10. W. Feller, An introduction to probability theory and its applications: Vol. II, 2nd ed., Wiley, 9.

New York, 1971.

11. J. Galambos, The asymptotic theory of extreme order statistics, Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1978. 12. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 5th ed., Academic Press, Boston, MA, 1994. 13. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, 2nd ed., University Press, Cambridge, 1952.

14. P. Hall, Representations and limit theorems for extreme value distributions, J. Appl. Probab. 15 (1978), 639-644. 15. I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff, Groningen, 1971. 16. R. LePage, M. Woodroofe, and J. Zinn, Convergence to a stable distribution via order statistics, Ann. Probab. 9 (1981), no. 2, 624-632.

17. V. V. Petrov, Sums of independent random variables, Ergeb. Math. Grenzgeb., vol. 82, Springer, Berlin, 1975. 18. A. Renyi, On the theory of order statistics, Acta Math. Acad. Sci. Hungar. 4 (1953), 191-227. 19. , Probability theory, North-Holland Ser. Appl. Math. and Mech., vol. 10, Elsevier, Amsterdam, 1970.

64

L. V. BOGACHEV

20. S. I. Resnick, Extreme values, regular variation, and point processes, Appl. Probab. Ser. Appl. Probab. Trust, vol. 4, Springer, New York, 1987. 21. G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian random processes: Stochastic models with infinite variance, Stochastic Model., Chapman & Hall, New York, 1994. 22. M. Schlather, Limit distributions of norms of vectors of positive i. i. d. random variables, Ann. Probab. 29 (2001), no. 2, 862-881. 23. I. Weissman, Multivariate extremal processes generated by independent non-identically distributed random variables, J. Appl. Probab. 12 (1975), 477-487. 24. V. M. Zolotarev, The Mellin - Stieltjes transformation in probability theory, Theory Probab. Appl. 2 (1957), no. 4, 433-460. 25. , One-dimensional stable distributions, Transl. Math. Monogr., vol. 65, Amer. Math. Soc., Providence, RI, 1986. DEPARTMENT OF STATISTICS, UNIVERSITY OF LEEDS, LEEDS LS2 9JT, UK

E-mail address: [email protected]

4 sot tr do Recherches Mathematiques 441A1 Proceedings and Lecture Notes l'-bum,, 42, 2007

Singular Continuous and Dense Point Spectrum for Sparse Trees with Finite Dimensions Jonathan Breuer ABSTRACT. Sparse trees are trees with sparse branchings. The Laplacian on some of these trees can be shown to have singular spectral measures. We focus on a simple family of sparse trees for which the dimensions can be naturally defined and shown to be finite. Generically, this family has singular spectral measures and eigenvalues that are dense in some interval.

1. Introduction This paper extends and complements the paper [5] in which the notion of sparse trees was introduced. Sparse trees are trees which have arbitrarily long "onedimensional" segments (by which we mean intervals of Z), separated by occasional non-trivial branchings. It is shown in [5] that, when these trees are spherically symmetric, one may decompose the Laplacian as a direct sum of Jacobi matrices which have sparse "bumps" off the diagonal. The spectral theory of these matrices is similar to that of one-dimensional Schrodinger operators with sparse potentials (see [11] and references therein). In particular, matrices of this type exist for which the spectral measures are singular with respect to Lebesgue measure. These ideas make it possible to construct simple examples of trees for which the Laplacian has interesting spectral behavior. Several examples with singular continuous spectrum were presented in [5].

In this paper we will be concerned with a family of sparse trees that "interpolates" between Z+ and the Bethe lattice. These trees can be obtained from the Bethe lattice by replacing an edge, at a distance n from the root, by a segment of length - y' for some fixed ry > 1. While the Bethe lattice is infinite-dimensional, a tree obtained in this manner can be shown to have dimensionality = log ryk/ log -y, where k is the connectivity of the original Bethe lattice. (For our definition of dimension see Section 3). Thus, by letting -y vary from 1 to oo, one gets a family of trees corresponding at one end ("ry = 1") to the Bethe lattice, and at the other end ("-y = oo") to Z+. 2000 Mathematics Subject Classification. Primary 47B15; Secondary 81.

This research was supported in part by the Israel Science Foundation (grant no. 188/02) and by grant no. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. This is the final form of the paper. ©2007 American Mathematical Society 65

66

J. BREUER

We shall analyze the spectral properties of the Laplacian on these trees with the help of the decomposition described above and some tools from the spectral theory of Schrodinger operators with sparse potentials. The constant branching, however, turns out to be a technical difficulty. We will bypass this difficulty by using an idea from [21], namely, we shall impose a certain probability measure on these trees and prove an "almost sure" result. It turns out that the situation for these finite-dimensional structures is markedly different from the one for Zd. These trees (generically) have purely singular spectral measures and some dense point spectrum. In addition to the new result described above, we also use this opportunity to expand the discussion on the basic setting and on some of the examples presented in [5]. Some basic facts that were briefly mentioned in that paper (such as the self-adjointness of the Laplacian on normal sparse trees), will be explained here in greater detail. We remark that graphs with singular continuous [18] and pure point spectrum [13] are known to exist. In this context, the family of sparse trees is interesting in that, when varying two parameter sequences (namely, the branching size and the distances between branchings), one encounters a rich spectrum of phenomena. We note, in particular, the existence of examples with spectral measures of fractional Hausdorff dimensions (see [5] and Theorem 4.4 below).

This paper is structured as follows. The next section presents the notion of sparse trees and the decomposition theorem that is basic for all that follows. Some simple results concerning spectral measures for the Laplacian on sparse trees are given in Section 3. Section 4 describes the construction of the finite-dimensional trees mentioned above and our results for them. As mentioned above, this paper uses some ideas and tools from the spectral theory of discrete one-dimensional Schrodinger operators. Relevant notions and results are presented in the appendix. We are grateful to Michael Aizenman, Nir Avni, Vojkan Jaksic, Yoram Last, Barry Simon, Simone Warzel and Andrej Zlatos for useful discussions. We also wish to thank Michael Aizenman for the hospitality of Princeton, where some of this work was done.

2. Sparse trees As noted in the introduction, basic to the analysis which follows is a certain decomposition of the Laplacian on a sparse tree. Since this is possible only when the tree has a certain spherical symmetry, we start with:

Definition 2.1 (Spherically homogeneous rooted tree). A rooted tree is called spherically homogeneous (SH) (see [3]) if any vertex v of generation j is connected with icy vertices of generation j + 1. A locally finite spherically homoBy locally finite we geneous tree is uniquely determined by the sequence mean that the valence of any vertex is finite.

In the definition above, a vertex is said to be of generation j if it is at a distance j from the root 0 (where the distance between two vertices is defined as the number of edges of the unique path between them). Thus, for spherically homogeneous rooted trees, the valence of a vertex depends solely upon its location with respect to the root. Let {L,,}' 1 and be two sequences of natural numbers such that kn > 2 for all n, and {L,,,}°°_1 is strictly increasing. We say that I' is a SH rooted

SPARSE TREES WITH FINITE DIMENSIONS

67

P,

FIGURE 1. An example of a SH rooted tree with rco = 1, r£1 = 2,

K2=rc3=rc4=1, rb5=2,... tree of type {Ln,

if

rc =

kn,

j = Ln for some n,

1

otherwise

We say that F is sparse if (Ln+1 - Ln) -> oo as n - oo. Since sparse trees are not regular (the coordination number is not constant), there are two natural choices for the Laplacian:

(Of) (x) _

f (y), y:d(x,y)=1

and

f (y) - #{y : d(x,y) = 1} f (x)

(of) (x) =

where #A for a finite set A is the number of elements in A (d(x, y) denotes the distance on the tree). For simplicity, we shall restrict our attention to O, though we note that all our results hold (when properly modified) for O as well. It is clear that if {kn}°°_1 is a bounded sequence then both 0 and O, on the tree, are bounded and self-adjoint. For unbounded coordination number, the issue of self-adjointness has to be addressed.

Definition 2.2. We call a SH tree of type {Ln,kn}°°_1-F normal if {kn} (Ln+I - Ln) > 1.

unbounded implies that lim supra-

The appendix has a proof that the Laplacians on normal SH rooted trees are self-adjoint. Clearly, any sparse tree is normal. The main technical tool in the spectral analysis of sparse trees is the following theorem:

Theorem 2.3 ([5, Theorem 2.4]). Let r be a normal rooted SH tree of type {Ln, kn}°n°-_I. Let

(2.4)

llj Iki -llj-1 c. n>1 Mn= k1-1 n=1 1

n=o.

J. BREUER

68

Furthermore, let R0 = 0 and Rn = Ln+1, for n > 1. Then A is unitarily equivalent to a direct sum of Jacobi matrices, each operating on a copy of P(7G+): 00

®(JnIDAID ..®Jn) n=O

where Jn = J({an(j)}° an(d)

(2.6)

1,

Mn times

{bn(j)}°°_1) with

_

j=Rm - Rn forsomem>n

-

otherwise

1

and

bn(j) - 0.

(2.7)

Remarks.

(1) The term Jacobi matrix stan ds for t he sem i-infinite matrix

J({a(j)}'1i

(2.8)

1) =

b(1)

a(1)

0

0

a(1)

b(2)

a(2)

0

0

a(2)

b(3)

a(3)

with b(j) E IIR, a(j) > 0. (2) For the case of a regular tree, a similar decomposition was discussed in

[2, 8,16] (see also [20] for a related result in the case of a metric tree). (3) As noted above, this theorem holds for A as well, with the decomposition: 00

_

(____ ... ®Jn)

(2.9) n=0

Mn times

where Jn = J({an(j)}°n°__1, {bn(j)}°n°__--1) with

an (j) = an(j)

(2.10)

and (2.11)

bn(j)

km,-1 j=Rn,,-Rnforsomem>n otherwise. -2

(4) Note that each Jn is a "tail" of Jn_1 in the sense that one can get Jn by deleting a finite number of rows from the top, and the same number of columns from the left, of Jn_1.

3. Singular measures on sparse trees In this and the next section we freely use terms (such as "transfer matrices") associated with the spectral theory of Jacobi matrices. The reader is referred to the appendix for their definitions, the notation we use, some basic results, and further references.

We start with a remark about the essential spectrum. Let I' be a sparse tree of type {Ln, kn}° 1. If kn --f oo, perturbation theory arguments show that the essential spectrum of A on I' is [-2,2]. If {kn} is bounded, then from remark 4 after Theorem 2.3, it is easy to see that the essential spectrum of A is contained in


69

the essential spectrum of Jo. Since the reverse inclusion is immediate, we have the 1o11owing

Proposition 3.1. Let F be a sparse tree of type {Ln, 1 and let Jo = Jo(F) be the corresponding Jacobi matrix appearing in Theorem 2.3. Let oeSs(0) be the rysential spectrum of 0 on F and 0ess(Jo) be the essential spectrum of J0. Then, ff either kn -> oo or {kn} is bounded, then (,l.1)

Qess(A) _ 0,ess(JO)

Now, let H be a self-adjoint operator on a separable Hilbert space 7-l, and VI E 7-l. The spectral measure associated with ') and H, µ.), is the unique measure on R satisfying

(0, (H - z)-14') _ J

dµ,p (x)

x-z

zEC\R

(see, e.g., [151).

Theorem 2.3 reduces the spectral analysis of 0 on a sparse tree to the spectral analysis of Jacobi matrices with sparse "bumps" off the diagonal. (For 0, we get "bumps" on the diagonal as well). An application of (suitably adapted) methods from the spectral theory of one-dimensional Schrodinger operators with sparse potentials (see [11] for a review of the relevant theory) to the situation at hand, allows us to establish interesting spectral behavior for the Laplacian on certain sparse trees. The following basic lemma was proved in the appendix of [1] for the case of the Bethe lattice. It holds for any SH rooted tree.

Lemma 3.2. Let F be a normal rooted SH tree with root 0. For any vertex v in r, let S, E 22(F) be the delta function at v, and y, the spectral measure associated with 0 and 5,,. Let (a, b) be an interval on which the absolutely continuous part of µo vanishes. Then, for any vertex v of IF, the absolutely continuous part of µ,, vanishes on (a, b).

Remark. Throughout this paper, "absolutely continuous" means absolutely continuous with respect to Lebesgue measure. PROOF. The proof is a simple consequence of the identification of the essential support of the absolutely continuous spectrum with the set of energies for which the Green's function has positive imaginary part, combined with the recursion relation (see [1]) for the diagonal elements of the forward resolvents (these are the resolvents of 0 restricted to the various forward subtrees of F). For a normal rooted SH tree F, let {Jn (F)}°°_0 be the Jacobi matrices appearing

in the decomposition of 0 on r, given by Theorem 2.3. Let µn be the spectral measure associated with Si E £2(Z+) and Jn. Then Lemma 3.2 above, says that, if we want to prove that all spectral measures associated with the Laplacian on F are singular, it suffices to prove that yo is singular. This will be useful later on. The following lemma is another simple tool for proving singularity of all the spectral measures. Its proof features the "bump" transfer matrix which will prove itself useful throughout the rest of this paper. (See equations (B.10)-(B.13) for the definitions of the transfer matrices that we use below).

J. BREUER

70

Lemma 3.3. Let Rm be a strictly increasing sequence of natural numbers such that, for m large enough, R,,,+1 - Rm > 2. Let J = J({a(j)}j_1,{b(j)}j_1)

be a Jacobi matrix such that

a(j) = rpm

(3.2)

l1

j = R,,, for some m otherwise

and b(j) - 0,

(3.3)

where p,,, > S > 0 for all m. Then, if {p,,,},°,°-1 is unbounded, then the spectral measure, p, associated with S1 E 22(Z+) and J, is singular with respect to Lebesgue measure.

PROOF. Assume limt-,,. pm, = 00, Rm, - Rm,-1 >- 2) R,,,,+1 - Rm, > 2 and fix E E R. Then (3.4)

S' , = TR,,,+1,R.,,,.,-1(E) =

(E)

= ((E2/ P m, - Pm,) E/pm,

-E/Pm, -1/pm,

It follows that (3.5)

max(1,pm, -E2) < JITR_,+l,R-,-1(E) and so, applying Proposition B.3 with mj = R,,,, + 1 and lj = Rm, - 1, (note that aRm, _1 = 1), we see that µ is singular on R.

Corollary 3.4. Let F be a sparse tree of type {L,,, k,,,}°O 1, with 1 unbounded. Then all the spectral measures for A on I' are singular with respect to Lebesgue measure.

PROOF. The statement follows from Theorem 2.3, Lemma 3.3 and the fact that for a general Jacobi matrix, the vector Sl is a cyclic vector.

On the other hand, a simple consequence of Proposition B.4 is the following

Lemma 3.5. Let R,,, be a strictly increasing sequence of natural numbers. Let

J = J({a(j)}j_1, {b(j)}'1) be a Jacobi matrix such that (3.6)

a(j) _

Pm

j = R,,,, for some m otherwise

and b(j) - 0,

(3.7)

where pm > 1 for all m. Let {/3,,,.}°n=1 be a sequence such that /3m > p,,,, and limm,,,.13m = oo and let Am = 1fl 1 01. If for some e > 0, (3.8)

- p Burn) hm sup (Rm+l >0 A(1+E) m--oo

then the spectral measure, µ, associated with Si E 22(Z+) and J, is continuous on

(-2,2).


71

PROOF. Consider an arbitrary closed interval I C_ (-2,2). We will show that 14(I n1 ) is continuous. From this it will follow that µ((-2, 2) n ) is continuous. For we have that I?,,, + 1 < j < SA(E) =

(E -1) 1

0

too that det (Sj (E)) = 1. Furthermore, define, as in the proof of Lemma 3.3 (3.9)

Sm (E) = TRm+1,Rm-1(E) = SR,,,.+1(E)'SRm (F')

_ ((E2/P. - Pm)

-E/p.

-1/Pm) Then we also have det(S,,,,(E)) = 1. Thus, if j2: Rn and j1 Rn for any in, n, E/pm

we have that

det(Tjl,j2(E)) = 1.

(3.10)

Note that if R,,,,+1 < j2 < j1 < R,,,,+1, then T11,j2(E) is just the transfer matrix for the free Laplacian, so that there is a constant CI, depending only on the interval

I, such that 1 < IlTjl,j2(E)I I < CI for any such 31, j2 and E E I. In addition, we have from (3.9) (3.11)

IISR-(E)II

Pm + 5.

Thus, for R,,, < j < R,,,,+1 we have

IITj(E')I1

(3.13)

A(1+E)

mZ

for some S > 0. Let M be chosen so that for all m > M, n > C. Now, for sufficiently large ml > M, we have, from (3.12), IITj(E)11-2 >

(3.14)

-A;C-2m1 > 2-C,-2M

j=R,,,,+1

Thus, the tail of the sum in (B.15) does not converge to zero. Therefore the sum is divergent and µ has no eigenvalues in I. This proves the lemma. Corollary 3.6. Let P be a rooted SH tree of type {Ln, kn}°O 1. Assume that, for some E > 0,

- Ln) > 0, lim sup (Ln+l A(

(3.15)

n-oo

r11_1

where A,, = for some sequence ,6n > kn, such that on - oo. Then any spectral measure for A on IF, is continuous on (-2, 2). PROOF. The statement follows from Theorem 2.3, Lemma 3.5 and the fact that for a general Jacobi matrix, the vector Sl is a cyclic vector.

J. BREUER

72

A simple consequence of Corollaries 3.4, 3.6 and Proposition 3.1 is the following theorem from [5]:

Theorem 3.7 ([5, Theorem 4.1]). Let {k,,}°°_1 be a sequence of natural numbers such that kn - oo as n -* oo. Let An = f 1 kj. Assume that (Ln+1- Ln) --> oo and let F be a SH rooted tree of type {Ln, kn}°°_1. Then the spectrum of A on F consists of the interval [-2, 2] along with some discrete point spectrum outside this interval. If for some e > 0, (3.16)

lim sup (Ln+1 - Ln) > 0, A(1+6) n_oo

then any spectral measure for A is purely singular continuous on (-2, 2).

Since the next section discusses trees with bounded kn, we quote the corresponding result from [5]. We sketch its proof here since some of the ideas will appear in the sequel:

Theorem 3.8 ([5, Theorem 2.2]). Let ko > 2 be a natural number and let kn - ko. Assume that (Ln+1 - Ln) --f oo and let F be a SH rooted tree, of type {L,,, kn}n'° 1. Then the essential spectrum of A on F contains the interval [-2, 2] and, provided (Ln+l - Ln) increase sufficiently rapidly, any spectral measure for A is purely singular continuous on (-2,2). By "sufficiently rapidly" we mean that (Ln+l - Ln) has to be made sufficiently large with respect to {(Li+1 - Li)}j 2 and take Ln = [yn] for some ry > 2. Denote the SH rooted tree of type {Ln, kn}°O 1 by I'k,.y. A simple calculation gives:

Proposition 4.1. Fix !y > 2 and NE) k > 2. Let F = I,k,,y and let Sr(r) _ {vi E V(F) I d(vi, 0) < r}, where V(F) is the set of vertices of r. Then log #Sr (r) _ log yk log #Sr (r) lim sup T--oo

log r

= lim inf

log r

log y

J. BREUER

74

Below, we shall refer to the quantity tlg ry as the dimension of Fk,.y. In the context of the analogy between sparse trees and Schrodinger operators with sparse potentials, described in the previous section, Fk,,y is analogous to a Schrodinger operator with bumps of fixed height placed at the sites [yn] of Z+. Zlatos deals with such operators in [21] and the analysis we present below is an adaptation of his methods (in particular [21, Section 6]) to the case at hand. While a large part of the argument translates word for word, there are a few significant changes, mainly having to do with the fact that the transfer matrices for our case are not, in general, unimodular. This is important for some of the arguments and, therefore, has to be bypassed to get the same results here. We discuss the changes below and give a sketch of the proof. However, we refer the reader to [21] for a more detailed discussion. First,

Definition 4.2. Let r be a rooted tree. For any self-adjoint operator H on 22(F), and -7r/2 < P < 7r/2, let

He = H - tan(g)Po where Po is the orthogonal projection onto the subspace spanned by the delta (4.1)

function at 0. We refer to He as H with boundary condition g. We also need:

Definition 4.3. Let µ be a measure on R. We say that µ has exact local dimension in I C_ ll if for any E E I there is an a(E) and for any e > 0 there is

6 > 0 for which µ ((E - 6, E + 6) n ) is both continuous with respect to (a(E) - e)dimensional Hausdorff measure, and singular with respect to (a(E)+e)-dimensional Hausdorff measure. We call a(E) the local dimension of the measure A. Let wn be a random variable uniformly distributed over

[-n,-n+ 1,...,n- 1,n]. Let (1, 1P) be the product probability space for all w,,,, n = 1, 2, .... Fix 1 < k E N and ry > 2 and for each w E 11 let Fk be the SH rooted tree of type {Ln, kn} 1 for Ln = [yn] + wn and kn =_ k. Clearly, Proposition 4.1 holds for any Fk F. The main result of this section is

Theorem 4.4. For P-a. e. w, all the spectral measures for 0 on Fk 7 are singular with respect to Lebesgue measure. Furthermore, let V(k) = (1 +k)2/(4k) and let (4.2)

I = (-

8('y -V(k)) /8(y-V(k)) try-1 2y-1

if y > V, and I = 0 otherwise. Then for P-a.e. w and for Lebesgue a.e. o E (-7r/2, 7r/2), the spectral measure, µ6o, associated with A. on F 7, and with the delta function at the root, is purely singular continuous in I with exact local dimension

log((4V(k) - E2/2)/(4 - E2)) (43)

1 -

log(y)

and it is dense pure point in the rest of [-2, 2].


75

Corollary 4.5. Assume ry > 4. Then if the dimension of Fk is at least 3, Art, have that I = 0 and so, for P-a.e. w and for Lebesgue a. e. o E (-7r/2,7r/2), the 4pretral measure pbo is dense pure point in [-2,2]. PROOF OF THE COROLLARY. This is a simple computation.

PROOF OF THEOREM 4.4. Theorem 2.3 and Lemma 3.2 imply that the theorem is an immediate consequence of Proposition 4.6 below.

Proposition 4.6. Fix k > 2 and 'y > 2. For any w E 1 and -ir/2 < o < it/2, h

i JQ = J({a"(j)}, {be(j)}) be a Jacobi matrix with

a"(j)-{vk

(44)

1

j=[y'']+w,,,,+1 for some m otherwise

and tan(g)

be(j) _

(4.5)

j=1 otherwise.

10-

Then, for IID-a. e. w, and for any o E (-ir/2, it/2), the spectral measure µe, associated

with the vector 61 E Z+ and with the Jacobi matrix J.', is singular with respect to Lebesgue measure. Furthermore, forIP-a.e. w and for Lebesgue a.e. o E (-ir/2, it/2), the spectral measure A. is purely singular continuous in I with exact local dimension given by (4.3), where I is as defined in Theorem 4.4, and it is dense pure point in the rest of [-2, 2].

A central role in the proof of the proposition will be played by the EFGP transform introduced in the proof of Theorem 3.8:

Fix k > 2 and let J = J({a(j)}, {b(j)}) be a Jacobi matrix satisfying

a(j) - lV 1

(4.6)

j=L,,,,+1 forsomen otherwise

for a sequence {L,,,,} satisfying L,,,+1 - L,,,, > 2, and

b(j) - 0.

(4.7)

For any E E (-2, 2), let 0 E (0, ir) be defined by 2 cos(o) = E, and let u2 cos(o) solve (B.4) for J, namely (4.8)

a(j)u(j + 1) + a(j - 1)u(j - 1) = Eu(j),

UE

j>1

with a(0) = 1. Recall that the EFGP variables [10] corresponding to u, r,(j) and 00(j), are defined through: rO(j) cos(O (j)) = uE(j) - cos(q)uE(j - 1) (4.9) r,(j) sin(O (j)) = sin(cb)UE(j - 1). (4.10) First, note that there are positive constants, C1(0), C2(0), such that (4.11)

C1(0)(ju(j - 1)12 + Iu(j)12) C ro(j)2 < C2(0)(Iu(j - 1)12 + Iu(j)12) We call r(j) the EFGP norm of u(j). Note, also, that for any j (j+2)

(j+2) (4.12)

Iu(i)12) < ro(j)2 + r1(j + 2)2 < C2(o)(

C1(0)( i=(.9-1)

Iu(i)12) i=(.9-1)

J. BREUER

76

For a function f : Z+ -i C and a sequence L = {L,,,}°°_1 of natural numbers, define

/

1/2

IIJIIL,L = I

\ j54 L,+2for 1<j 2, and

b(j) - 0. Assume that for some E E (-2,2), u is a solution of (4.8) whose EFGP norm (4.27)

satisfies (4.28)

r(Ln + 3) = e

where Qn = Ej' 1(Zj + Xj) with 0 < dl < Zj < d2 < oo and E; 1 Xj = o(n). Then there exists a subordinate solution v of (4.8) for E, such that for any d < dl and for all sufficiently large n, the corresponding EFGP norm -p - satisfies (4.29)

p(Ln + 3) < e-dn

PROOF. Let v be any solution of (4.8) different from u and let p be its EFGP Since the transfer matrices TLn+2(E) are unimodular, the argument of [10, Theorem 2.3] applies to show that there exist E-dependent constants, Cl, c2 such that norm.

(4.30)

Cl max(pL,.+3, rL,.+3) < II TLn+2(E) 11 < c2 max(pL,+3, rL, +3)


79

Note, further, that there exists a constant B > 0 such that (4.31)

II< B

IITn+1,..(E)II =11

and that det(T,,,+1,,(E)) = 1 as well. Thus, it follows that

0

(4.32)

ITn,n-1(E)112 < o0

n=1

rug one can apply [12, Theorem 8.1] to get a vector v E ][82 such that

for any other vector v E R2. From this point, the proof follows the proof of 121, Lemma 2.1], word for word, to show that v generates the claimed solution.

Appendix A. Self-adjointness of the Laplacian on normal SH rooted trees Proposition A.1. Let IF be a rooted SH tree of type {Ln, k,}°° 1. Then the operator Al defined over

D(A1) = {u E £2(r) u is of compact support}

(A.1)

via the equation (A.2)

(Alu)(x) _

u(y), y:d(x,y)=1

is symmetric. If lim supra-. (Ln+1 - Ln) > 1 then A -the closure of Al - is selfadjoint. The same statement holds for Al and 0 (defined over the same domain), with equation (A.2) replaced by (A.3)

(A1u)(x) = E u(y) - #{y : d(x, y) = 1} u(x). y:d(x,y)=1

PROOF. Since the proof for A and A is precisely the same, we use A.

It is trivial to see that Al is symmetric, so in order to show that A is selfadjoint, all we have to show is that ker(A* + i) = {0}.

Assume that Au = iu, then it follows that AU = -iu (where -for a complex number denotes complex conjugation). Let nj be a subsequence for which Lni +1 < Ln;+i For a vertex v with Ivl - d(v, 0) = Lnj + 1, let us denote its unique forward neighbor by v. One can verify that an analogue of Green's formula (see, e.g., [4, Chapter VII, formula 1.4]) holds and we have:

2iIvI!5L.i+1 E 1 u(v)12 = E u(v) .

U(V)

-

u(v) . u(v)

+1

so

Iu(v) I . lu(v) I.

Iu(v) I2 1

(B.3)

a(1)u(2) + b(1)u(1) = Eu(1).

Since, for a given E E R, all solutions to (B.2) - (B.3) are linearly dependent (determined by u(1)), it suffices to study U1,E which is the solution satisfying u1,E(1) = 1.

It is convenient to define a(0) = 1 and to extend (B.2) to j = 1 by demanding U1,E (0) = 0. Thus u1,E is the unique solution to (B.4)

a(j)u(j + 1) + a(j - 1)u(j - 1) + b(j)u(j) = Eu(j), j > 1

with (B.5)

U1,E(0) = 0,

u1,E(1) = 1.

We further define u2,E as the unique solution to (B.4) satisfying (B.6)

U2,E(0) = 1,

U2,E(1) = 0.

Note that any solution, u, to (B.4) with u(1)

0 can be viewed as U1,E for a slightly modified Jacobi matrix. Namely, u solves (B.2) - (B.3) for the same set of parameters except with b(1) changed to (b(1) + u(0)/u(1)). This remark is basic for the analysis of Section 4. We say that u1,E is subordinate if lim IIu1,EIIL = o

(B.7)

L-oo IIu2,EIIL

where

IIfllL =

[

(:If(j)I2+(L-

[L])If([L]+1)12) 1/2

j=1

The Gilbert - Pearson theory of subordinacy [7] says that the singular part of p is supported on the set of energies where u1,E is subordinate, and that the absolutely continuous part of p is supported off this set. The Jitomirskaya-Last extension of this theory [9] analyzes further the singular part of p according to its singularity/continuity with respect to dimensional Hausdorff measures (see [9] for the concept of Hausdorff measures and dimensions). In Section 4 of the paper we use the following results from [9]:

Proposition B.1 ([9]). Assume that 1 < a(j) < M for some M > 1. Assume that for some 1 < ,3 < 2 and every E in some Borel set A, every solution u of (B.4) obeys 2

(B.8)

lim sup IIuIIL < 00

L-oo V

Then µ(A n ) is continuous with respect to (2 -13)-dimensional Hausdorff measure.


81

Proposition B.2 ([9]). Assume that 1 < a(j) < M for some M > 1. If liminf IIu1,EIIi

(If.9)

L-oo

L

_- 0

for every E in some Borel set A, then µ(A fl ) is singular with respect to adimensional Hausdorff measure.

The next results we quote relate the properties of p to the properties of the transfer matrices corresponding to J. These are the 2 x 2 matrices 1j(E) = SA(E)S _1(E) ... S1(E),

(B.10)

where (B.11)

((E - b(j))/a(j) -a(j - 1)/a(g)) 1 0 J

S, (E) =

It isn't hard to see that (8.12)

TA(E) = (u1, (E(j) 1)

u

1)

2(E(j)

so it is not surprising to find that the behavior of Tj (E) is related to the behavior of the eigenfunctions. For any jl, j2i we use the shorthand Tj1j2 (E) = Tj1(E)Tj2 (E)-1.

(B.13)

The following is a generalization of [12, Theorem 1.2] relating the behavior of Tj(E) with the existence of absolutely continuous spectrum.

Proposition B.3. Let mj, lj be arbitrary sequences of natural numbers and let r

AI = { E liminf 1 IITmj,i;(E)II < oo j-'°° ai;

(B.14)

111

Then AI supports the a.c. part of a in that µac (R \ Al) = 0. PROOF. Note that det(Ti3 (E)) = 1/al, so that IITi;(E)-1II = ai3IITi;(E)II

Thus, we have that

1 IITm; (E)Tij (E)-1II lu(1)12 + lu(O)12 JITm(E)MM2

References 1.

M. Aizenman, R. Sims, and S. Warzel, Stability of the absolutely continuous spectrum of random Schrodinger operators on tree graphs, Probab. Theory Related Fields 136 (2006), no. 3, 363 - 394.

2.

C. Allard and R. Froese, A Mourre estimate for a Schrodinger operator on a binary tree, Rev. Math. Phys. 12 (2000), no. 12, 1655-1667.

3.

H. Bass, M. V. Otero-Espinar, D. Rockmore, and C. Tresser, Cyclic renormalization and automorphism groups of rooted trees, Lecture Notes in Math., vol. 1621, Springer, Berlin, 1996.

4.

Ju. M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Transl. Math. Monogr., vol. 17, Amer. Math. Soc., Providence, RI, 1968.

5.

6.

J. Breuer, Singular continuous spectrum for the Laplacian on certain sparse trees, Comm. Math. Phys. 269 (2007), no. 3, 851-857. R. Carmona and J. Lacroix, Spectral theory of random Schrodinger operators, Probab. Appl., Birkhauser, Boston, MA, 1990.

7.

D. J. Gilbert and D. B. Pearson, On subordinacy and analysis of the spectrum of one-

dimensional Schrodinger operators, J. Math. Anal. Appl. 128 (1987), no. 1, 30-56. V. Georgescu and S. Golenia, Isometries, Fock spaces and spectral analysis of Schrodinger operators on trees, J. Funct. Anal. 227 (2005), no. 2, 389-429. 9. S. Ya. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. I: Half-line operators, Acta Math. 183 (1999), no. 2, 171-189. 10. A. Kiselev, Y. Last, and B. Simon, Modified Prefer and EFGP transforms and the spectral analysis of one-dimensional Schrodinger operators, Comm. Math. Phys. 194 (1998), no. 1, 8.

1-45. 11. Y. Last, Spectral theory of Sturm-Liouville operators on infinite intervals: A review of recent developments, Sturm-Liouville Theory: Past and Present (W. O. Amrein, A. M. Hinz, and D. B. Pearson, eds.), Birkhauser, Basel, 2005, pp. 99-120. 12. Y. Last and B. Simon, Eggenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrodinger operators, Invent. Math. 135 (1999), no. 2, 329-367. 13. L. Malozemov and A. Teplyaev, Pure point spectrum of the Laplacians on fractal graphs, J. Funct. Anal. 129 (1995), no. 2, 390-405. 14. D. B. Pearson, Singular continuous measures in scattering theory, Comm. Math. Phys. 60 (1978), no. 1, 13-36. 15. M. Reed and B. Simon, Methods of modern mathematical physics. I: Functional analysis, 2nd ed., Academic Press, New York, 1980. 16. R. V. Romanov and G. E. Rudin, Scattering on the Bruhat - Tits tree. I, Phys. Lett. A 198 (1995), no. 2, 113-118.


83

It. Simon, Spectral analysis of rank one perturbations and applications, Mathematical Quantum Theory. II: Schrodinger Operators (J. Feldman, R. Froese, and L. Rosen, eds.), CRM 1'roc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109-149. , Operators with singular continuous spectrum. VI: Graph Laplacians and Laplace 11citrami operators, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1177-1182. It, Simon and G. Stolz, Operators with singular continuous spectrum. V: Sparse potentials, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2073-2080. 1) M. Solomyak, On the spectrum of the Laplacian on regular metric trees, Waves Random Media 14 (2004), no. 1, S155-S171. 21. A. Zlatos, Sparse potentials with fractional Hausdorff dimensions, J. Funct. Anal. 207 (2004), no. 1, 216 - 252. INSTITUTE OF MATHEMATICS, THE HEBREW UNIVERSITY OF JERUSALEM, JERUSALEM, 91904, 101tAEL


Centre de Recherches Mathematiques t'!iM Proceedings and Lecture Notes V.,lume 42, 2007

Some New Estimates on the Spectral Shift Function Associated with Random Schrodinger Operators Jean-Michel Combes, Peter D. Hislop, and Frederic Klopp ABSTRACT. We prove some new pointwise-in-energy bounds on the expectations of various spectral shift functions (SSF) associated with random Schrodinger operators in the continuum having Anderson-type random potentials in both finite-volume and infinite-volume. These estimates are a consequence of our new Wegner estimate for finite-volume random Schrodinger operators [5]. For lattice models, we also obtain a representation of the infinite-volume density of states in terms of the expectation of a SSF for a single-site perturbation. For continuum models, the corresponding measure, whose density is given by

this SSF, is absolutely continuous with respect to the density of states and agrees with it in certain cases. As an application of one-parameter spectral averaging, we give a short proof of the classical pointwise upper bound on the SSF for finite-rank perturbations.

1. Introduction: The Wegner estimate and the spectral shift function Some recent analyses of random Schrodinger operators have involved three related concepts: the Wegner estimate for the finite-volume Hamiltonians, the spectral shift function (SSF), and the integrated density of states (IDS). In this note, we prove some new pointwise bounds on the expectation of some SSFs that occur in the theory of random Schrodinger operators in the continuum. These bounds result from an improved version of the Wegner estimate [5]. In earlier work [4, 8],

we used LP-bounds on the SSF in order to obtain better estimates on the IDS. In our most recent work, we obtain an optimal Wegner estimate directly without using the SSF and found, as a consequence, new pointwise bounds on the expectation of the SSF. It has often been conjectured that in the case of ergodic, random, Schrodinger operators of the form considered here the SSF for a local single-site perturbation should be in L' (1l) once it is averaged over the random variables on which the disordered potential depends. We prove this in this note. We mention that these types of bounds on the SSF also play a motivating role in the fractional moment method for proving localization in the continuum [1]. For lattice models, 2000 Mathematics Subject Classification. Primary:

35P25, 81Q10; Secondary:

47A40,

81U20.

The work of the second author is supported in part by NSF grant DMS-0503784. This is the final form of the paper. ©2007 American Mathematical Society 85

J.-M. COMBES ET AL.

86

the pointwise bounds on the SSF are a simple consequence of the fact that the corresponding perturbations are finite-rank (cf. [2,22] and Section 4.3). We first recall a special case of the Wegner estimate proved in [5] that will be used for the bounds on the SSF. We refer to [5] for the general statement, valid for arbitrary bounded processes on Zd, and the proofs. The family of Schrodinger

operators H, = Ho + V,, on L2 (Rd), is constructed from a deterministic, ZGdperiodic, background operator Ho = (-i0 - A0)2 + V0. We consider an Andersontype random potential V, constructed from the single-site potential u as (1.1)

VV, (x) _ E wju(x - j). jEZd

The family of random variables is assumed to be independent, and identically distributed (iid). The results are independent of the disorder provided it is nonzero. We define local versions of the Hamiltonians and potentials associated with bounded regions in Rd. By AL(x), we mean the open cube of side length L centered

at x E Rd. For A C Rd, we denote the lattice points in A by A = A n Zd. For a cube A, we take Ho and HA (omitting the index w) to be the restrictions of Ho and Hu respectively, to the cube A, with periodic boundary conditions on the boundary OA of A. We denote by Eo (.) and EA(.) the spectral families for Ho and HA, respectively. Furthermore, for A C Rd, let XA be the characteristic function for A. The local potential VA is defined by (1.2)

VA(x) = VW (x)XA(x),

and we assume this can be written as (1.3)

VA(x) = EWju(x

j).

jEA

For example, if the support of u is contained in a single unit cube, the formula (1.3) holds. We refer to the discussion in [4] when the support of u is compact, but not necessarily contained inside one cube. In this case, VA can be written as in (1.3) plus a boundary term of order IaAj and hence it does not contribute to the large JAI limit. Hence, we may assume (1.3) without any loss of generality. We will also use the local potential obtained from (1.3) by setting all the random variables to one, that is, (1.4)

VA(x)

uj(x),

jEA

where we will write uj (x) = u(x - j). We will always make the following four assumptions:

(Hl) The background operator H0 = (-i0 - A0)2 + VO is a lower semi-bounded, zd-periodic Schrodinger operator with a real-valued, Z' -periodic, potential V0i and a Zd-periodic vector potential A0. We assume that VO and A0 are sufficiently regular so that H0 is essentially self-adjoint on Co (Rd) (H2) The periodic operator H0 has the unique continuation property, that is, for if (Ho - E)O = 0, and if q any E E R and for any function 0 E vanishes on an open set, then 0 - 0. (H3) The nonzero, non negative, compactly supported, bounded single-site potential u E Lo (Rd), and it is strictly positive on a nonempty open set.

NEW ESTIMATES ON THE SPECTRAL SHIFT FUNCTION

87

(114) The random coupling constants {wj j E Zd}, are independent and identically distributed. The probability distribution po of wo is compactly supported with a bounded density ho E Lo (]R). These imply that the infinite-volume random Schrodinger operator Hu, is ergcxlic with respect to the group of Zd-translations. Our results also apply to the randomly perturbed Landau Hamiltonian H, (A) _ !!!,(B) + AV,, for A $ 0, where Vu, is an Anderson-type potential as in (1.1). The I

Landau Hamiltonian HL (B) on L2 (]R2) is given by (1.5)

HL(B) = (-i0 - Ao)2,

with Ao(x, y) =

B

(-y, x)

The constant B 0 is the magnetic field strength. Under these assumptions, the Wegner estimate necessary for our purposes has the following form.

Theorem 1.1. We assume that the family of random Schrodinger operators H4, = Ho + V, on L2(Rd) satisfies hypotheses (H1) -(H4). Then, there exists a locally uniform constant CW > 0 such that for any E0 E R, and E E (0, 1], the local Hamiltonians HA satisfy the following Wegner estimate (1.6)

]F{dist(a(HA), Eo) < e} < lE{Tr EA([Eo - e, Eo + e])}
0, and A uniformly distributed on [0,1]. Birman and Solomyak proved a relation (cf. [19]) between the averaged, weighted, trace of the spectral

J.-M. COMBES ET AL.

88

of H(A), and the SSF for the pair H(1) - H(\ = 1) and H(O) projector H(\ = 0) = Ho. For any measurable A C R, this formula has the form (2.2)

fdA

V1/2EA()V1/2

JdE(E;Ho + V, Ho),

whenever all the terms exist. For example, if V is relatively Ho-trace class, then all the terms are well-defined. We apply this formula as follows. First, we must also make a stronger hypothesis on the probability distribution than (114). We will assume: (H4') The random coupling constants {wj j E Zd}, are independent and identically distributed. The probability distribution po of wo is the uniform I

distribution on [0, 1].

As, for some C > 0, one has 0 < >j uj < C, it follows trivially that

0. We write H for HA with wj = 0. The Birman-Solomyak formula (2.2) then has the form f1

H + uj, H ).

dwj

(2.4)

fA T aking the expectation of (2.3) and using formula (2.4), we obtain

l (2.5) jEA

jEA

111 o

JJ1

< CE{TrEA(A)} C0JAj JAI,

where we used the result of the proof of Theorem 1.1 on the last line. We conclude from (2.5) that ( l f H + uj, H )}1 Co. (2.6) I fo dE { I

0

I

111

AI jEA

JJ

If the spatially averaged expectation of the SSF is L oc(ll) in E, we can conclude a pointwise bound from (2.6), for Lebesgue almost every energy E, of the form (2.7)

lE{ 111

Il

(E; H + uj, H ) } < Co. jEA

JJJ

In [8], we proved that the SSF for local Schrodinger operators with compactlysupported perturbations is locally-L1, so this pointwise bound (2.7) holds. Finally, we observe that due to the periodic boundary conditions on 9A and the Zd-periodicity of Ho, we have that for any j, k E A (2.8) 1E{6(E; H3 L + uj, H3L)} =1E{6(E; Hk + Uk, Hk )}, and consequently it follows from (2.7) that for any j E A, (2.9)

lE{6(E; H3 L + uj, H3,)}

< Co.


89

Theorem 2.1. Under the hypotheses (H1) - (H4'), the expectation of the spectral shift function, corresponding to the variation of a single site of the finite-volume Hamiltonian, is uniformly locally bounded in energy. That is, for any bounded en-

ergy interval, there is a constant CI > 0, independent of A, so that for Lebesgue almost every E E I, we have ]E{e(E; H3 , + uj, H3,)}

(2.10)

< CI,

for any j E A.

In the lattice case, the perturbation uj is rank-one, so by the general theory (cf. [2,22], or see Section 4.3), we have the bound e(E; H3 , + uj, H3, ) < 1,

(2.11)

for any j E A, uniformly in E E R.

3. Bounds on the spectral shift function for infinite-volume Hamiltonians We consider the thermodynamic limit of the SSF in (2.7). The Birkhoff Ergodic

Theorem implies that the limit of the right side of (2.7) is the expectation of the SSF corresponding to the pair of infinite-volume Hamiltonians (Ho-, Ho- + uo) if we replace l;(E; H3 , + uj, H3,) by e(E; Hjl + uj, Hjl), where Hjl is the infinitevolume Hamiltonian with wj = 0. This is the content of the next proposition.

Theorem 3.1. Let Hol be the infinite-volume random Hamiltonian Hu, with wo = 0 and assume hypotheses (H1) - (H4'). Then the SSF l; (E; How + uo, HO-L) is well-defined and ]E{t; (E; Hol + uo, HO-L) } E Loo (R).

(1)r We begin with the integrated expression (2.6) and write

PROOF.

(3.1)

ICI

fodE]E{

ICI Et;(E;H +uj,H ) }

111

jEA

JJ

JAI

f dEE{ Inl,l;(E;Hjl +uj,Hjl)} + £iQO)

where the error term is

(3.2) £A(0) -

fdEE{((E;H+ u, H) - (E; H1 jEA

We will prove below that £A(A) -4 0 as IAI -> oo. Assuming this for the moment, it follows from the fBirkhof Ergodic Theorem and (3.1) that (3.3)

Limo

IQI

JQdEE{ l

Il

(E;Hi±+uj,Hj1)}

l

jEA

JdEE{(E;Ho± + uo, How) } < Cr < oo. O In order to justify the interchange of the expectation and the infinite-volume limit, we note that the nonnegative series in brackets on the first line of (3.3) converges pointwise a.e. to the integrand on the second line of (3.3). As the SSF t;(E; Hjl + =I

I

-

uj, Hjl) E L o(R), the term in the brackets on the right of the first line of (3.3)

J.-M. COMBES ET AL.

90

is uniformly bounded, so the exchange is justified by the Lebesgue Dominated Convergence Theorem. We apply the Lebesgue Differentiation Theorem to the second line of (3.3), since the SSF is in L' (lR), and obtain the pointwise bound in Theorem 3.1. (2) It remains to prove the vanishing of the error term in (3.2) in the infinitevolume limit. Using the identity on the first line of (2.5), we obtain l (3.4)

£A(A) = IES

1

}.

I

l jEA We define a local nonnegative measure iA by (3.5)

'cA(A) =

)))

'{yprUl/lEA(A)Uj/2

II 7EA

and the nonnegative measure kA, defined similarly but with the spectral projection E(.) for the infinite-volume Hamiltonian H,. In terms of these local measures, we can express the right side of (3.4) as (3.6)

£A(A) = Inl [NA(A) - RA (A)]

converges vaguely to zero by computing the We first prove that the measure Laplace transform of the measure. The Laplace transform £(£A)(t) is easily seen to be given by (3.7)

£(£A)(t) = IIIE{Tr

VA(e-tai, - e-tHA)}.

Using the Feynman-Kac formula for the heat semigroups, for example, one easily shows, as in [12], that

lim £(£A)(t) = 0,

(3.8)

IAA--goo

for t > 0 pointwise, for a reasonable expanding family of regions A. This implies the measure £A(.) converges vaguely to zero which, in turn, implies that the right side of (3.6) converges to zero.

A consequence of this result is an apparently new relationship between the infinite-volume SSF and the DOS for lattice models. The analogous relation for continuum models defines a new measure absolutely continuous with respect to Lebesgue measure and to the DOS measure. These results follow easily from the proof of Theorem 3.1.

Corollary 3.1. Let v be the DOS measure for the random Hamiltonian H. For lattice models, for any Borel set A C IR, we have (3.9)

v(A) = f dE E{!; (E; H0, + uo, Hol) }.

For continuum models, there is a nonnegative measurer, absolutely continuous with respect to the DOS measure and Lebesgue measure, with distribution given in (3.13), so that (3.10)

r(A) =

r

JA

dE IE{e(E; Hol + uo, How)}.


91

For any closed bounded interval I C I[8, there are constants 0 < cI < CI < oo, so that for any Borel set A C I, we have 0 < c(A) < cIJAJ

(3.11)

and 0 < n(A) < Civ(A).

PROOF. From the Birkhoff Ergodic Theorem, and expression (2.4), we can express the integral on the right in (3.3) in terms of a positive measure c as follows (3.12)

dEIE{ (E; How +uo,How)} =IEI Iii oo

JAI

I

jEA

where

K(0), is the nonnegative measure with distribution function given by

(3.13)

K(E) - IE{Truo/2P(E)uo/2},

where P(E) is the spectral family for H,. For the lattice case, this measure is just How +uo, H0.L)} is a representation the DOS measure, since uo = bo, so that of the DOS. It follows immediately from (3.12) and Theorem 3.1 that for any closed bounded interval I C I[8, there exists a finite constant 0 < CI < oo, so that for any Lebesgue measurable set A C I, we have (3.14)

0 < c(A) = J

+uo,Hol)} < CI JAI.

A

It remains to prove that rc is bounded above by the DOS measure. This implies the absolute continuity with respect to v. We simply note that there exists a constant 0 < Co < oo, depending only on u, so that (3.15)

0<E

)u'/2 < Co TrEA(A),

jEA

and recall the definition of the DOS measure. This implies that 0 < c(A) < Cjv(A),

forACICR. This measure rc is similar to the DOS measure for continuum models. The distribution function for the DOS for continuum models is given by N(E) = IE{Tr'XA1(o)P(E)XA1(o)}. The measure rc is equivalent to the DOS measure v if the single-site potential satisfies COXA1(0) 0, and it is equal to v in the special case that u = XA1(o). The equivalence of measures means that there are constants CO, co > 0 so that (3.16)

cov(A) < c(A) < Cov(A),

for all Borel subsets A C R.

4. Comments We make three comments on various other results concerning the SSF associated with random Schrodinger operators that have recently occurred in the literature related to random Schrodinger operators. For deterministic Schrodinger operators, pointwise bounds are known only in a few specific cases, such as finite-rank perturbations (cf. [2,22] and Theorem 4.2 below) or perturbations of the Laplacian on L2 (Rd) by sufficiently smooth potentials [20].

J.-M. COMBES ET AL.

92

4.1. Related results on the averaged SSF. Bounds on the LP-norm of the SSF, for 0 < p < 1, were proved in [8] and improved in [11]. More recently, Hundertmark et al. [10], obtained some new integral bounds on the SSF that indicate that one cannot expect that, in general, the SSF is locally bounded. Indeed, Kirsch [13,14] proved that if the Dirichlet Laplacian in AL, a cube of side length L centered

at the origin, is perturbed by a nonnegative, bounded potential supported inside the unit cube Al, then the corresponding finite-volume SSF, at any positive energy, diverges as L -+ oo. Raikov and Warzel [18] considered the SSF for the Landau Hamiltonian (1.5) and a perturbation by a compactly-supported potential. They showed that the SSF diverges at the Landau energies. The averaged SSF is expected to be better behaved. In addition to the pointwise bounds of Theorems 2.1 and 3.1, Aizenman et al. [1] proved an interesting bound on a spectral shift function related to the ones treated here. They consider the SSF

(t, E) for a pair of Hamiltonians Ht = Ho + tV and Ht + U, where V and U are nonnegative bounded potentials such that V is strictly positive on a neighborhood of the support of U. Specifically, for any S > 0, we define the set Qs - {x E Rd I dist(x, supp(U)) < 6}. We then require that V be strictly positive on Q.

Theorem 4.1. For any 0 < s < min(2/d, 1/2), there is a finite positive constant C8,6 > 0f so that the SSF (t, E) satisfies the bound (4.1)

J0

1

E)Jsdt < C3,sIJUIJ.(1 + IE - Eol + jVjj.)23(d+1)

where E > Eo - inf a(Ho).

4.2. Spectral shift density. Kostrykin and Schrader [15,16] introduced the spectral shift density (SSD) that is closely related to the integrated density of states. The SSD is the density of a measure ° obtained by the thermodynamic limit (4.2)

f g(A) d8(A) = lim f 9(A)

(E; Hon XAVw)

JI Note that the size of the perturbation XAVW is of order Al. JThey prove that the SSD (E) is given as (4.3)

(E) = No(E) - N(E),

a.e. E E R,

where No(E) is the IDS of Ho and N(E) is the IDS of Hw.

4.3. A pointwise bound on the SSF for finite-rank perturbations. We consider the SSF for a finite-rank perturbation. Let B > 0 be a nonnegative finiterank operator with rank N. Let H3 = Ho+sB be the one-parameter perturbation of a self-adjoint, lower-semibounded operator Ho. The variable s E [0, 1] is uniformly distributed. We consider the SSF e(E; H1, Ho) and recover the classical pointwise upper bound N usually obtained by other methods cf. [2,22].

Theorem 4.2. The spectral shift function for the pair of self-adjoint operators (H1, Ho), where 0 < B = H1- Ho is a finite-rank operator of rank N < oo, satisfies the bound (4.4)

0 < e(E; H1i Ho) < N.


93

PROOF. Let f E Co (R) and consider the formula for the SSF:

Tr(f(Hi) - f (Ho)) = - JR

H1, Ho) dE

1

=

Jo ds Tr f (HS)

ds

r1

= J ds TrB1/2 f'(HS)B1/2 0

N

1

E f ds (01, B1/2f'(Hs)B1/20j) 7=1

0

Let ES(.) be the spectral family for H3. The matrix element in (4.5) is written as (4.6)

(0j, B1/2f'(Hs)B1/2 ) = f f'(A) d(B1/2

,

E.(A)B1/2

)

= f f'(A) dpoi (A), where Oj - B1/2Oj and µH9 is the corresponding spectral measure for H3 and 'j. We divide the support of f' into p subintervals Ok and bound the absolute value of the integral over A in (4.6) from above as P

f f'(A) dp / (A)

I f'(xk) I N' / (Ok), k=1

where xk E Ok is such that I f (xk)I = supXEOk I f'(x)I. Inserting this into the integral over s in (4.5), we see that it remains to estimate (4.8)

f 1 µH9 (Ok) ds =

J(Bh/2, Es(k)B2) ds.

o

For bounded probability distributions with compact support, the integral on the right in (4.8) was estimated in [3]. The result is 1

f(Bh12j,Es(k)Bh/2j)ds

(4.9)

< 1. Combining this bound with (4.5) - (4.6) and recalling the approximation (4.7), we obtain since

(4.10)

f

N

p

, (x, y) is not analytic in A if and only if

AE E°U{AE (0,oo):1-/3I(A)=0}. In the case AE (0,oo)with 1-/3I(A)=0, the resolvent kernel has a pole at the point A A0 (/3) > 0, where I (Ao (/3)) = 1//3. According to (3.3) this positive pole exists if d = 1, 2 for all /3 > 0 but by (3.4), for d > 3, it exists for /3 > /3cr(d) = 1/I(0+). By the Spectral Theorem, if Ao(/3) and lip are the principal eigenvalue and its normalized eigenfunction, respectively, for HQ, then (3.7) R.x (x, y) _ Q(x)00(y) ( + Jf o

d(E..61, bx)

-- 00(x)'bo(y) + J A -A0(/J)

fo

u.,y(dz)

-z A-z 4d where µ,,,.y(dz) = d(EZby, 6x) is the spectral measure. In particular, for x = y, the spectral measure py,y(dz) is positive and finite. Comparison of (3.6) and (3.7) gives A-A0

)

4d

0,3(x) = cI (Ao (,3), x),

with a normalizing constant c such that 1100112 = 1. (One can also prove that limn-o Im RA+is (x, y) = Im RA(x, y) is a bounded continuous function of A on [-4d, 0] which means that the spectral measure is a.c. on [-4d, 0]. This fact won't

0

be used however.)

From now on we shall write /3or instead of /3Cr(d), suppressing its dependence on d. We now turn to the proof of Theorem 3.1. PROOF OF THEOREM 3.1. Defining

l

1

b(t, x) = Ex [exP{/3 fbo(xy) dsl6o(xt))] o

we see that Zp,t =

V) (t, x) XEZd

Notice that the Feynman-Kac formula gives (3.8)

a0,0

(t, x) + /38o(x)0(t, x), t (t' x) = 0

't/'(0, x) = b0(x)

Consequently (t, x) = p,3 (t, 0, x) is the fundamental solution of the heat equation for the operator HQ given at (3.1). If 1 denotes the vector with all components equal to 1, it then follows that (3.9)

Zp,t = (p' (t, 0, ),1) _

PO (t, 0, x) xEZd

From (3.9) we see that the asymptotic behavior of ZQ,t as t -4 oo is closely related to

the spectral properties of the operator W. Notice HQ is a bounded operator with 11H'3 11 < 2d + /3. The spectrum of H° is E0 = [-4d, 0]. The Fourier representation

M. CRANSTON AND S. MOLCHANOV

102

of H° is given by multiplication by its symbol. Namely, select an orthonormal basis

e1,e2, .. , ed of R and setting qj = (0, ej), d

A f (0) = 2 E(cos Oj - 1) f (0).

(3.10)

j=1

Let's return now to (3.8). Taking the Laplace transform in time of both sides, and writing MA(x) = fo p,3 (t, 0, x)e-at dt, we obtain O.,MA(x) + (3bo(x)MA(x) = )Ma(x) - So(x). Solving gives,

MA(x) = R,\ (x, 0).

(3.11) Since

ZP,t = E PO (t1 0, x) XEZd

we have by Plancherel, (3.12)

f

r

Zp to-at dt = (MA, 1) = (2-7r)d f 1'Ia(0)8 (0) dO d

(2ir)d

= (27r)dNla(0) = (27r)dC1-i3I(\)1

+ Al

A(1 -,3I(A))

Inversion of the Laplace transform now gives the formula

j( e(o+is)t ds

1

Zp,t

which is valid for any o > )o(/3). By deformation of contour, (3.12) yields the asymptotic formula

=

eao(R),(2lr)d

Z13, t

+ D(ebt),

,3Ao (a) II (Ao (a)) I

for any b > 0. Consequently, if A0 ((3) > 0, then lim 1 t-oo t In Zo,t = Ao (0)

If A0(3) = 0, then 1

lim In Zo,t < 6, t-.co t for any b > 0. On the other hand, since Zo,t > pp (t, 0, 0) > po (t, 0, 0) - ct- 42, it follows that lim 1 In ZQ,t > 0.

t+oo t Thus, in the case A0(3) = 0, we have

lim 1 In Zp,t = 0.

t-- co t

Finally, regardless of the value of )o(/3), we have proved 1

lim t-*00 t In Zo,t = Ao (3)

HOMOPOLYMERS

103

The analyticity of F(0) for 0 > /3cr follows from the analyticity of I(A) and the relation /3I(Ao(/3)) = 1.

4. Markov description of polymer in globular phase We now give a description, in the globular phase 0 > /3cr, not just of the distribution of the endpoint x(t), but of the initial part of the path xT, 0 < r /3cr, define the Doob transform kernel, (4.1)

rp(t, x, y) =

PO (t, X, Y)bP(Y) e-A0(0)t.

*3 (x)

Then ro(t, x, y) is the transition density for an ergodic, pure jump, Markov process on Zd with invariant probability distribution

ir(x) =0p(x), and generator

A0f (x) = E ap(x, y) (f (y) - f (x)) 1Y-s1=1

where

if Ix - yJ > 1,

0,

00(y)

a/ (x, y) _

if I x - yI = 1

*3 (X)

if y = x. The endpoint of the polymer, xt, with respect to Pp,t satisfies Ao(/3) + 2d - /360(x),

(4.2)

,3 (x)

lim Pp,t(xt = X) _

t

00

(0a, U`)

For any finite T, the distribution of the process xs, s E [0, T], with respect to the measure P0,t converges weakly to the Markov process with transition density rp as t -+ oo. The asymptotic invariant distribution of any initial segment of the polymer in the globular phase is thus given by ir(x) _ '/ (x), x E Zd, in the sense that, (4.3)

lira lira P/,t(xs = x) = 02 (x). 8-00 t-00

Also,

(4.4)

Zp,t = eA0(a)t 0/(0) E'p(x) + O(ebt),

for all 0 < 6 < A0(/3).

XEZd

PROOF. We've already established (4.4). Using (4.5)

p/3(t, x, y) = Oa(x)03(y)e''0(3)t + O(ebt),

for all 0 < 6 < Ao(/3)


104

we get (4.2) since (4.6)

Pp,t(xt = x) =

-

PO (t, 0, x)

Zp,t

00(13)t 1(0) Q(x) + O(eat) + O(elt)

e1,o(13)tV)p (0) >xEZ' V)1(x)

since ExEZd 013(x) _ (00, 1). Write the semigroup generated by pp as

Qtf (x) = E P0 (t, x, y) f (y) = et"' f (x) yEzd

where the reader will recall that H13 = A+(3So and Qt acts on the space of bounded functions. Since 00 is the eigenfunction corresponding to Ao(Q),

Qt'p (x) =

etao (a)pp (x)

Now the kernel pp (t, x, 000(y) e-co(a)t

rp (t, Xy) =

013(x)

generates a semigroup, which we denote by Rt with Rtl = 1. The generator AO of Rt as calculated by the formula Rhf (x) - f (X) A0f (x) = lim h\,0 h

results in the expression

ap(x,y)(f(y) - f(x)),

Apf(x) =

claimed in the theorem. Using the asymptotic formula p,3 (t, x, y) -

013(x)op(y)e,\0(p)t

together with the definition of rp (t, x, y) we see that both

tl

rp (t, x, y)

(y)

(x)rp(t,x,y)

0)3' (y).

and

xEZd

In addition, (4.7)

rp(t, x, z)/p 1(z) = Epp(t, x, z),p 1(x)e-a0(p)t z

z

_

eA0(13)tO1(x)0Q(z)Op 1(x)e-ao(p)t + o(1)

= (013,J) +o(1).

This yields the asymptotic finite dimensional distributions of the initial part of the polymer trajectory in the globular (0 > 13cr) phase. Namely, if 0 = To < Tl <
oo.

j=1

In order to obtain weak converge of the measures PQ,t on the space of trajectories up to time T we need to establish tightness. According to [2], this requires control of the oscillations. Set WT (x, [ti-1, ti)) -

sup

s,tE[ti_i,ti)

Ixt

- xsl

and

wT(x, S)

-inf Max WT (x, [ti-1, ti)), {ti} 1 E)

t > To.

But, we have (4.11) PQ,t(WT(x, 6) > E) = Zp tEo[ea.fo 6o(xe) ds1{w, (x,6)>E}]

Zp tEo[ePfo bo(xs)dsl{wT(x,b)>E}ExT[ePfo-T 60(xs)d3] Eo[epT /3,t

e,3TPo(WI(x,6)>E)->0,

t-400.

This proves the convergence in distribution of the process on [0, T] under Pp,t as t -> oo. It also provides the invariant distribution statement (4.3) since t

lim Pp,t (xs = x) = rp (s, 0, x).

and

lim rQ(s, 0, x) _p(x). 300 This proves the theorem.


106

An immediate consequence is a relative of a result of Alexander and Sidoravicius [1].

In that paper the notion of pinning is defined in terms of the local time,

Lx,t = fo bx(xs) ds. The polymer is said to be pinned if for some 6 > 0, Lot,t

>S =1.

Corollary 4.2. Let Lx,8 = ff Sx(x,.) dr, be the local time of the polymer at x up to time s. Then, if Q > Qcr, for every e > 0, Lx,s

lim lim Pp,t

s--.0o t-,C.0

2

Va(x)

s

We now turn attention to the behavior of the polymer trajectory in the diffusive

phase, namely 0 < 0cr(d), d > 3. Since this should behave like the free polymer, we consider briefly the random walk with transition probability po(t, x, y). Since po satisfies (2.1), the Fourier transform, PO (t, 0, y) _ >zEZd po(t, z, y)ei(O,') satisfies

asoo (t, 0, y) _

(4.12)

0, y),

Po(0, 0, y) = ei(0,y).

So, Po(t, ', y) = ei(0,y)e-t4«). Setting y = 0 and inverting gives po(t, x, 0) = P0 (t, 0, x) =

e-2dt

1 e2t j_1 cos -i(0,x)

1

(27r)d

Td

dO

An explicit formula may be given in terms of Bessel functions which leads to the following results, E

E

\

A I -+

xt = l

/

d

47r

/ -l ' dy. e JA

We expect similar results hold for the polymer in the non-globular phase 0 < 3cr. This expectation arises from the "depinning" of the polymer, in this regime since

the attraction of the potential isn't sufficient to make typically transient paths repeatedly visit the origin. As a result, the paths under the polymer measure are very much like the free paths since they eventually stop visiting the origin.

Theorem 4.3. In the high temperature regime 0 < 13cr, d > 3, as t -+ oo Zp,t -+

1

Y-- -)31(0) E

(4.14)

Covp,t Xt - tI xt

PQ,t C

VI-'t

EA

--+

e-lyl2 dy

1 d

47r

JA

HOMOPOLYMERS

107

PROOF. Assuming Q < Ncr, the Fourier transform of the fundamental solution p¢y(t, 0, x) satisfies (4.15)

at

(t, 0, 0) _ -4,(0)p0 (t, 0, 0) + Qpp (t, 0, 0),

A 3 ( 0 ,0 ,0 ) = 1.

Thus, (4.16)

+ 31

P0 (t, 0,

0, 0) ds.

0t

Inverting this equation we have (4.17)

pp (t, 0, x) = po (t, 0, x) + )3

ft

po (t - s, 0, x)pp (s, 0, 0) ds.

J0

Introduce the Laplace transforms of po (t, 0, 0) and po (t, 0, 0) using the notation

PP(A) = fe_Atp(t,o,0)dt and

Po(i) _ f0

e-Atpo(t, 0, 0) dt.

Then using P0 (t, 0, 0) = e-t1(0) and the Fourier inversion formula po(t, 0, 0) =

jpo(t, 0, 0) do

1

one obtains Po(A) =

d

1

I (A).

(2ir)d IT A + 'D(O)

It follows from this and (4.17) that PQ(A)

P0 (A)

- /Po(A)

Now, as we're in the case ,3 < ,3cr(d), it follows that

I

= P(0)

(4.18)

1-,31(0)

1 +,0 J

0

p,3 (s, 0, 0) ds,

as t

cc

0 1

= 1 - /3I(0) . The asymptotic variance is easy to handle, (4.20)

E P0 (t, 0, XEZd

x)2 = DA(


108

Then, by (4.17) we have t (t - s)pp (s, 0, 0) ds)

DR (t) = Zp,i 1 121 t +,3

\

o

Since Zp,t -+ 1/(1 - 131(0)), this implies

Da(t) _ (1 -

/3

f

t p,3 (s, 0, 0) ds)

_

Thus, in the case 13 < /3Cr, the polymer has linear variance (like the free polymer.) The statement about the covariance is handled in an entirely analogous manner. In order to derive the Central Limit Theorem, evaluate the Fourier transform pp (t, 0,

at 0 = T/., 0 E R. Since -b(1) ti 1012 as 0 -+ 0, and using (4.16) we see that (4.21)

po(t, 0, 0) = (1 +

o(1))e-1

12 +

(1 + o(1))Q

f

t e-1

12(1- t

po(s, 0, 0) ds

0

e-1012

1

1 - /31(0)'

ast -moo.

From (4.21) and (4.19) follows immediately the Central Limit Theorem for the distribution of xt/f with respect to the Gibb's measure PQ,t as t -* oo..

5. Behavior of the polymer at A =)3,r(d) Our analysis in this case rests on the asymptotic behavior of _ 1 do I (A)

A+4)(0)'

as A -- 0. We examine this in the next lemma for d = 3. Higher dimensions can be treated in an entirely analogous manner.

Lemma 5.1. If d = 3, there is a constant c3 > 0 such that as A -* 0,

I(A) = 1(0) - c3v(1 + 0(1)). PROOF. This was established in [3].

According to the lemma, the Laplace transform satisfies (5.1)

1

(A) PBcr (A)

-13 r I (A)

c13c(0)

=

c

A -+ 0.

This last asymptotic expression is valid not only for real A but also for A E C. Inverting gives 1

00

P0- (t, 0, 0) = 22-7r f P13- (A + ix)et(A+zx) dx,

and by deformation of contour, we get pact (t, 0, 0) =

2,7r

f Pacr (z)etz dz, 7

where ry is the curve with image the imaginary axis above i and below -i and the semicircle x2 + y2 = S2, x > 0. By the smoothness of PP_ (z) for I Im z I > b, we obtain

HOMOPOLYMERS

Lemma 5.2. For

109

I3cr, d = 3, there is a positive constant c2 such that pp_ (t, 0, 0) _

t -, oo.

,

As a consequence,we can derive the limiting distribution and variance of xt with respect to Ppcr,t as t -p oo.

Theorem 5.3. For d = 3 and /3 = Ncr as t --> oo, ZQcr,t - 2c20crVt, EQcr,t[(c,xt)2] ^ (5.2)

COOVpcr,t(xt)

3SI2t'

^' 2tI,

PP-,t (Vt- E ) ---+ P( E .) where

is a random vector with characteristic function 1

1

f(0) = 2 J a PROOF. First substitute (5.3)

e-112(1

U)

-

lu du

0 E Rd.

= 0 into (4.16) and use Lemma 5.2 to get

/t

Zpcr,t = pacr (t, 01 0) = 1 + /3cr J P13- (s, 0, 0) ds - 2c2Qcr Vt,

t ->oo.

0

As before, (5.4)

Epcr ,t [

xt)

2]

x)2

= ZPcr,t E po_ (t, 0, x) XEZd

j

/ft

=ZZ't(S 2t+OcrlS[ I2J (t-413-(s, 0, 0)ds tf1

(2c2/3cr

Vt)-1IS I2t (1

+ /3crc2t1/2

J0

(1 - u)u-1/2du) ,

t -+ 00

N

s II2t. A similar argument handles the covariance claim. As a result, we see that the proper normalization of xt for a limiting law would be xtl \. Interestingly, the

limiting distribution in the critical case 0 = /3cr is a mixture of Gaussians. Evaluate

(4.16) at O/' with 0 E R to get (55)

cr

(t0,

=e

After normalization by

I2 (1

+ 0(1)) + c2/3cr

j(i

o(1))e-I0I2(1-s/t)s-1/2 ds.

+

2c2/3cr / it follows that du.

E&-t[e 0

VrU

That completes the proof. We now examine the approach to the mixed Gaussian phase at /3 = /3cr. That is, given a large t, we give a value /3(t) < Ncr close to /3cr such that the distribution of

xt/\ under Pp(t),t is approximately the mixed Gaussian appearing at the critical parameter Or. The result is the following


110

Theorem 5.4. If /3cr-/3(t) = o(t-1/2) as t -* oo then the distribution of xt/\ approaches the mixed Gaussian with characteristic function 1 fo

e-1012(1-u) du//i.

PROOF. For 0 < Q < Or define for 0 < s < t, Op (s) = pQcr (s, 0, 0) - pQ (s, 0, 0).

Now fix a large value of t and let 13 = li(t) depend on tin such a way that j3r-,3(t) _ o(t-1/2). From (4.17) we derive the formula (5.6)

AP(s) _ (Ncr - /d)°o(s) +Q fpo(s - r, 0, 0)AQ(r) dr,

0 < s < t.

Evaluating at s = t and iterating, we get with to = 0, (5.7)

oa(t) = (13cr - 13)

x Coo(t)

3n

+ n=1

t

tn_1 n

t1

11 po(ti - ti-1, 0, 0)Ao(tn) dtn ... dtll

f f ... f 0

0

0

l

i=1

And using the simple bound jAo (s) I < 1, it follows that (5.8)

1 A0(t)1

co

r/

1, then there exists a meromorphic or entire solution of (1.1). After Poincare, solutions of (1.1) are called the Poincare functions admitting a multiplication theorem (cf. [16]). Later on, G. Valiron [15, 16] elaborated the case, where R(z) = P(z) is a polynomial, i.e., f (Az) = P(f (z)), z E C, and obtained conditions for the existence of an entire solution f (z). Furthermore, he derived the following asymptotic formula for M(r) = maxi, ,=r. If (z) I: (1.2)

(1.3)

log M(r) - rPQ I

llO

)

,

r

Co.

Here Q is a 1-periodic function bounded between two positive constants, p = log m/ log JAI and m = deg P(z). An interesting example of such equations, which stems from the description of Brownian motion on Sierpinski's gasket, has been studied in [1, 7, 8]. This is the functional equation (1.4)

f(5z) = 4f2(z) - 3f(z).

Similar equations arise also in the theory of branching processes (cf. [2,9]). Different aspects of the Poincare functions are discussed in papers [3, 5, 6,13] closely related to the present one. 2000 Mathematics Subject Classification. Primary 30D05; Secondary 39B32, 30D15, 37F10. The second author is supported by the Austrian Science Foundation FWF, project S9605, part of the Austrian National Research Network "Analytic Combinatorics and Probabilistic Number Theory". This is the final form of the paper. ©2007 American Mathematical Society 113

G. DERFEL ET AL.

114

In our paper we derive further results of Valiron's type. Namely, in addition to (1.3) we find asymptotics of entire solutions f(z) on various rays argz = 79 of the complex plane. It turns out that this heavily depends on the arithmetic nature of A.

For instance, the following statement (Theorem 4.5, below) is proved: If arg. = 2irf and 0 is irrational, then f (z) is unbounded along any ray arg z = 19. Moreover, if we denote cp(z) = log If (z) I (where the main branch of logarithm is taken) then there exists a sequence r,,, - oo, such that the limit (1.5)

lim

p(r"e

79

)

=L

exists and L > 0. On the other hand, if 0 is rational (and, in particular, if 0 = 0, i.e., A is real) f (z) may be bounded on some rays and even in whole sectors. Nevertheless, for rational 0, the limit (1.5) still exists under some additional assumptions. Denote Q = t/s and suppose that t, s are relatively prime. Put q = A (note that 1 < q E I8) Here we have the following result (see Theorems 5.2 and 5.3 below): Suppose that either JAI > m2 or s > 2p. Then f (z) is unbounded on any ray, and one can find a geometric progression r,,, = q'ro, (ro > 0), for which the limit (1.5) exists and L > 0. Further refinements are possible when A > 1 is real and P(z) = pmzm + + piz + po is a polynomial with real coefficients. Namely, we prove the the following statement (Theorem 6.3):

Assume that: (a) A > m; (b) pi > 0, for i > 2; (c) All preimages of 0 under P, i.e., P-n ({0}) are real. Let f (z) be an entire solution of (1.2) such that f (0) and f'(0) are real and f (0) > 0 and f'(0) > 0. Then f (z) tends to infinity along any ray in the sector 0 < Jt9J < 7r/2. Moreover, cc(rel') ,., r' Q,g

(1.6)

(2-),

r

cc

where Q y (z) is 1-periodic and bounded between two positive constants.

Condition (c) plays an important role in the last statement. If P(z) is quadratic polynomial (a case, arising in some applications) it is possible to give an exact criterion for reality of P-I({0}) (see Lemma 6.7 below): Let

P(z)=az(z-w), 034 wER

(1.7)

All preimages of 0 under P are real, if and only if the following condition is fulfilled

awl >

(1.8)

r2 for w > 0 S(4

forw max{e, 2mK}.

Denote

Mo = max W(z) and mo = min co(z). zEe"90 I

zEe"'O I

Then the following is an immediate consequence of Theorem 4.1.

Theorem 4.3. Suppose that the assumptions of Theorem 3.1 (or Theorem 3.9) are satisfied, ,3 = t/s is rational and (4.3) holds. Then cp(rel'90) - rPQ

(4.4)

log r log q J '

where Q is a continuous 1-periodic real function of a real variable bounded between two positive constants: (4.5)

Vt E ]18:

mo - 3/4 < Q(t) < Mo + 3/4 ro

(gro)P

Remark 4.4. The theorem of Valiron [16] cited in Section 1 follows from Theorem 4.3.

PROOF. Suppose that f (z) is an entire solution of (3.1). Denote

M(r) = max lf (z) zj=r

and

0(r) = log M(r).

Then arguments analogous to those which led us to (3.12) and (3.13) yield (4.6)

'(I) Ir) = mV)(r) + R(r)

with

JR(r)J < M(r).

(4.7)

Thus the conditions of Theorem 3.9 are satisfied for the real scaling factor JAI in (4.6). Clearly, also M(r) > max{e, 2mK} for r large enough, which implies (4.3). From this we obtain

log M(r) - rPQ which implies Valiron's theorem.

log r log JAI

ASYMPTOTICS OF THE POINCARE FUNCTIONS

123

4.2. Irrational angle. In this case we cannot claim the existence of geometric progressions along a given ray, for which (4.1) holds. Instead we will prove that for any ray rei5 there exists a sequence of positive real numbers (sn) with sn -i 00 such that (si19)

lim

n--oo

SP

=L>0

exists. In particular, this implies that in the case of irrational /3 solutions of the Poincare equation are unbounded on any ray. As far as we know this phenomenon has not been mentioned in the literature before. An alternative proof of the latter statement based on the Phragmen-Lindelof principle will be given in Section 5. Theorem 4.5. Suppose that the assumptions of Theorem 3.1 (or Theorem 3.9) are satisfied and 0 = 1/(27r) arg A is irrational. Then on any ray arg z = 0 there exists a sequence (snei") with sn --- 00 such that the limit L(zo)

(4.9)

exists and satisfies (3.7).

PROOF. Let zo satisfy (3.4). Then there exists an e > 0 such that If (z)I > max(e, 2mK) holds for all z E B(zo, E). By Lemma 3.8

Li(z) = n-*oo lim

(4.10)

w(Anz)

mn is a continuous function of z on B(zo, E) and convergence is uniform in B(zo, e) as indicated in the proof of Lemma 3.8.

By the density of the set in arg A I n c N} on the unit circle there exists a sequence of integers (nk)kEN such that lim arg Ank = V - arg zo

k--oo

Let Sk=lzolei19

mod 27r.

Al nk CIAI)

Then clearly limk-,,,, (k = zorand we have lim (AnkSk) mnk

= lim (IzOIe"JAInk)

k-*oo

k-.oo

mnk

__ Ll(z0)

by uniform convergence. Denoting Sk = IzoIIAI nk we have then (skei'v)

lim

k-*oo

SP

= lim

o(ske

") =

k-oo (IZOIIAIn k)'

Li(zo) Izoll

= L(zo)

0

5. Conditions for the unboundedness of solutions along rays Note that Theorem 3.1 and Corollary 4.1 are results of conditional type. In general, we do not know whether condition (3.4) is satisfied in a specific point zo = roe'50. In this section we give some conditions under which any nontrivial entire solution f (z) of (3.1) is unbounded on any ray arg z = 19. Denote as before A = IAle27rip.

We can assure the unboundedness of f (z) along all rays under the following conditions either on I A I or on arg A: (a) Al Iis large compared to m = deg P(z)

G. DERFEL ET AL.

124

(b) 3 is rational,

t/s (in lowest terms) and s is large compared to p =

log m/ log JAI

(c) /3 is irrational. The proofs of (b) and (c) are based on the Phragmen-Lindelof principle, or more precisely on the following corollary.

Corollary 5.1 (Phragmen-Lindelof, [10]). Assume that f (z) is a nonconstant entire function of order p > Suppose that n rays emanate from the origin and split the complex plane C into n angular regions each of which has angle less than 7r/p. Then f (z) is unbounded on at least one of these rays. 2.

5.1. Large JAI. Theorem 5.2. Suppose that JAI > m2, i.e., p < 2. Then (1) f (z) is unbounded along any ray arg z = 19 (2) For any 19 there exists a sequence rn = rn(19) - oo such that the limit lim

(5.1)

n--.co

orne' (

rn

)

= L(19)

exists.

(3) These limits L(19) are bounded between two positive constants for all 19 E [0, 2-7r].

(4) Under the additional assumption that /3 = 1/(2ir) arg.\ is rational (/3 = t/s in lowest terms) further refinements are possible. Denote q = AS (and note that q > 1). Then rn can be chosen as a geometric progression rn = rogn independent of the direction V.

PROOF. Unboundedness of f (z) along any ray follows from the assumption p < 12 (see [14, 8.73]). Moreover there exists a sequence rn - oo such that m(Tn) = min Iz If (z) I -* oo.

-r

In particular, one can find a circle Co = {z E C I IzI = ro} such that condition (3.4) is satisfied for all z E Co. Then (2) and (3) follow either from Corollary 4.1 (if /3 is rational) or from Theorem 4.5 (if /3 is irrational). Applying Corollary 4.1 once again yields (4).

5.2. Rational /3 with large denominator. Theorem 5.3. Suppose that /3 = 1/(2.7r) arg, is rational (/3 = t/s in lowest terms). Suppose further that s > 2p. Then (1) f (z) is unbounded on any ray arg z = V. (2) Furthermore, for any 19 one can find a geometric progression rn = rn(t9) _ gnro(19) (with q = \S > 1) such that the limit (5.1) exists and L(t9) > 0.

PROOF. Suppose that there exists a ray arg z = 191 (r > 0) such that f (z) is bounded along this ray. Then in view of equation (3.1) f (z) is bounded along all rays arg z = 191 + 2k-7rt/s mod 2-7r (k = 0, ... , s - 1). The angle between two consecutive rays is 27r/s < 7r/p. From Corollary 5.1 it follows that f is constant. Assertion (2) follows by applying Corollary 4.1 as in the proof of Theorem 5.2.

ASYMPTOTICS OF THE POINCARE FUNCTIONS

125

5.3. Irrational /3. Suppose that /3 = 1/(2ir) arg A is irrational. It was already proved in Section 4.2 as an immediate consequence of Theorem 4.5 that every nontrivial entire solution f (z) of (3.1) is unbounded along any ray argz = 19. Here we present an alternative proof of this statement using again the PhragmenLindelof principle. However the existence of the limit (4.9) cannot be obtained in this way.

Theorem 5.4. If /3 is irrational, then every nontrivial entire solution f (z) of (3.1) is unbounded along any ray arg z = 19.

PROOF. Suppose that f (z) is bounded on a ray arg z = 191. Then again by (3.1) f (z) is bounded along all rays arg z = 191 + 2kir/3 mod 27r for k E N. Since the angles 191 + 2k7r/3 mod 27r are dense in [0, 2ir], one can find n such rays with angles between consecutive rays less than -7r/p. Corollary 5.1 implies then that f (z) is constant.

6. Real case: real A > 1 and polynomial P(z) with real coefficients Throughout this section we will assume that ) > 1 is real and all coefficients of P(z) are real. Also we use the notations of Section 2 above. Immediately, from Lemma 2.5 (points (3) and (4)) we derive

Corollary 6.1. Suppose that (1) pi > 0 for i > 2 (2) fo > 0 and fn,o > 0. Then

(a) fi > 0 for all i (b) there are infinitely many i for which fi > 0 (c) f (z) > 0 for all z E I[8+. From this we obtain

Corollary 6.2. Under the assumptions of Corollary 6.1 f (z) -* oo in 1[8+. Moreover,

f (r) = M(r) = max If (z) I I=I=T

and

log f (r) - rPQ C log r

(6.1)

for r -+ oo,

g

where Q is a continuous 1-periodic function bounded between two positive constants.

Further refinements are possible under additional assumptions (cf. also [3, Theorem 1]).

Theorem 6.3. Assume that

(1) A>m (2) pi > 0 for i > 2 (3) all preimages of 0 under P are real.

G. DERFEL ET AL.

126

Let f (z) be an entire solution of (3.1) such that f (0) and f'(0) are real and f (0) > 0 and f'(0) > 0. Then f (z) tends to infinity along any ray in the sector I arg z < 2 Moreover, log If (re"9) 1 - rpQ,9

(6.2)

(lolog g

r)

for r --> 00,

where Q,y is a 1-periodic function bounded between two positive constants.

The proof is based on the following three lemmas.

Lemma 6.4. Suppose that all preimages of 0 under P are real. Let f (z) be an entire solution of (3.1) such that f(0) and f'(0) # 0 are real. Then f(z) has only real zeros.

PROOF. Suppose that f (z) = 0 has a nonreal solution zo. Then P(n)

(zoA-1)) = 0 = .f (zo) = P(f (f (zoA -n)) = which implies that f (zo.\-n) is an nth preimage of 0 under P and is therefore real by our assumptions. Using Taylor's formula we obtain lim An(f (z) -n) - f (0)) = f'(0)z.

ncc

Here the left-hand side is real and the right-hand side is not, which gives a contradiction. Lemma 6.5. Suppose that the assumptions of Lemma 6.4 are satisfied. Assume

further that

(1) pi>0fori>2

(2) f(0)>0 and f'(0)>0. Then f (z) has nonpositive real zeros only.

PROOF. The assumption f'(0) # 0 implies that no = 1. Furthermore, we have

fn,, > 0. Thus by Corollary 6.1 f (z) > 0 on ]IB+. Thus the zeros have to be nonpositive.

Lemma 6.6. Assume that f (z) is an entire function of order 0 < p < 1 with only negative real zeros. Then f (z) tends to infinity faster than any power of z along any ray in the sector I argzI < 7r/2.

PROOF. Because 0 0 are the negative zeros of f. Write z = x + iy and S,,={zECCI Iargzl 0

for an arbitrary function g(t), say compactly supported. The significance of the power dissipation condition is as follows. We suppose that the internal energy of the TDD system at time t is given by (1.24)

Ho

2

f(f(z, t)2 +

t)2) dx,

with f, (x, t) as in (1.21). Then (1.25)

dtHo(t)

_

('Ya-0(x, t)ata.0(x, t) + f,,(x, t)atff (x, t)) dx '

t) OXf(x, 7l

J

t) + f, (x, t)at7r (x, t)

R /p

f,(x,t) J

er

x(x,T)atf7r(x,t

T)

dT) dx

0

J

R

1a2

(x' t) OX f7l(x7 t) + If71

(x,t)5X2 `Y(x,t)

7

/per

+ f7C(x, t) f (x, t)

f,1(x, t) J

t

T) dTJ dx

0

= J f (x, t)ato(x, t) dx - f JO X(x, )a(x, t - ) d55(x, t) dx, R

A. FIGOTIN AND J. SCHENKER

136

where in going from the third to the final line we have used integration by parts and the assumption that 8xo and f, vanish at spatial infinity. The first term on the r.h.s. is the rate of work done on the system by the external force f. Similarly, we interpret the second term as the rate of work done by the dissipative force, f x(T)8t O(x, tT) dT. Thus, power dissipation amounts to the requirement that, for any trajectory, the total work done by the dissipative force is negative. Let us note that the right hand side of (1.23) is equal to (1.26)

x(x, 0) J 'G g(t)2 dt + J g(t) J 8Tx(x, T)g(t - T) dT dt 'G 00 00 0 00 f {x (x, 0)6(t1

t2) + 2 [0, X1 (x, I t1

t2I)}g(t1)g(t2) d2t,

2

which follows from integration by parts after noting that 8tg(t - T) = -8Tg(t - T). Here S(t) is the Dirac delta function. Thus the power dissipation condition is equivalent to the statement that for every x the (generalized) function DX(x, t)

(1.27)

x(x, 0)6(t) + z [0,X] (x, It

is positive definite in the sense of the classical Bochner's Theorem, see [7, Theorem IX.9]. Thus, by Bochner's Theorem, the time Fourier transform of D is a nonnegative measure,

f

00

(1.28)

x(x, 0) + z x(x, 0) +

J0

["T x] (x, It1)e'wt dt

ax(x,T) COs(WT) dT

OC

=wJ

T) sin(wT) dT = w Im k(x, w) > 0.

X

0

Since x is real, B,, (x, w) is symmetric under w H -w, and indeed Bochner's Theorem shows that a symmetric measure of suitably bounded growth is nonnegative if and only if it is the Fourier transform of a real positive definite distribution. The simplest physically relevant example of a susceptibility satisfying the power dissipation condition is obtained with x(x, T) = a > 0, a positive constant. Then

f" (x, t) + a J "0 f, (x, t - T) dT = 7r (x, t).

(1.29)

0

Using the equation of motion f, = 8to, we find that (1.30)

r x t)

8

x t)

x tT dT=B

x t)

x t),

where we have applied the boundary condition limt-_o O(x, t) = 0. Combined with the equation of motion 8tir = y8xo + f we obtain (1.31)

8t 0(x, t) + a8tq(x, t)

y8.20(x, t) = f (X, t),

which is the dynamical equation for a driven damped string, with damping force per unit length -aatq(x, t). Note that a is dimensionally an inverse time and 1/a is the characteristic time for the damping of oscillations. A more realistic model for damping is obtained by supposing x(x, T) to be a nontrivial function of T as in (1.21). To allow for damping restricted to only a part

DISPERSIVE DISSIPATIVE STRING

137

of the string, we suppose that x depends on x as well. For instance, we could take the Debye susceptibility (1.32)

x(x,T) = a(x)e

v(x)T

with a(x) > 0 and v(x) > 0 nonnegative functions of x. This results in the following integral-differential equation for 0 00

(1.33)

e-"(')'8t O(x, t - T) dT - ry8xO(x, t) = f (x, t),

8t O(x, t) + a(x) J 0

or after integration by parts (1.34)

+ a(x)v(x)2

DX(x, w) = a(x) J

0

e-"(x)'O(x, t - 7) d7 - ry5x2O(x, t) = f (x, t).

J0

F

Then (1.35)

a(x)v(x)o(x, t)

at o(x, t) + a(x)atq(x, t)

W

e- )Tw sin(wT) dT = a(x)

2

> 01

v(x)2 + w2 -

so the Debye susceptibility satisfies the power dissipation condition. The results of [2, 3] show that under the above conditions it is possible to find

a coupling function ;(x, s), s E f8, such that solutions to the TDD equations are generated by solutions to the following extended system fr(x,t)

at0(x,t)

(136)

8t7r(x, t)

'YBX20(x,

=

8t V) (x, s, t)

t) + f (X, t)

0(x, s, t)

810(x, s, t) =

8s,0 (x,

s, t) + c(x, s) f, (X, t),

with (1.37)

f,, (x, t)

7r (x, t)

J

c(x, s) V) (x, s, t) ds. Do

That is, given a solution (0, 7r, b, 8) to (1.36), at rest at t = -oc and with f, given by (1.37), the first two coordinates (0, 7) obey (1.22) with f,, given by (1.21), and conversely any solution to (1.21) - (1.22) with the string at rest at t -oc arises in this way. Note that for each fixed x the additional variables 8, may be interpreted as describing the oscillations of a "hidden string" with coordinate s, displacement b (x, s, t), momentum 0(x, s, t) and driven by external force c(x, s) f, (x, t). Thus (1.36) is precisely the system of coupled strings described above and illustrated in Figure 1. The extended system, consisting of the physical and hidden strings is Hamiltonian with symplectic form (1.38)

J(01, 71, W11 B1; 02172 021 02) =

f

{01(x)72(x) - 02(x)71(x)} dx

+ JR2 {01(x, s)02 (x, s) - b2 (x, s)01(x, s)} ds dx


138

and Hamilton function (1.39)

xf(0,7r, ,0,t) =

f

2

f(x,t)O(x)dx

{f"(x)2+'}'(ayO(x))2}dx-J

+2

s))2} ds dx,

{9(x, s)2 +

where f, is given by (1.37), that is f, (x) = 7r(x) - f c(x, s),O(x, s) ds. The dynamical equation for an excitation of the hidden string at x is a driven wave equation (1.40)

s, t) = 0s,0 (x, s, t) + c(x, s) f, (x, t).

at,0 (x,

The solution to (1.40) with the hidden string at rest at t

-oo is easily written

down, see (1.3): 1

(1.41)

b (x, s,

[,fs+T

00

210

s-T

c(x, r) dr] f,I (x, t - T) dT.

Thus (1.42)

fc(z s)(x, s, t) ds

l[fc(xs)f s +T

J

2

0

-T

Compari ng (1.21) and (1.37) we see that for the extension to reproduce the TDD system upon elimination of and 0 it is necessary and sufficient that the coupling ,;(x, s) satisfy (1.7), that is +T

/f

x(x, T)

(1.43)

1 2

JR

c(x, r) dr ds.

c(s, x) IS-T

The existence of such a function, which is unique under a natural symmetry assumption, is guaranteed by the power dissipation condition [2, 3]. Indeed, if we let (x, o,) denote the s-Fourier transform of c, s(x, Q) = J c(x, s)e'o's ds,

(1.44)

then (1.43) is equivalent to (1.45)

Dx (x, w) _ x, w)s(x, -w),

so it suffices to take [2]

x, o,) =

(1.46)

2DX (x, a).

Furthermore, this choice of x, o,) is unique if we ask further that x, o,) > 0 and that or F- (x, a) be symmetric. Then, (1.47)

c(x, s) =

J

2DX(x,

o,)e-'as do

=

JR

2DX(x, o,) cos(o s) da,

is symmetric and positive definite. For example, in view of (1.28), (1.35), and (1.47), the following coupling produces the Debye susceptibility, (1.48)

c(x, s) = v2a (x) asW (v(x)s).


139

where (1.49)

'I'(s)

f °O

1

27ri

1

/

wZ

f"' w v 1+ w2

e

s1

iu)s

/'1

1

1 -U2 s 1 171/2 e-Isl sin(O) do. IS II I

That is, (1.50)

2a /x\ ((s) S

S(x s) =

v(()

fo

/2

sin

(O)ev(x)Isl sin(O) do2.

1

Local energy and momentum conservation in the extended system We interpret the Hamiltonian no with f - 0 as the internal energy of the damped string system consisting of the coupled physical and hidden strings. We have conservation of energy in the extended system, in the form

ff(xt)at(xt)dx,

(2.1)

dt no i.e., the rate of change of no is the rate of work done on the system by the external force.

A significant advantage of working with the extend system is a transparent interpretation of the energy of the dissipative string as a sum of contributions from

the physical and hidden strings. That is, it is natural to break the no into two pieces (2.2)

no(0,7r,0,9,t)

Ho(0,7r,')+Hhs(0,9)

the energy of the physical string and hidden strings respectively, (2.3) (2.4)

Ho(0, 7r,

2

f{f)2 +'Y (OXO(z))2} dx

Hhs (0, 8) = 1 f {8(x, s)2 + 2

(8s'O (x, s)) 2 } ds dx,

with f.7r given by (1.37).

The internal energy no can be written as the integral of a local energy density (2.5) E(x, t) = Eo(x, t) + Ehs(x, t) with

Eo(x,t) = 2{(atO(x,t))2 +_Y(a.0(x,t))2}

Ehs(x,t)=

2

f {(5

(x,s,t))2}ds.

The energy conservation law (2.1) has the following local expression (2.7)

8tE(x,t) + a J(x,t)

f(x,t)at0(x,t)

with the energy current (2.8) J(x, t) = -'y f,(x, t) = -ry8tO(x, t). It is interesting to compute the time derivatives of Eo and Ehs alone: (2.9) atEo(x, t) + a,,J(x, t) = f(x, t)8tO(x, t) - 810(x, t)atO(x, t) (2.10) atEhs(x, t) = 8tA(x, t)at0(x, t)


140

with (2.11)

A(x, t) _ 7(x, t)

f, (x, t) = J S(x, s)V)(x, s, t) ds

f x(x,)Ot4'(x,tT)dT, ,

J0

where we have used (1.42). The first of these equations (2.9) is simply the local version of the energy law for the TDD string (1.25). From the second (2.10), we see that the energy of the hidden strings, which is the energy lost to dissipation up to time t, is Ehs(x,t)

ft at'

(x,t')

x(x,T)at,O(x,t'-T) dT dt'.

00

(2.12)

0

t

00

ft00

DX(x, tl

t2)at1O(x,

t1)at24'(x, t2) dtl dt2.

If the susceptibility x and hence the couplings is independent of x, then the extended system is invariant under spatial translations. Associated to this symmetry is a local conservation law

atp(x, t) + axT (x, t) _ f (x,

(2.13)

t)ax0(x, t),

for the wave momentum density (2.14)

p(x, t) _ 7(x, t)axO(x, t)

_

f 9(x, s, t)Ox2)(x, s, t) ds

{atO(x, t) + 0(x, t)}axO(x, t)

f atV) (x, s, t)ax0(x, s, t) ds,

with wave momentum flux, called stress, (2.15)

T (x, t) = Eo (x, t) + A(x, t)at0(x, t) +I{(at0(x,s,t))2

(asV(z,s,t))2}ds.

When the driving force vanishes, f = 0, the total wave momentum

P=

(2.16)

J

p(x, t) dx

is a conserved quantity. The wave momentum density p - p0 + phs is again a sum of contributions (2.17)

po(x, t) = -{ato(x, t) + o(x, t)}ax0(x, t)

= - J from the physical and hidden strings. Likewise we separate the stress (2.18)

phs(x, t)

(x, s, t)ax2b (x, s, t) ds

T(x, t) = To (x, t) + Ths (x, t)

(2.19)

into two pieces, (2.20) (2.21)

To(x, t) = Eo(x, t) + 0(x, t)at0(x, t) Ths(x, t) =

2J

{(at2p (x, s, t))2 - (as2b (x, s, t))2} ds.


141

Then (2.22) (2.23)

atpo (x, t) + OXTo (x, t) _ -f (x, t)OX0(x, t) + OXA(x, t)at0(x, t)

atPhg(x, t) + a.Thg(x, t) _ 0.0(x, t)at0(x, t).

When we study eigenfunctions below, it will be convenient to work with complex valued solutions. In the above expressions, terms which are quadratic in the field variables should be modified in the complex case by the replacement

ab - Re ab. That is Eo(x,t) = 2I {Iato(x,t)I2+'YlaXO(x,t)12} (2.24)

Ehs(x,t) =

2

f{at(x s, t)2 +

J(x, t) _ --y Re atq(x, t)axo(x, t) PO (X, t)

= - Re at0(x, t)Ox0(x, t) - Re 0(x, t)ax0(x, t),

etc.

3. The eigenfunction equation It is useful and interesting to study steady state solutions to the extended system (1.36), for example solutions which are periodic in time e-'" (0, (x), 7r", (x), (x, s), 0, (x, s)). We refer to the spatial component -1),, (x, s) _ (0,, (x), 7r, (x), (x, s), 0, (x, s)) of such a time periodic solution as an eigenfunction for the linear system (1.36) with eigenvalue w. Thus an eigenfunction satisfies

iwo,(x) = f, (x) 'YaX2

-iwxrL, (x) _

(3.1)

OL, (x)

s) = 0,,(x, s) iwO (x, s)

as ), (x, s) + c(x, s) f r (x),

with s) ds - 7ru, (x) = 0.

f 7l (x) + J c(x,

We see that the displacement of the hidden string at position x satisfies as ), (x, s)

(3.3)

w20 , (x, s) _ iwc(x, s)0, (x),

so

(3.4)

0, (x, s)

a(x) cos(ws) + b(x) sin(ws)

a(x) cos(ws) + b(x) sin(ws) +

/'W

hi 0, x 1o

2

x

P. V. J

J

sin(Li Is' 00

e-'ors

2

1

w2 S(x, o-) do-

s1)s(x, s') ds',

142


with a(x) and b(x) undetermined functions of x. Here, P. V. denotes the "principle value" integral, (3.5)

P. V. f

e-ias

1

w2

Q2

(x, a) do,

= lim

e-'as

boo

{a:1a2-w2 >b}

= lim 1

e '0"'

61o2JIR

Qz

1

- w2

x, Q) d,

1

1_.

1

Q2-(w+i

By (3.4), we see that (3.6)

fc(xs)w(x,s)ds = a(x)x, w)

-

iw

(x) P. V.

a2

27r

wz

(x, a)s(x, -a) do,.

Recalling (1.45), that s(x, Q)S(x, -a) = Dx(x, a) = 2a Im x(x, a), we see that the final term on the r.h.s. can be expressed (3.7)

- 02(x) P. V. J/

"0

(;-22i 2 Q Im x(x, or) do-

io,(x) 27r

=

P.V. J

1

00

a-w

+

-a 1

aImx(x,a)da

w JJ)

iw Re x(x, w)0 , (x),

where we have used the symmetry a Im x(x, a) = -a Im x(x, -Q) and the "KramersKronig relation" 00

(3.8)

wRex(x,w) =P.V. l 7r

J

1

00 01 -w

QImx(x,Q)do-.

Plugging this result into (3.2) and combining the first two equations of (3.1) we find that the string displacement 0 solves the following elliptic problem (3.9)

ry82(x) _

-w2 (1 + Re x(x, w))O", (x) - iwa(x)s(x, w).

In analyzing (3.9) we should distinguish two cases: (1) c(x, w) 0 for all x. (2) S(x,w) nonzero for x in an open set.

In the first case, which is nongeneric, the medium described by the hidden strings is not absorbing at the given frequency w, that is x(x, w) is real for every x. Thus (3.9) is a very strong restriction on 0,, namely that it should satisfy the Schrodinger equation (3.10)

(-ryax+V(x))q",(x) =AoW(x) with potential V(x) _ -w2Z(x,w) and spectral parameter A

w2. In the second case, the physical string displacement may be decomposed as follows (3.11)

OL, (x) _

0(1)

(x) + 0(2) (x)

where 0,(1)(x) is any solution to the nondissipative eigenfunction equation (3.12)

waX2O(l)(x)

= -w2 (1 + Re k(x, w))0wl) (x),


143

and is an arbitrary function with support in {x : s(x, w) only choose a to be

0}. Indeed, we need

(l

(3.13)

a(x) =

(x) + w2 + Re x(x, w)) 0(2) (x) wc(x, w)

Thus, the physical string displacement for an eigenmode 0,(x) may be chosen arbitrarily within that part of the string which is absorbing at the given frequency w.

We want to emphasize the significance of this fact, since the formal eigenvalue problem (3.14)

'Y8xOu, (x) = -w2 (1 + X(x, Lo)) OU, (x)

does not allow for such arbitrariness in the choice of 0,(x). The resolution to this apparent contradiction lies in recognizing that a TDD system is an open part of a larger conservative Hamiltonian system. It is only for the extended Hamiltonian system that the eigenmodes I are unambiguously defined with e;Wt4)u, a stationary solution to the canonical Hamiltonian evolution equation. The "legitimate" eigenmodes for the original TDD string consist of projections 0, of the eigenmodes -1), onto the subspace of the physical string. Thus a TDD string, being an open system has as many eigenmodes, i.e stationary solutions, as its the minimal conservative extension introduced and described in [1-3]. In particular, a finite-dimensional TDD system typically has infinitely many stationary solutions and, hence, infinitely many eigenmodes. Another view on the construction of eigenmodes follows. The eigenfunctions written down above, involving as they do arbitrary excitations of the hidden strings, may not be relevant to the dynamics (1.36) with the external force acting on the physical string. Indeed, note that the coefficient b(x) does not appear in the effective equation (3.9) for 0 . To see which eigenfunctions appear in the expansion of a general solution to (1.36) with a compactly supported external force, it is convenient to introduce the Fourier-Laplace transform (3.15)

e

e'(th(t) dt,

defined for

(1) (E C if h - 0 super exponentially fast as tj - oc, for instance if h is compactly supported in time. (2) Im c > 0 if h 0 super exponentially fast as t -oc. (3) Im c < 0 if h - 0 super exponentially fast as t oc. Taking the Fourier-Laplace transform of (1.36) results in the following equations

(x, ) = (3.16)

Y8

(x, O + f (X1

-i0b(x, s, ) = 8(x, s, S, O + c(x,

(3.17)

f1(x,

*(x, ) -

J

00

c(x,

8) 1,

(x,

s, O ds.

00

Due to our consideration of solutions with the strings at rest at t = -oc, we expect the quantities in (3.16) to be well defined only for Im ( > 0. However,


144

if the driving force is compactly supported in time then f (x, O is defined for all ( E C, and we may extend (x, O to ( E C \ I1 by solving (3.16). Following the eigenfunction analysis, we obtain, with £ = sign of Im (, i(O(x, 0

(3.18)

Je-,as

27r

E

1

D2-

O(x

(2 s(x Q) do,

s') ds'

2

-(2 (1 + X(x, eO) (x, O + f (x,

(3.19)

where k(x, O is defined for Im ( > 0 as 00

TSX(x, r) dT.

(3.20)

02

For a function h vanishing at +oo or -oc we have the Fourier inversion formula

(3.21)

h(t) =

J

e-"t:F?lth(w + ir) dw,

(t lim h(t) =

)

with rl > 0 arbitrary. Inverting the solution to (3.16) with rl > 0 or < 0 produces distinct solutions to (1.36): for rl > 0 we obtain the desired causal solution with the strings at rest at t -oc, while for rl < 0 we obtain the anti-causal solution with the strings at rest at t +oc. In a certain sense we are only interested in the causal solution to (1.36) and thus to the solution to (3.16) only for ( in the upper half-plane. However, since (3.16) involves a source term, this solution, often called the resolvent, is not directly expressed as a superposition of eigenfunctions. However, there is a general procedure for decomposing the solution to (3.16) into a superposition of eigenfunctions. Namely for w E IE8 we define

(x, s))

(3.22)

ho

(x, w - i6), ir(x, w + iS) - ir(x, w - is), 2/J(x, S, w + 1(S)

V)(x, S, w

i(5), 8(x, S, w +

(b(x, S, w

Is))

.

Since f (x, O is continuous at each w E R, it follows from (3.16) that (&W, 7, V), 0,) is an eigenfunction for each w. To recover the resolvent for ( in the upper half-plane from the eigenfunctions (3.22), suppose that the external force is supported in the set {t : t > to}. Then (3.23)

f

(M to, < Ce- Im

(Im > 0).


145

Thus exp(-i(to) f (x, O is bounded in the upper half-plane, and by analyticity we have, 2

(x s ) 8(x, s, ))

(3.24)

_

e'((-w)to

1

27ri JR

(cbw (x), xw (x), 0,, (x, s), 0,,, (x, s)) dw,

(Im ( > 0)

expressing the solution to (3.16) as a superposition of eigenfunctions. (There is a similar formula for the advanced solution with Im ( < 0, involving an upper bound t1 on the support of the external force.) Now let us fix w and suppose that we have an eigenfunction of the form (3.22). Then by (3.18) and (3.19), (3.25) (3.26)

2J

4Pw(x, s) = io

(X)

sin(wIs' - sI)s(x, s') ds' + gw2x)

w)

00

'Y8x0w (x) _ -w2 (1 + Re 5 (x, w)) 0, (x) - iw2 Im X(x, w)gw (x),

with (3.27)

w + i6) + (x, w

gw (x) _ Eli

i6)}.

In other words, the eigenfunctions which appear in the expansion (3.24) are of the form (3.9) with b(x) = 0. We refer to such eigenfunctions as spectral eigenfunctions. By (3.24), the spectral eigenfunctions form a complete set for the description of the dynamics of the extended system (1.36). Thus, the freedom to choose gw in (3.26) provides us with a rich enough family of solutions to describe the dynamics of the TDD string. Note that a solution to (3.16) satisfies the eigenvalue equation away from the spatial support of the driving force. Thus, one may try to produce an eigenfunction as the limit of a sequence or solutions to (3.16) with a sequence of driving forces supported farther and farther from the origin. When this procedure works to produce a nontrivial eigenmode, the result is of the form (3.26) with a special choice of the arbitrary function gw. We can obtain in this way, (1) The causal eigenfunctions with gw(x) _ Ow(x), (3.28) (3.29)

(x, s) _ _Ya.2

0w (x

e'wls'-sls(s', x) ds' -00

Ow (x) _ -w2(1 + X(x, w))Ow(x),

corresponding to a driving force in the distant past and the causal bound-

ary condition with the strings at rest at t = -oc. (2) The anti-causal eigenfunctions with gw(x) _ -Cpw(x), (3.30) (3.31)

w( x s) = wx Y8xOw(x)

e iws

sly( si

x)ds i

w2(1 + X(x, w))Ow(x),

2For a general linear system the limit on the right hand side of (3.22) is a defined only as a vector valued measure and the integral on the right hand side of (3.24) should be interpreted as the integral of 1/(w - () against this measure. For the systems of coupled strings considered here, however, these boundary measures are always absolutely continuous, so (3.24) holds with (0,,,, 7,,,, /i,,,, 8,,,) defined for almost every w by (3.22).


146

corresponding to a driving force in the distant future and the anti-causal, or advanced, boundary condition with the strings at rest at t = +oc.

4. Energy flux in an eigenfunction 0, is The energy density E in an eigenfunction, at a point x with (x, w) typically infinite due to the contribution from the hidden strings. This result is to be expected physically as the eigenfunction represents the idealized steady propagation of a monochromatic wave through an absorbing medium with infinite heat capacity. However, due to the decoupling between the hidden strings, energy can flow only through the physical string and we expect the energy flux J to be finite. Furthermore as we shall see 8 J, which by (2.7) is formally equal to -ate, can be nonzero at a point x, in which case the eigenfunction is a steady state in which energy is dissipated to or absorbed from the medium at x at a constant rate. By (2.8), the energy flux for an eigenfunction is (4.1)

J(x,t) = -ryRe{iwOW(x)BxOW(x)} = rywImO'(x)OXO'(x).

Thus, by (3.26), (4.2)

(x)}.

-8xJ(x) = -wlmOW(x)ryBx0W(x) = w3Imx(x,w)

for a spectral eigenfunction, where gw(x) is the arbitrary function describing the excitation of the medium, as represented by the hidden strings. The energy density of the physical string (X) 12 + 8 (x)2} (4.3) Eo(x) = 1{w2l0",

,

is constant in time. We have a constant rate of dissipation at each x with -8xJ(x) 54 0,

(4.4)

8tE(x)

8tEhs(x)

w3

(x)}

Thus the eigenfunction represents a steady state situation in which energy is flowing into or out of the hidden strings at a constant rate at each point x with Im k(x, w) j 0.

For a general spectral eigenfunction (3.26), the dissipation 8tE(x) may be positive or negative, however for a causal eigenfunction (3.29), with gw = 0, we have (4.5) 8tE(x)

8tEhs(x)

w3 IM X(x, w) 0, (x) 12 > 0,

(causal eigenfunction)

corresponding to a steady dissipation of energy from the physical string into the medium, represented by the hidden strings. Similarly, for an anti-causal eigenfunction (3.31) there is a steady flux of energy out of the medium and into the physical string (4.6)

9tEhs(x)

-w 3lmx(x,w)jOL,(x)I2 < 0,

(anti-causal eigenfunction).

5. Plane waves and momentum flux in an homogeneous system Suppose that the susceptibility x(x,T) x(T) is constant for the whole range of x G R. Then it is interesting to look for plane wave solutions e,(1x-"t) (001701 ,Oo(s), 80(s)) with Oo, 7ro constants and mo(s), 8o(s) independent of x, i.e., eigenfunctions a e'kx.


147

A first observation is that there are no causal or anti-causal eigenfunctions of this form, at least at frequencies with Im x(w) > 0. Indeed a causal solution would satisfy (5.1)

-"yk20o = -w2 (1 + X(w)) Oo,

and the only nontrivial solutions to this equation are exponentially growing as x - foc unless x(w) is real. Such evanescent waves play a role in constructing scattering states for in-homogeneous systems but are physically irrelevant in the homogeneous system. Thus to find plane wave solutions it is necessary to look beyond causal eigenfunctions. For a plane wave spectral eigenfunction, see (3.26), we have (5.2)

(5.3)

-'k20o = -iw2go ImX(w) 00 (s) = 20 cos(ws)S(w) +

w2 (I + Re x(w)) Oo, loo

J

sin(wls'

sl)S(s') ds',

00

with go an arbitrary constant describing the excitation of the hidden strings. A bounded solution results only for k real, so

go=ic o,

(5.4)

aCR,

must be a pure imaginary multiple of 00. Furthermore, we must have 1 + Re x(w) - a Im k(w) > 0.

(5.5)

Then setting yk2

(5.6)

a Im x(w))

= w2 (1 + Re x(w)

we obtain a nontrivial plane wave solution.

If the medium is not dissipative at frequency w, so Im '(w) = 0, there is a dispersion relation between k and w (5.7)

yk2

(Im k(w) = 0).

= w2 (1 + Re k(w)),

At frequencies w with nontrivial dissipation, that is Im k(w) j 0, there is no relation between k and w. Indeed k may be chosen arbitrarily provided we take yk2 1 + Re x(w) (5.8)

a-

Im k(w)

w2 Im x(w)

By (4.1), the energy flux for a plane wave (5.9)

J = -'wkIOo 12

is constant. Thus a,,J = 0 and plane wave solutions represent a steady state flow without dissipation of energy. More precisely, the dissipation of energy to the medium, as described by the hidden strings, is exactly balanced by the energy emitted from the medium. The energy density of the physical string is Eo = (w2 + yk2) Oo 2 (5.10) and the energy density of the medium, described by the hidden strings, is of course infinite. As the system is homogeneous, we can also consider the wave momentum density and stress in a plane wave eigenfunction. By (2.17) the wave momentum density

of the physical string is (5.11)

po = wklool2 - kImAooo,


148

where

Do =

(5.12)

f

(s) ds = iwoo (a Im k(w) - Re x(w))

.

T hus 3

po =wk{1+Re x(w)

(5.13)

00I2

The wave momentum density of the hidden strings is infinite, since (5.14)

Phs = wk J e

V)0 (s) I2 ds

by (2.18). Of more interest is the stress, which by (2.19) - (2.20) is

T=Eo+wIM Dogo+Ths,

(5.15)

where Eo given by (5.10) and Ths is formally 00

(5.16)

{w2I4'o(s)I2

Ths = 2 J

-

aso (s)

2}

ds.

The integral on the r.h.s. of (5.16) does not converge absolutely, but we can regularize it by defining Ths = limbo Ths(6) with (5.17)

Ths(6) = 2 f "0

las,0o(s)12} ds.

e-6lsl{w21V)o(S)12

To evaluate the integral in (5.17) it is useful to write (5.18)

Fas

o(S) 2

1a2

-Re Vo(s)a,2,,Oo(s)

+w2100(S)12 + W,; (S) IM V)0 W00,

gas where we have used (3.3). Thus (5.19)

Ths(6) =

2

f

slsI

e

f I as 00

=2

1V)o(0)12

42 J

IV)o(8)12

+ L,),; (s) Imo(s)o} ds

e-slsll'0o(s)I2ds

00

2w

f

e-SIslc(s) ImV)o(s)0o ds. 00

It follows that (5.20)

Ths =

00

l oThs(6) = 2 wf c(s)ImV)o(s)Oods = 2wlm0o00, 1

00

if, for instance, Ic(s)l is integrable. Thus (5.21)

T = Eo + I w Im 0o0o 1

= I (w2(1+Re x(w)-aImX(w))+7yk2) which is identical to the result for the undamped string!

00

2

= yk210012,


149

6. Scattering eigenfunctions for a semi-infinite medium To close, we would like to illustrate the power of the above method by unambiguously constructing the scattering states representing an idealized description of the following experiment. Imagine that we have an long, i.e., infinite, string whose right half (x > 0) is subject to dispersion and dissipation with susceptibility X(T). That is, the the susceptibility for the whole string is

x(x,T)x(T) x>0 Suppose we drive the left end of the string, at x = -oc, periodically with frequency w, sending an incoming wave toward the TDD right half of the string. After some time a steady state is reached in which a certain fraction of the incoming wave is absorbed by the dissipative half of the string and a certain fraction is reflected. The steady state eigenfunction describing the above experiment is a causal eigenfunction since the source is at x = -oc and started in the distant past. Thus by (3.29), the string displacement satisfies

-w2(1+I[x > 0]x(w))0,(x),

(6.2)

where I [x > 0] = 1 for x > 0 and 0 otherwise. Indeed, this is the naive equation that one might right down for the scattering states of a TDD string. However, it is only within the context of the extended Hamiltonian system that we understand this to be only one of many possible eigenfunction equations, the choice of which is dictated by the physics under consideration, namely a source at spatial infinity in

the distant past. By (6.2), we have

W elk<x + OL,(x=

re-ik< x,

k< _

w

,

(x < 0)

on the left half of the string, up to an over all multiple which we fix to be 1 without loss. In (6.3) the term eik<x is the incoming wave and re-'k>x is the reflected wave. To find the reflection coefficient r we need to solve for the form of the eigenfunction on the right half of the string. Again by (6.2), we have (6.4)

'o2OL (x)

w2 (1 + X(w)) OL, (x),

(x > 0)

for x > 0. Furthermore, since we require the solution to be bounded this determines 0 uniquely up to an over all multiple

ve'k'x,

qw(x)

(6.5)

where v is an, as yet, undetermined transmission coefficient and w2 k>

1/

(1 + k(w)).

7


150

denotes the unique square root of z in the upper half-plane.3 Since Imk> > 0, at least if Im x(w) 0, we see that 0,(x) decays exponentially in the dissipative half of the physical string (x > 0). The excitation of the hidden strings, which are restricted to x > 0, is given by (3.28): Here

V ik>x 0w(x s) = 2e

e

iwjs-S'I(6.7) ;(s') ds',

(x > 0).

Oc

To determine v and r we note that the eigenfunction equation (6.2) forces 0, and axo,, to be continuous functions of x, in particular at x = 0. Thus, (6.8) lim O,(x) =I+ r = v = lim Ou, (x) X10

xTO

(6.9)

limax0,(x) = kv = limax0,(x). xTO

X10

We conclude that

=1

(6.10)

1 - p(w)

2 (

)

V

r

-'+ p(w),

,

where (6.11)

k> k
0, i + x(w) w 0 we have L,) Re V1 + x(w) > 0,

(6.12)

and thus Rep (w) > 0.

(6.13)

This implies that

r

4

1+

P(w)

2

Re p(w) = v12Rep(w) =

V

2Rek> > U, k
x

x0.

31f x(w) is real and (1+x(w)) > 0 then the medium transmits at frequency w and k> should be determined as the limit

(w+i5)2 610

V

'Y


151

Furthermore, we see that the energy flux is nonnegative, representing a flow of energy from the source at x = -oc, and the rate of dissipation, -55xJ(x) 54 0, is nonzero for every x > 0, (6.17)

-aaxJ(x) = 2Jw2(1 - jr 2) Im k>e-2Imk>x

(x > 0).

Finally, we note that the energy density of the physical string is (6.18)

Eo(x) = x < 0,

Ir12)

w21(1+ I+X(w)j) V w2(1 + r12)

e_2Imk>x

2e-2Im k>x

x > 0,

x0,

where we have noted that by (6.10)

(6.19)

1+lr12=

l+IP(w)12H2

1+11 x(w)I

V

2

In summary, the scattering eigenmode (6.20)

0(X, t.)

_

f eiW(x/V;i-t) + re-iW(x/

+t)

veiW(P(W)xlV;i-t)

(6.21)

x0,

O(x, s t) = ve"(P(W)x/v/5-t) f elW1s-s'Ic(s') ds',

(x > 0),

describes a steady state in which the incoming wave is partially reflected, with the remainder an evanescent transmitted wave that penetrates the dissipative part of the string with an exponential profile resulting in an excitation of the hidden strings accounting for dispersion and dissipation. The total rate of dissipation in the TDD portion of the string is 00

(6.22)

f axJ(x) dx = J(0)

fw2(1

r

2)

0

References 1. A. Figotin and J. H. Schenker, Spectral theory of time dispersive and dissipative systems, J. Stat. Phys. 118 (2005), no. 1-2, 199 263. 2. , Hamiltonian treatment of time dispersive and dissipative media within the linear response theory, J. Comput. Appl. Math., to appear, available at arXiv:physics/0410127. 3. , Hamiltonian structure for dispersive and dissipative dynamical system, submitted, available at arXiv:math- ph/0608003. 4. V. Jaksic and C.-A. Pillet, Ergodic properties of classical dissipative systems. I, Acta Math. 181 (1998), no. 2, 245 282. 5. P. M. Morse and H. Feshbach, Methods of theoretical physics, McGraw-Hill, New York, 1953. 6. L. Rey-Bellet, Open classical systems, (Grenoble, 2003), Open Quantum Systems. II: The Markovian Approach (S. Attal, A. Joyce, and C.-A. Pillet, eds.), Lecture Notes in Math., vol. 181, Springer, Berlin, 2006, pp. 41 78. 7. M. Reed and B. Simon, Methods of modern mathematical physics. I: Functional analysis, 2nd ed., Academic Press, New York, 1980.


152

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA

IRVINE, IRVINE, CA 92697-

3875, USA

E-mail address: [email protected] DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY, EAST LANSING, MI 48824,

USA Current address: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540,

USA


Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes Volume 42, 2007

Localization at Low Energies for Attractive Poisson Random Schrodinger Operators Rancois Germinet, Peter D. Hislop, and Abel Klein ABSTRACT. We prove exponential and dynamical localization at low energies

for the Schrodinger operator with an attractive Poisson random potential in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity.

1. Introduction and main results The motion of an electron moving in an amorphous medium where identical impurities have been randomly scattered, each impurity creating a local attractive potential, is described by a Schrodinger equation with Hamiltonian (1.1)

HX :_ -A -I- VX

on L2(Rd),

where the potential is given by (1.2)

u(x -

VX(x) SEX

with X being the location of the impurities and -u(x < 0 the attractive potential created by the impurity placed at (. Since the impurities are randomly distributed, it is natural to model the configurations of the impurities by a Poisson process on Rd [35, 37].

The attractive Poisson Hamiltonian is the random Schrodinger operator Hx in (1.1) with X a Poisson process on Rd with density g > 0; Vx being then an attractive Poisson random potential. The attractive Poisson Hamiltonian Hx is an Rd-ergodic family of random self-adjoint operators; it follows from standard results (cf. [27,37]) that there exists fixed subsets of R so that the spectrum of Hx, as well as the pure point, absolutely continuous, and singular continuous components, are equal to these fixed sets with probability one. Poisson Hamiltonians have been known to have Lifshitz tails, a strong indication of localization, for quite a long time [6, 12, 31, 33, 37, 41, 43]. In particular, the 2000 Mathematics Subject Classification. Primary 35P05, 81Q10; Secondary 35J10, 47A10. The second author was supported in part by NSF Grant DMS-0503784. The third author was supported in part by NSF Grant DMS-0457474. This is the final form of the paper. ©2007 American Mathematical Society 153

F. GERMINET ET AL.

154

existence of Lifshitz tails for attractive Poisson Hamiltonians is proved in [33]. But up to recently localization was known only in one dimension [42]; the multidimensional case remaining an open question (cf. [34]). We have recently proved localization at the bottom of the spectrum for Schrodinger operators with positive Poisson random potentials in arbitrary dimension [16,17]. We obtained both exponential (or Anderson) localization and dynamical localization, as well as finite multiplicity of eigenvalues. In this article we extend these results to attractive Poisson Hamiltonians, proving localization at low energies.

Localization has been known for Anderson-type Hamiltonians [1, 7, 20, 28, 30, 32,36]. In random amorphous media, localization was known for some Gaussian random potentials [13,34,44]. In all these case there is an a priori Wegner estimate in all scales (e.g., [7-11,13,26,30,36]). Bourgain and Kenig's proved localization for the Bernoulli-Anderson Hamiltonian, an Anderson-type Hamiltonian where the coefficients of the single-site potentials are Bernoulli random variables [3]. They established a Wegner estimate by a multiscale analysis using "free sites" and a new quantitative version of unique continuation which gives a lower bound on eigenfunctions. Since they obtained weak probability estimates and had discrete random variables, they also introduced a new method to prove Anderson localization from estimates on the finite-volume resolvents given by a single-energy multiscale analysis. The new method does not use the perturbation of singular spectra method nor Kotani's trick as in [7,40], which requires random variables with bounded densities. It is also not an energy-interval multiscale analysis as in [14,29,45], which requires better probability estimates. To prove localization for Poisson Hamiltonians [17], we exploited the probabilistic properties of Poisson point processes to use the new ideas introduced by Bourgain and Kenig [2, 3].

Here we study attractive single-site potentials, which we write as -u, where u is a nonnegative, nonzero L°°-function on Rd with compact support, with (1.3)

u_XA,_ (o) < u < u+XA,+(o)

for some constants u+, 6+ E ] 0, oo[.

(AL(x) denotes the box of side L centered at x E Rd.) It follows that HX is essentially self-adjoint on C°°(W') and o (HX) = ll. with probability one [6,37]. We

show that the conclusions of [17] hold in an interval of negative energies of the form ]-oo,Eo(o)] for some Eo(o) < 0. We obtain both exponential (or Anderson) localization and dynamical localization, as well as finite multiplicity of eigenvalues. For a given set B, we let XB be its characteristic function, Po (B) the collection

of its countable subsets, and #B its cardinality. Given X E Po (A) and A C B, we set XA := X n A and NX(A) :_ #XA. We write I A I for the Lebesgue measure

of a Borel set A C lRd. We let AL (x) := x + (-L/2, L/2)d be the box of side L centered at x E Rd. By A we will always denote some box AL(x), with AL denoting

a box of side L. We set xx := XA, (x), the characteristic function of the box of side 1 centered at x E ][8d. We write (x) := 1 + x12, T(x) := (x)" for some fixed v > d/2. By Ca,b,..., ca,b,..., Ka,b,..., etc., will always denote some finite constant depending only on a, b..... A Poisson process on a Borel set B C W1 with density o > 0 is a map X from a probability space (f2, P) to Po(B), such that for each Borel set A C B with IAA < 00 the random variable Nx(A) has Poisson distribution with mean gJAI, i.e.,

ATTRACTIVE POISSON RANDOM SCHRODINGER OPERATORS

P{Nx(A) = k} _

(1.4)

(pl`4I)k e-eIAI k!

155

for k =0,1,2,...,

and the random variables {Nx(Aj )}; 1 are independent for disjoint Borel subsets {A;}" 1 (e.g., [25,39]).

Theorem 1.1. Let Hx be an attractive Poisson Hamiltonian on L2(I[8d) with density p > 0. Then there exists an energy E0 = Eo(p) < 0 for which the following holds IP-a.e.: The operator Hx has pure point spectrum in ]-oo, Eo] with exponentially localized eigenfunctions, and, if 0 is an eigenfunction of Hx with eigenvalue E E ]-oc, Eo] we have, with mE := g z Eo - E < mEo := s - 2 Eo that

IX
1 and s E ]0, 1[ such that for all eigenfunctions V), 0 (possibly equal) with the same eigenvalue E E ]-oo, Eo] we have (1.6)

JJXx

MM

JXycbM < CxIIT-1011 JT

for all x, y E Zd.

10IIe(y)Te-Ix-yls

In particular, the eigenvalues of Hx in ]-oo, Eo] have finite multiplicity, and Hx exhibits dynamical localization in ]-oo, E0], that is, for any p > 0 we have (1.7)

II(xPe-itxxX]-.,Eo](Hx)Xo112
0 (20), as in (1.1)-(1.3). We start by showing that the attractive Poisson Hamiltonian is self-adjoint and we have trace estimates needed in the multiscale analysis.

Proposition 2.1. The attractive Poisson Hamiltonians HX and Hy are essentially self-adjoint on C0o(Rd) with probability one. In addition, we have

tr{T-le-tHXT-1} < oo for all t > 0 ]P-a.e.,

(2.4)

and

tr(T-'X]_.,E](HX)T-1) < oo for all E E ll P-a.e.

(2.5)

Since the potential is attractive, it may create infinitely deep wells. This is controlled by the following estimate.

Lemma 2.2. Given a box A we set (2.6)

116+,A :_ (26+7.d) f1 A.

There exists L* = L* (d, o, 6+), such that for any L > L* we have (2.7)

P{Ny(A26+(j)) < ologL Vj E J6+,AL} > 1 - L-2(loglogL)/3

and (2.8)

]P{11 xALVyii

u+o log L} > 1 -

L-2(loglogL)/3.

It follows that for P-a. e. w we have VX(,,J) (x) > VY(.) (x) > -c,, log(1 + JxJ)

(2.9)

for all x c ]18d

where c, > 0 (depending also on p, 5+,0, u+). PROOF. We may assume o log L >_ 1. Standard bounds on Poisson random variables (cf. [17, Eq. (2.7)]) give (2.10)

P { NY (A26+ (X)) > o log L }
Li (d, o, S+). It follows that for L > L* (d, o, 6+) we have

P{NY(A2b+(j)) < ologL Vj E 16+,AL} ? 1 -

>1-L

2L)d L -3(log log L)/4 b 2(log logL)/3


157

Now, for any x E AL there exists j E .Ub+,AL s.t. Ab+ (x) C A26+ (j). Hence, we also have (2.12)

-L-2(log)°g L)/s

P{Ny (A5+ (x) fl AL) 1

But if the event in (2.12) occurs, it follows from (1.3) that (2.13) JVY (x) < u+Q log L Vx E AL, and (2.8) follows, since 0 > Vx(x) > VY (x)

(2.14)

because of (2.2). The Borel-Cantelli Lemma now gives (2.9). In the one-dimensional case an estimate similar to (2.9) can be found in [24]. PROOF OF PROPOSITION 2.1. In view of (2.9), it follows from the Faris-Levine Theorem [38, Theorem X.38] that Hx and HY are essentially self-adjoint on C°° (Rd) with probability one.

The trace estimate (2.4) follows from Gaussian bounds on heat kernels [4, Lemma 1.7], which hold lF-a.e. in view of (2.9). As a consequence, we have (2.15)

tr(T-1X]_o°,E] (Hx)T-1) tr(T-le-Hxe

= < e2E

2HxX]-.,E](Hx)e-HxT-1)

tr(T-1e-2HxT-1)

< oo

for all E E R P-a.e.

Given two disjoint configurations X, Y E Po (Rd) and ty = {t(}SEY E [0, 1] Y, we set (2.16)

Hx,(Y,t,) :=-A+ Vx,(Y,ty),

where Vx,(y,ty)(x) := Vx(x) + E t(u(x - ). (EY

In particular, given Ey E {0,1}Y we have, recalling (2.1), that (2.17)

Hx,(Y,EY) = Hxux(Y,EY)

We also write H(Y,ty) := Hg,(Y,ty) and (2.18)

H4J := Hx(w) =

3. Finite volume The multiscale analysis requires finite volume operators,which are defined as follows. Given a box A = AL(x) in Rd and a configuration X, we set Hx,A :_ -DA + Vx,A on L2 (A), where AA is the the Laplacian on A with Dirichlet boundary condition, and (3.1)

(3.2)

VX,A := XA VXA

with Vx as in (1.2).

The finite volume resolvent is RX,A(Z) :_ (HX,A - Z)-'-

We have DA = VA VA, where VA is the gradient with Dirichlet boundary condition. We sometimes identify L2(A) with XAL2(Rd) and, when necessary, will use subscripts A and Rd to distinguish between the norms and inner products of L2(A) and L2(Rd). Note that we always have (3.3)

XAVx,A = XA Vx,Al,

F. GERMINET ET AL.

158

where

A = AL(x) := AL-6+(x)

(3.4)

with S+ as in (1.3),

which suffices for the multiscale analysis. The multiscale analysis estimates probabilities of desired properties of finite volume resolvents at energies E E R. (Lp1 means for some small S > 0. We will write U for disjoint unions: C = A U B means C = A U B with A f1 B = 0.)

Definition 3.1. Consider an energy E E ll and a rate of decay m > 0. A box AL is said to be (X, E, m)-good if II Rx,AL (E) II < eL1

(3.5)

and (3.6)

IIXXRx,AL (E)XyII
-u+o log L for all x E A,

and (3.18)

H(Y,t,,) A > -u+ologL for all ty E [0,1]Y.

PROOF. The estimate (3.16) is an immediate consequence of (3.9). It also follows from (3.9), by the same argument used for (2.13), that for all A-acceptable' configurations Y we have the lower bound (3.17), from which we get (3.18) since H(Y,ty),A > HY,A > VYA.

Lemma 3.6. There exists a scale L = L(d, 0, 6+) < oo, such that if L > L we have P{52(°) } > 1

(3.19)

- L-('-g log L)/2

PROOF. Recalling (2.7), we have (3.20)

IED{S2(°) } > 1

- L-z(iog log L)/s - 4d0(Ld-1 + Ld)T1L - 202Ldrlid,, 0

and hence (3.19) follows for large L.

Lemma 3.6 tells us that inside the box A, outside an event of negligible probability in the multiscale analysis, we only need to consider A-acceptable configurations of the Poisson process Y. Fix a box A = AL(x), then (3.21)

XAY

Nx(A17L(j)) = Ny(A,11L(j))

for all j E JJA

F. GERMINET ET AL.

160

introduces an equivalence relation in both QA °') and QA °); the equivalence class of

X in Q(°') will be denoted by If X E Q°), then [X]A = [X]A n QA°) is its equivalence class in Q. Note that [X]' = [XA]'. We also write [A]A := U [X]A

(3.22)

for subsets A C Q(o)

XEA

The following lemma [17, Lemma 3.6] tells us that "goodness" of boxes is a property of equivalence classes of acceptable' configurations: changing configurations inside an equivalence class takes good boxes into just-as-good (jgood) boxes. Proceeding as in the lemma, we find that changing configurations inside an equivalence class takes jgood boxes into what we may call just-as-just-as-good (jjgood) boxes, and so on. Since we will only carry this procedure a bounded number of times, the bound independent of the scale, we will simply call them all jgood boxes.

Lemma 3.7 ([17]). Fix E0 > 0 and consider an energy E E [0,E0]. Suppose the box A = AL (with L large) is (X, E, m) -good for some X E Q(O'). Then for all Y E [X]' the box A is (Y, E, m) jgood, that is, (3.23)

IIRY,A(E)II < eLl

+OL/4

eL'

and

(3.24)

IIxxRY,A(E)XyII

L

0

Moreover, if X, Y, X U Y E Q(O') and the box A is (X, Y, E, m)-good, then for all X1 E [X]' and Y1 E [Y]' we have X1 U Y1 E [X U Y]'A, and the box A is (Xi, Y1, E, m) -jgood as in (3.23) and (3.24).

We also have a lemma [17, Lemma 3.8] about the distance to the spectrum inside equivalence classes.

Lemma 3.8 ([17]). Fix E0 > 0 and consider an energy E E [0, E0] and a box A = AL (with L large). Suppose dist(E, Q(HX,A)) < TL for some X E Q(O') , where

7- « TL < 2. Then (3.25)

dist(E, a(HY,A))
Ld- for all boxes AL'- CAL.

(3.29)

We also set (3.30)

CA,B := CA,B,g = [B]A.

Definition 3.10. Given A = AL (x), a A-bevent (basic event) is a subset of Q(O) of the form (3.31)

CA,B,B',S :_ {Y E [B U B' U S]A} n {X E CA,B,s} n {X' E CA,B',s},

where we always implicitly assume B U B' U S E JA. In other words, the A-bevent CA,B,B',S consists of all w c Q(O) satisfying Nx(w) (AIL (j)) = 1 Nx'(w) (A,7L (j)) = 1

(3.32)

1

NY(,,,) (AqL (j)) = 0

if j E B, if j E B', if j E S, if j E UA\(B U B' U S).

CA,B,B',S is a A-dense bevent if S satisfies the density condition (3.29). In addition, we set (3.33)

CA,B,B'

CA,B,B',O = {X E CA,B} n {X' E CA,B'}.

The number of possible bconfsets and bevents in a given box is always finite. We always have (3.34)

CA,B,B',S C {X E CA,B,S} n Q(O),

(3.35)

CA,B,B',S C CA,m,m,BuB'us = {Y E [B U B' U S]A}.

Note also that it follows from (3.15), (3.26) and (3.33) that (3.36)

S2A°)

=

U

CA,B,B'

{(B,B');BUB'E,7A}

Moreover, for each S1 C S we have (3.37)

CA,B,S =

U CA,BUS2,S\S1, s2 c s1

(3.38)

CA,B,B',S = U CA,BUS2,B'u(S1\s2),S\S1 S2CS1

Lemma 3.7 leads to the following definition.

Definition 3.11. Consider an energy E E R, m > 0, and a box A = AL(X)The A-bevent CA,B,B',s and the A-bconfset CA,B,S are (A, E, m)-good if the box A is (B, S, E, m)-good. (Note that A is then (w, E, m) jgood for every w E CA,B,B',S)

Those (A, E, m)-good bevents and bconfsets that are also A-dense will be called (A, E, m)-adapted.

F. GERMINET ET AL.

162

Definition 3.12. Consider an energy E E R, a rate of decay m > 0, and a box A. We call QA a (A, E, m)-localized event if there exist disjoint (A, E, m)-adapted bevents {CA,B_B,sv}i=1,2,...,r such that z

(3.39)

QA = U CA,B,,B

,s,.

i=1

If 11A is a (A, E, m)-localized event, note that 11A C QA(0) by its definition, and hence, recalling (3.38) and (3.33) , we can rewrite S1A in the form J

QA = U

(3.40)

j=1

where the {CA,A3,A' }j=1,2,,..,J are disjoint (A, E, m)-good bevents. We will need (A, E, m)-localized events of scale appropriate probability.

Definition 3.13. Fix p > 0. Given an energy E E R and a rate of decay m > 0, a scale L is (E, m)-localizing if for some box A = AL (and hence for all) we have a (A, E, m)-localized event PA such that lP{1 A} > 1 - L-P.

(3.41)

4. A priori finite volume estimates Given an energy E, to start the multiscale analysis we will need, as in [2, 3], an a priori estimate on the probability that a box AL is good with an adequate supply of free sites, for some sufficiently large scale L. The multiscale analysis will then show that such a probabilistic estimate also holds at all large scales.

Proposition 4.1. Let HX be an attractive Poisson Hamiltonian on L2(Rd) with density o > 0, and fix p > 0. Then there exists a scale Lo = Lo(d, u, o, p) < 00, such that for all scales L > Lo, setting (4.1)

6L :_ (o-'(p +d+1)logL)11d,

EL := -2u+ologL,

and

EL - E < mL := 2

- 2 EL for all E E ]-oo, EL], the scale L is (E, mL,E)-localizing for all energies E E ]-oo, EL]. (4.2)

mL E

2

2

PROOF. Let A = AL(x), and let 8L and EL be as in (4.1). If Y E Q(O), it follows from (3.18) and the Combes-Thomas estimate (e.g., [19, Eq. (19)]) that for all E E ]-oo, EL] and all ty E [0, 1] Y we have, with mL(E) as in (4.2), that (4.3)

JR(Y,tY),A(E)II 4v' d.

We now require L > 8L + 6+, and set (4.5) (4.6)

J:= { j E x+ SLzd n A; A5L (j) C All JA :_ {S E JA; Ns(ASL (j)) > 1 for all j E J}.


163

If S E ,7A, the density condition (3.29) for S in A follows from (4.6), and it follows

from (4.3) and (4.4) that CA,o,o,s is a (A, E, mL,E)-adapted bevent for all E E ]-oo, EL] if L > L1(d, u, o, p). We conclude that

QA = U CA,O,O,S = U {Y E [S]A}

(4.7)

SEJA

SEJA

is a (A, E, mL,E)-localizing event for all E E ]-oo, EL]. To establish (3.41), let S'L := 6L/2d and consider the event Q(') := {Ny (A,L (j)) > 1

(4.8)

for all

j c J}.

We have

(4.9){)} > 1 - ()de_2 if L >

ee/(P+d).

Since QA

=1

(p + d + 1)LP+1 log L > 1

LP 1,

Q(t) f1 Q(0), (3.41) follows from (4.9) and (3.19) for

L > Lo (d, u, CQ, p).

5. The multiscale analysis and the proof of localization The Bourgain-Kenig multiscale analysis, namely [3, Proposition A'], was adapted to Poisson Hamiltonians in [17, Proposition 5.1]. To apply the latter to attractive Poisson Hamiltonians we must show that the requirements of this multiscale analysis are satisfied. More precisely, we must show that attractive Poisson Hamiltonians satisfy appropriate versions of Properties SLI (Simon - Lieb inequality), EDI (eigenfunction decay inequality), IAD (independence at a distance), NE (average number of eigenvalues), and GEE (generalized eigenfunction expansion); see [18]. The Wegner estimate is proved by the multiscale analysis; it is not an "a priori requirement." Since events based on disjoint boxes are independent, we have Property IAD. Property GEE is satisfied in view of (2.4), and we also have (2.5), which is needed in the multiscale analysis.

But Properties SLI, EDI and NE require some care and modification. In a box AL we always work with AL-acceptable configurations X, whence the potential VX,AL satisfies the lower bound (3.17). Inside the box AL, Properties SLI and EDI (see [3, Section 2; 21, Theorem A.1]) are governed by the same constant YE,L given in [21, Eq. (A.2)], and hence for AL-acceptable configurations we have (5.1)

SUP

'YE,L 0 and p = gd-. Then there exist an energy E0 = Eo(o) < 0 and a scale Lo = Lo(g), such that setting mE := a 2Eo - E < mE,, 4 -2Eo, the scale L is (E, ME) -localizing for all L > Lo and E E] - oo, Eo]. Theorem 1.1 now follows from Proposition 5.1 as in [17, Proposition 6.1].

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2.

3.

4. 5.

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J.-M. Combes, P. D. Hislop, and F. Klopp, Holder continuity of the integrated density of states for some random operators at all energies, Int. Math. Res. Not. 2003, no. 4, 179-209. 9. J.-M. Combes, P. D. Hislop, F. Klopp, and S. Nakamura, The Wegner estimate and the integrated density of states for some random operators, Proc. Indian Acad. Sci. Math. Sci. 112 (2002), no. 1, 31-53. 10. J.-M. Combes, P. D. Hislop, and E. Mourre, Spectral averaging, perturbation of singular spectra, and localization, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4883-4894. 11. J.-M. Combes, P. D. Hislop, and S. Nakamura, The LP-theory of the spectral shift function, the Wegner estimate and the integrated density of states for some random operators, Comm. Math. Phys. 218 (2001), no. 1, 113-130. 12. M. Donsker and S. R. S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (1975), no. 4, 525-565. 13. W. Fischer, H. Leschke, and P. Miiller, Spectral localization by Gaussian random potentials in multi-dimensional continuous space, J. Statist. Phys. 101 (2000), no. 5-6, 935-985. 14. J. Frohlich, F. Martinelli, E. Scoppola, and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), no. 1, 21-46. 8.

15. J. Frohlich and T. Spencer, Absence of diffusion with Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), no. 2, 151-184. 16. F. Germinet, P. Hislop, and A. Klein, On localization for the Schrodinger operator with a Poisson random potential, C. R. Acad. Sci. Paris Ser. I Math. 341 (2005), no. 8, 525-528. 17. , Localization for Schrddinger operators with Piosson random potential, preprint. 18. F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), no. 2, 415-448. 19. , Operator kernel estimates for functions of generalized Schrodinger operators, Proc. Amer. Math. Soc. 131 (2003), no. 3, 911-920. 20. , Explicit finite volume criteria for localization in random media and applications, Geom. Funct. Anal. 13 (2003), no. 6, 1201-1238. 21. , A characterization of the Anderson metal-insulator transport transition, Duke Math. J. 124 (2004), no. 2, 309-351. 22. , New characterizations of the region of complete localization for random Schrodinger operators, J. Stat. Phys. 122 (2006), no. 1, 73-94.


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, in preparation. 24. I. W. Herbst and J. S. Howland, The Stark ladder and other one-dimensional external field problems, Comm. Math. Phys. 80 (1981), no. 1, 23-42. 25. J. F. C. Kingman, Poisson processes, Oxford Stud. Probab., vol. 3, Oxford Univ. Press., New 23.

York, 1993.

26. W. Kirsch, Wegner estimates and Anderson localization for alloy-type potentials, Math. Z. 221 (1996), no. 3, 507-512. 27. W. Kirsch and F. Martinelli, On the ergodic properties of the spectrum of general random operators, J. Reine Angew. Math. 334 (1982), 141-156. 28. W. Kirsch, P. Stollmann, and G. Stolz, Localization for random perturbations of periodic Schrodinger operators, Random Oper. Stochastic Equations 6 (1998), no. 3, 241-268. 29. A. Klein, Multiscale analysis and localization of random operators, Random Schrodinger Operators: Methods, Results, and Perspectives, Panor. Synthese, Soc. Math. France, Paris, to appear. 30. F. Klopp, Localization for continuous random Schrodinger operators, Comm. Math. Phys. 167 (1995), no. 3, 553-569. 31. , A low concentration asymptotic expansion for the density of states of a random Schrodinger operator with Poisson disorder, J. Funct. Anal. 145 (1997), no. 2, 267-295. 32. , Weak disorder localization and Lifshitz tails: continuous Hamiltonians, Ann. Henri Poincare 3 (2002), no. 4, 711-737. 33. F. Klopp and L. Pastur, Lifshitz tails for random Schrodinger operators with negative singular Poisson potential, Comm. Math. Phys. 206 (1999), no. 1, 57-103. 34. H. Leschke, P. Miiller, and S. Warzel, A survey of rigorous results on random Schrodinger operators for amorphous solids, Markov Process. Related Fields 9 (2003), no. 4, 729-760. 35. I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the theory of disordered systems, Wiley-Intersci. Publ., Wiley, New York, 1988.

36. F. Martinelli and H. Holden, On absence of diffusion near the bottom of the spectrum for a random Schrodinger operator on L2(IR"), Comm. Math. Phys. 93 (1984), no. 2, 197-217. 37. L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Grundlehren Math. Wiss., vol. 297, Springer, New York, 1992. 38. M. Reed and B. Simon, Methods of modern mathematical physics. II: Fourier analysis, selfadjointness, Academic Press, Boston, MA, 1975. 39. R.-D. Reiss, A course on point processes, Springer Ser. Statist., Springer, New York, 1993. 40. B. Simon and T. Wolff, Singular continuum spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75-90. 41. P. Stollmann, Lifshitz asymptotics via linear coupling of disorder, Math. Phys. Anal. Geom. 2 (1999), no. 3, 279-289. 42. G. Stolz, Localization for random Schrodinger operators with Poisson potential, Ann. Inst. H. Poincare Phys. Theor. 63 (1995), no. 3, 297-314. 43. A.-S. Sznitman, Brownian motion, obstacles and random media, Springer Monogr. Math., Springer, Berlin, 1998. 44. N. Ueki, Wegner estimates and localization for Gaussian random potentials, Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, 29-90. 45. H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124 (1989), 285-299. DEPARTEMENT DE MATHEMATIQUES, UNIVERSITE DE CERGY-PONTOISE, SITE DE SAINT MARTIN, 2 AVENUE ADOLPHE CHAUVIN, 95302 CERGY-PONTOISE CEDEX, FRANCE

E-mail address: germinet@math. u-cergy . f r DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KENTUCKY, LEXINGTON, KY 40506-0027,

USA

E-mail address: [email protected] DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA -IRVINE, IRVINE, CA 92697-

3875, USA E-mail address: [email protected]


On the Influence of Random Perturbations on the Propagation of Waves Described by a Periodic Schrodinger Operator Yuri A. Godin, Stanislav Molchanov, and Boris Vainberg ABSTRACT. We study the variance of the propagation matrix of a periodic Schrodinger operator assuming that the monodromy matrix undergoes small random perturbations over each period. It is shown that if the frequency of propagation lies inside the band, then the total variance is proportional to the sum of the variances of single perturbations, i.e., obeys the probability law. However, if the frequency of propagation is close to the band edge, the resulting variance is proportional to the sum of cubic roots of the variances that make propagation highly sensitive to small perturbations.

1. Introduction The product IIN of N >> 1 random unimodular (or symplectic) matrices is a typical object in the theory of random media (localization, propagation of electromagnetic waves through periodically stratified media, evolution of the magnetic field in turbulence flows of a conducting fluid, and more). One can find some general facts on IIN in [4, 7]. There are two approaches to the asymptotic analysis of IIN, N -> oo. One can study the a.s. behavior of IIIINII2 = [tr(IINll It is known (the Furstenberg theorem) that with the minimum of assumptions there exists so called quenched (or a.s.) Lyapunov exponent which is the (positive) limit

'fq = lim

112 > 0. NN Another approach requires the prior averaging over the ensemble of realizations and the study of the norm moments of IIN:

N-oo

In I

rya(p) = lim N- oo

In E I I IIN I I2

N These limits are called the annealed Lyapunov exponents of order p, and due to Jensen's inequality (p + 1) 1'Y(P+1) > p 1'Y(P) >

ryq.

2000 Mathematics Subject Classification. Primary 35R60, 34F05; Secondary 35Q60, 35B27. This is the final form of the paper. ©2007 American Mathematical Society 167

YU. A. GODIN ET AL.

168

The goal of this paper is to study how a small random perturbation may affect transmission through a long slab of a periodic media, in particular, near the band edge of the spectrum. Let H be a Schrodinger operator on L2(R) with a potential q = q(x) supported on an interval (0, f): z

H V) = -

(1.1)

+ q(x) V) (x),

dx2

and let V) be the scattering solution for H, i.e., (1.1) holds and e"'x + r(w)e-'Wx,

4'(x) =

t(w)e'WX,

x < 0; x > f.

Then the transmission coefficient t(w) satisfies the following relation (see, for example, [3]) : 4

It(w)I2 -

(1.2)

IITW(o,f)II2+2' where TW (a, b) is the Pruffer transfer matrix for the interval (a, b) TW (a, b) [w

,oa(a)]

-

[W -10b1

Ml

for any solution _ V), (X) of HV)W = w20W, and I I T 112 is the Hilbert - Schmidt norm of the matrix T = [ti,j]

IITII2 = trTTT = Eti,7 We will also consider 1D Schrodinger operator with a periodic potential q(x):

HO =

(1.3)

_dz 0

d0

+ q(x),O (x),

q(x + L) = q(x),

on L2 (1I) and operator HN with truncated potential qN containing N periods: z

(1.4)

HNO

ax2

+qN(x),O,

qN(x) = I[O,NL)(x)q(x)

Spectral properties of operators (1.3) or (1.4) depend on the monodromy operator MW (transfer matrix over the period L) (1.5)

W(0)

MW Lw

1

(0)]

_

AW(L)

-

( L)

(in the Pruffer form). Here 0,,, is an arbitrary solution of HO = w2o. If tN(w) is the transmission coefficient for the operator HN then (1.2) implies (1.6)

ItN(w)I2 =

II4

IIMN 2 +2

Thus, transmission through the truncated periodic medium depends on the upper bound of IIMW 112-

Let us suppose now that the monodromy operators through each period [(n - 1)L, nL], n = 1, 2, ... , N, contains some noise (defects, absorption, etc.), i.e., the corresponding propagator through the period M,,,,, = TW([(n - 1)L, nL]) is a

RANDOM PERTURBATIONS OF A PERIODIC SCHRODINGER OPERATOR

169

small random perturbation of the monodromy operator M. Formula (1.6) takes the following form in this case ItN(w)12 =

4 11

2

lln=1 Mw,n112

We will consider two models where the perturbation is either additive or multiplicative. In the first case, the propagator Mw,n has the form (1.8)

Mw,n = Mw + QWn,

] indepenwhere a is a small parameter, and Wn = [ fn(n 77" /Ln . Here Sn, ?7n, (n, An are dent (for all n) N (0, 1) random variables depending only on w. In the multiplicative case we assume that

(1.9)

Mw,n = Mw (I + UV,),

n- ] . Here Sn, 77n, yn are random variwhere a is a small parameter, and Vn = [ En S n -fn ables which are not necessary independent, but have a joint Gaussian distribution with the covariance (1.10)

tn

B = [bi,7] = E SnTM

n(n

n'7n

en (n

ln

77n (n

77nr

cn

sn

Clearly matrix (1.8) can be written in the form (1.9) and vice versa. The difference between the additive and multiplicative perturbations is in the assumptions on the random variables. In the first case we assume that the entries of the perturbation matrix are independent identically distributed random variables (isotropic case). This assumption makes the problem simpler. However, the disadvantage of this model is that det Mw,n - 1 for small a has the order 0(a). Since the determinant

of the transfer matrix for operator (1.1) is equal to one, the additive model is applicable only to the situation when q(x) is complex valued (lossy medium) or the operator (1.3) contains a term with first derivative. In the multiplicative case, the

entries of the matrix Vn are not independent and the trace of Vn is zero. This leads to the fact that det Mw,n = 1 + O(a2),

or--+ 0,

which makes the multiplicative perturbations more realistic than the additive ones. We are going to show now that the multiplicative model appears as a first order (in a) approximation in the case of a small random perturbation of a deterministic potential. Consider the perturbation of periodic Schrodinger equation by the white noise e(x) (1.11)

-y" + [V(x) + a(x)]y = w2y,

V(x + L) = V (x),

where a is a small parameter. We represent two linear independent solutions yi (x), i = 1, 2, of (1.11) in the form (1.12)

yi(x) ='0i (x) + a 1 G(x, s)e(s)yi(s) ds,

YU. A. GODIN ET AL.

170

where G(x, s) is the Green's function G(x, s) = V)2(x)01(s) -01(x)02(s),

(1.13)

0,

s < x, s > x,

and 01(x), 02(X) is the fundamental set of solutions of the unperturbed equation (1.11). Within the linear approximation in a, solution of (1.11) is given by ds + O(a2),

yi(x) = 4bi(x) + Q J x G(x,

(1.14)

i = 1, 2.

0

If we denote

/L

77ij = J

(1.15)

ds,

i=1,2,

then the values of y(L) and y'(L) in the linear approximation can be express from (1.14) as follows

yi(L) = Yi(L)

'/1

1Ji(L) _ i(L)

Q{ui1iW2(L)

-gi201(I')],

i=1,2.

Therefore, the monodromy matrix M of the perturbed equation (1.11) has the form

(1.16) M - [W yly

_

y2(L))j W)2(L)

(L)

b1(L)

- [W-1Oi(L)

V2 (L)

+ Q [W

= MG, (I+O' [

rj11 , (L) - ij12 01 (L) 1114 2(L) - W ,

2 (L) - wrj2 z 1(L) 77120'(L) - q22 (L)

-7712

W-1 ?711

7112

where Mu, is the unperturbed monodromy matrix '1(L)

(1.17)

z(L)

W

The problem under investigation depends on two parameters: Q -4 0 and N oo. Our goal is to find the error in the transfer matrix over N periods, N (1.18)

IIN = fl M,,,,,,, n=1

due to a random perturbation of the monodromy matrix on each period. We deal with the second moment, or to be more precise, we estimate the scalar variance of IIN, (1.19)

DN = Etr(IINIIN) - tr(HNnN)Ia=o,

as a -p 0, N oo. If DN is small, then the variance of any smooth function of 11N (and the covariance of any smooth vector or matrix function of IIN) is small. In particular, the variance of the transmission coefficient has the same order as DN. In the linear theory, the error in the process of summation is increasing slowly (as the function of the sample volume). If SN = a 1 + ... a N where (j are independent N(0,1) random variables, then the variance of SN is equal to U2 N, i.e., the error due to summation of N small random perturbations is negligible if


171

(12N 1 in the gaps (open intervals separating

the bands), equations 0(w) = ±1 give the edges of the bands. The dispersion relation k = k(w) (w belongs to a band) is defined by the formula A(w) = cos k(w).

(2.1)

The points where the edges of two neighboring bands coincide are called degenerate

gaps. It is well-known (see [2] or [6]) that the monodromy operator has a Jordan block with a superdiagonal and a multiple eigenvalue ±1 when the frequency w coincides with a nondegenerate bandgap edge. There is a simple expression for the Hilbert - Schmidt norm of powers of the monodromy operator through the dispersion (2.1) (see [5]): (2.2)

IIM. 112 = (IIM- 112 - 2)

sine Nk(w) + 2.

sin k(w)

Note that det M, - 1 implies that x = 11M,112 - 2 > 0,

(2.3)

and

x > 0 at the band edges.

(2.4)

In fact, IIMw 112

a2 + 02 + y2 + b2 = (a

- 6)2

+ (Q + y)2 + 2 det Mu, > 2, = and the equality is possible only if a = S, 3 = -'y. One can easily check that the latter cannot occur at the band edges since M, has a nontrivial Jordan cell

structure there.

YU. A. GODIN ET AL.

172

3. Error in the transmission due to additive perturbations Theorem 3.1. The variance DN has the following asymptotic behavior when w is fixed, a -+ 0, and N is bounded or growing not very fast. If w is an interior point of a band and a2N < C < oo, then (3.1) DN = c(w)a2N(1 + 0(1)). If w is a spectral band edge and a213N < C < oc, then DN = c(w)a2/3N(1 + 0(1)).

(3.2)

The proof of this statement is based on the following two simple propositions. Let

AN = E(HNfN), xN = XN(w, a) = tr AN, N>1, and A0 = I (where I is the identity matrix), x0 = tr I = 2. Then (3.3)

DN = xN (w, a) - xN (w, 0),

(3.4)

N>1.

Proposition 3.2. Let N = 1 and therefore, Al = E(M,,1MW1). Then Al = M'MT + a2x0I. PROOF. In fact, for W 1 = [

µ

]

we have

2+772 2+11z Al=M"MT+a2E(W1WT)=MWMT+a2E 1c (+77A +/1 J

= M'MW + 2a21. Proposition 3.3. If A = AT is a real nonrandom symmetric matrix then E[MW + aWn]A[MT + aW,a] = MWAMT + a2(tr A) 1. PROOF. In fact,

E[ML, + aWn]A[MT + aWn] = MWAMT + a2 E(WnAW,a).

It remains to note that for W = [ s µ ] , WT = [ 77 C ] , A = [ b d ], we have P E(WAWT) = E [a( + by b( + d1] Lk µJ

a+d 0

0

_ trA

a+d - [

0

0

trA]

PROOF OF THEOREM 3.1. Applying successively Proposition 3.3 and using Proposition 3.2, we get

An = MAn_1MT +a2(trAn,_1)I = M[MAn_2MT +a2(trAn,_2)I]MT +a2(trAn_1)I = M2An_2(MT)2 + a2(trAn-2)MMT + a2(tr An-1)I n-1

= Mn(MT)n + a2E(tr As s=0

)Mn-s-1 (MT)n-s-1 , l


173

where n > 1, and A0 = I. We apply the operator tr to both sides of the equation above and put µs =trMS(MT)S. This implies

xn = An + Q2(xoAn-1 + ... + xn-1µo)

(3.5)

In particular,

xo=µo=2, 2

x1 =,Ul + or X0A0'

X2 = Y2 + a2 (X0µ1 + x1µ0).

Let us introduce generating functions of sequences {xn} and {µn}: cc

X (Z)

00

_

µ(z)

xnzn,

_

n=0

/dnzn n=o

The sequence {µn} is bounded for each fixed value of w0 (see (2.2)). Then (3.5) implies that IxnI < C(n+1)Ixn_ll. Thus, the generating functions are well defined. It follows from the recursive relation (3.5) that X (Z) = µ(z) + za2X (z)µ(z),

and therefore,

X(z) _

/2(z)

1 - za2µ(z) Now we calculate µ(z). We rewrite (2.2) in the form

As -

xsin2 sk sine k

+2.

where x > 0. Hence sine sk

°O

2

µ(z)=Exsin2kz s +1-z s=0

Calculating the sum of geometric series 00

CK)

pn cos nx = Re E pneinx = n=0

n=o

1 -pcosx 1- 2pcosx+p2'

we get

_ A(Z)

xz(1 + z)

2

1-z+ (1 - z)(1 - 2z cos 2k + z2) 2(1 - 2zcos2k+z2) + .z(1 +z)

(1-z)(1-2zcos2k+z2) The function X(z) = µ(z)/(1 - a2zµ(z)) = P2(z)/Q3(z) is a rational function. Here (3.6)

P2(z) = 2(1 - 2z cos 2k(w) + z2) + xz(1 + z),

(3.7)

Q3 (z) = (1 - z) (1 - 2z cos 2k(w) + z2) - a2zP2 (z).

Assume that the denominator Q3(z) has three simple roots 0, z1, zl. Then P2 (z) Q3 (z)

_

CO

z0 - z

+

cl

cl

zl - z

z1 - z

YU. A. GODIN ET AL.

174

where P2(zo)

c = c1

Z 1 )

2 0 . 2 )

°

=

Z 1 )

P2(zl)

(1 + 2a2)(zo - zl)(zl - zl)

Expansion of each simple fraction into geometric series implies CO

cl

61

x,v = N+1 + N+1 + N+1. z° zI zl This explicit formula allows to find the asymptotic behavior of XN as a ---> 0, (3.8)

N-+oo. Consider now two cases when w belongs to a band and when w is a spectral band edge. If w belongs to a band then I cos 2k (w) < 1 and therefore the roots of the polynomial Q3 with or = 0 are z = 1 and z = e+2ik("). Clearly, when or -p 0, the roots have the form zo = 1 + ao(w)a2 + 0(a4), z1 = e2ik + al + 0(a4) Z2 = z1 (w)a2

Thus

xN(w a) = /

co(w)e-ao(W)°2N

(I + o(1)) + 2 Re

c1(w)e-a1(W)a2N (1

+ 0(1)),

a-p0, a2N 0 due to (2.4). Thus, the roots of Q3 can be found from the equation 1 - z + µW (z) = 0,

µ=

a2/3ei2j7r/3

_ (zP2(z))1/3,

where cp is analytic in a neighborhood of z = 1, and cp(1) = (2x) 1/3 > 0. Hence the roots of Q3have the form z., = 1 + aj (w)a2/3 + 0(a4/3) j=0,1,2, aj(w) = ao (w)ei2j7,/s when a -+ 0. Here zo is real, z2 = z1. Now (3.8) implies that for or -+ 0, a2/3N < C, (3.9) XN(W, a) =

co(w)e-a°(-)2/3N (1

+ 0(1)) + 2 Re c1(w)e-",

(")0r2/3N

(1+0(1)),

and this leads to the second statement of the theorem.

4. Multiplicative perturbation Theorem 4.1. If w is an interior point of a spectral band, then the first statement (3.1) of Theorem 3.1 holds. If w is a spectral band edge, then the second statement of Theorem 3.1 (3.2) is valid provided that the correlation matrix B(w) is nondegenerate at that frequency. PROOF. In the multiplicative case, perturbation of the monodromy matrix over

the period has the form (1.9) and the variance is defined by (1.19), where IIn is given by (1.18). Similar to the additive case, An is evaluated by induction.


175

The following formula plays the role of Proposition 3.2 (4.1)

Al =

E(M41(I + aV1)(I + 0,VTI )MT) LO

= M,(I+U2E(V1VT))MW. Formula (3.3), with N = n, after evaluating the expectations with respect to random variables with subindexes s < n, becomes (4.2)

An = E(M, (I + oVn)An_1(I + o'V,n )MW ),

n > 1.

Now (4.1) - (4.2) can be written as follows (4.3)

n > 1,

An, = To(I+a2T1)A.-1,

Ao = I,

where To and T1 are linear operators in the 3D space S of symmetric matrices. In fact,

ToA = MWAM., T1A = E(V"

(4.4) (4.5)

Thus, (4.6)

An = [To(I+o'2T1)]nAo = [To(I+v2T1)]n.

Let Aj = Aj (w, a), j = 1, 2, 3, be eigenvalues of the operator

T = To(I + a2T1).

(4.7)

Then 3

(4.8)

xn, (w, Q) = tr An _

An (w, a). j=1

To proceed the proof, we need

Lemma 4.2. If the matrix M, has the eigenvalues \1 and A2 then the eigenvalues of To in (4.4) are A2, a2, and Al A2. In particular, if w is inside a spectral and the eigenvalues of To are e+2ik(w) and 1. band, then A1,2 = If w is a nondegenerate band edge, then Al = A2 = ±1, and To has the eigenvalue A = 1 of multiplicity three with one-dimensional eigenspace.

PROOF. Let M,xj = \jxj and A = xixT, i, j = 1, 2. Then (4.9)

MWAMT = M xixTMT =

A AjxixT = AiAAA.

Similarly, AT = xjxT also satisfies (4.9). Then matrix A + AT is symmetric and is an eigenmatrix of To with the eigenvalue AiAj. Consider now w being a nondegenerate band edge. Then matrix M, is similar to a Jordan block [2] r

(4.10)

ML, = C-1JC,

J = [0

11

1]

, detC:0, p = f1.

The first statement of Lemma 4.2 provides existence of eigenmatrix A = x1xT for To with the eigenvalue equal to one. Let us show that the eigenspace of To is one-dimensional. To this end, it suffices to compute the first and the second adjoint

YU. A. GODIN ET AL.

176

eigenmatrices. The monodromy matrix Mu, has an eigenvector x1 and an adjoint vector y1 such that (4.11)

MWxl = µx1,

(4.12)

M",Y1 = µY1 + x1

Then one can easily check that

X2 = (x1xT + X, YT + Y, XT).

(4.13)

2

is the first adjoint eigenmatrix. Indeed, (4.14)

M,X2ML = z ML (xix1 + x1Y1 + Y1x1 )MLU 2(M,xl(Mr xl)T +

Mwxl(MYi)T + MLYi(Mwxl)T)

= 2 (xlxl + xl (YT + xT) + (Yl + xT)xT) = X2 + X1. Similar to (4.14), one can show that X3 given by (4.15)

X3 = 2(Xix + Yly1 + x1Y1 + Y1xT),

satisfies the equation

M,X3MW = X3 + X2.

(4.16)

Now we complete the proof of Theorem 4.1. If w is inside the spectral band, then due to Lemma 4.2 eigenvalues of To are distinct and equal to 1, ef2ik(w) Hence, the eigenvalues of the operator T have the form (4.17)

Al = e2ik(w) + g(w)a2 + 0(a4),

(4.18)

A2 = Al,

(4.19)

A3 = 1 +

h(w)a2+0(0,4).

This, together with (4.8) and (3.4), leads to (3.1). It remains to show that if det B 54 0 (the covariance matrix is not degenerate) then Ai(w, a) = 1 + ci(w)a2/3 + 0(a413),

(4.20)

a ---> 0, ci(w)

0.

Let us calculate the characteristic polynomial of T (4.7). Observe that due to Lemma 4.2 (4.21)

det(To - AI) = (1 - A)3.

Expansion (4.20) is apparent if we show that (4.22)

det(T - I) = det(To(I + a2 T1) - I) = c(w)a2 + 0(a4), or -+ 0, with c(w)

In an orthonormal Jordan basis the monodromy matrix M, has the form (4.23)

J=[

µ] ,

d 4 0,

to which it can be reduced by an orthonormal matrix 0. Operator T defined by (4.7), has the form (4.24)

TX = MXMT +0' 2 ME(VXVT)MT.

0.


177

In the Jordan basis it has the following representation TX = O(MOTXOMT + a-2ME(VOTXOVT)MT)OT (4.25) = JXJT + 0,2J E(VXVT)JT, where V has the form (4.26)

66

t_

V=OVOT_

-S1]

[S3

I

and Z are linear combinations of random variables z (entries of random potential V) such that tr V = tr V = 0. Let (4.27)

_ b11 B = b21

b12

b13

b22

b23

b31

b32

b33

be the covariance matrix of random variables 2, i = 1, 2, 3. Since the linear mapping e is nondegenerate, det B > 0. Now we represent symmetric matrix X = [ y Z ] as a vector [ x y z ] T and evaluate

det(T - I) (4.22). In this notation, operator T has the form of a 3 x 3 matrix T, and one can easily find that operator To is given by the matrix 2d d21 1 d

11

To = 0

(4.28)

0

j

so that (4.29)

0

1

= To(I + 2T1),

where Tl is given by (4.27). Finally 2d 0

d2

0 0

0

0

0

(4.30)

det(T - I) = det

d

= 2633 U2 + 0(a4),

+ a'2

1

2d

d2

b11

b12

b13

0

1

d

b21

b22

b23

0

0

1

b31

b32

b33

a , 0.

Here b33 > 0 since det B > 0. This implies (4.20) and completes the proof of Theorem 4.1.

5. Numerical example As an example we consider perturbation of periodic delta-function potential [5] (5.1)

V(x) = A>S(x - nL), n

where A > 0 is a constant. The monodromy matrix in this case has the form ra sinwL 1 0 (5.2) M = I ,, Q6 _ [-coswL sin wL cos wL] [A/W 11 _ coswL + (A/w) sinwL sinwL [(A/w) cos wL - sin wL cos wL '

YU. A. GODIN ET AL.

178

so that in (5.8) - (5.9)

a = cos(wL) + A sin(wL),

(5.3)

w

= sin(wL),

(5.4)

(5.5)

7 = A cos(wL) - sin(wL),

(5.6)

8 = cos(wL).

w

is reduced to evaluation of the

Calculation of the covariance matrix B = following integrals

f (5.7)

B

(s)' 2 (s) ds

i (s)1i2(s) ds

(s) ds

fL/2(s)ds

-fo /)l(s)%z(s)ds

- fL 211(s)40I2 (s) ds

fL 0i (s) ds

fo y'l(s)42(s)ds - fo

- fL '0i

(s) ds

fo

where 01(s) = cos(ws) and 02 = sin(ws). For numerical calculations it is more convenient to work with vectors rather than two-dimensional matrices. Therefore, we will represent symmetric matrices A,,, by vectors. Corresponding matrices of the linear operators in ][83 defined by (4.3) are denoted by To and T1 a2

To

(5.8)

= a-/

2aO

02

2a8 - 1

,38 I

278

82

72

.

The elements of the perturbation matrix T1 are expresses through the covariance matrix (5.7) as follows b11

(5.9)

T1 = b13 b33

2b12

b22

-b11 + b23

-b12

-2b13

b11

Figure 1 illustrates the sensitivity of the scalar variance of the transfer matrix over one period on the propagating frequency w for L = 1 and A = 6. If w = 2.6 is

0.6

0.4

0.2

0.040.0

0.02

0.04

a2

FIGURE 1. Dependence of the scalar variance of the transfer matrix over one period on the perturbation variance a2 for L = 1 and A = 6. The lower curve corresponds to the frequency w = 2.6 in the middle of the first band, while the upper one matches w = 1.977 near the left edge of the first band.


179

1.01

X 0.5 1

0.0 2.0

FIGURE 2. Dependence of the scalar variance of the transfer matrix over one period on the frequency w for the perturbation vari-

ance a2 = 0.04 and L = 1, A = 6. Band edges are denoted by vertical lines.

close to the middle of the band, then the variance of the transfer matrix is increasing linearly with a2 having the same order (lower curve). If, however, w = 1.977 is close to the band edge then the eigenvalue undergoes drastic changes with a2 (cf. (3.1) (3.2)). Dependence of the variance of the transfer matrix on the frequency w is shown

in Figure 2 for the perturbation variance a2 = 0.04 and L = 1, A = 6. As the frequency approaches the band edges, the variance of the transfer matrix increases dramatically.

References 1. A. Figotin and I. Vitebskiy, Oblique frozen modes in periodic layered media, Phys. Rev. E (3) 68 (2003), no. 3, 036609. 2. B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, translated by A. Feinstein, Transl. Math. Monogr., vol. 39, Amer. Math. Soc., Providence, RI, 1975. 3.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the theory of disordered

systems, Wiley-Intersci. Publ., Wiley, New York, 1988. 4. S. Molchanov, Lectures on random media, Lectures on Probability Theory (Saint-Flour, 1992) (P. Bernard, ed.), Lectures Notes in Math., vol. 1581, Springer, New York, 1994, pp. 242-411. 5. S. Molchanov and B. Vainberg, Slowing down and reflection of waves in truncated periodic media, J. Funct. Anal. 231 (2006), no. 2, 287-311.

6. S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of solitons: The inverse scattering method, Plenum, New York, 1984. 7. L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Grundlehren Math. Wiss., vol. 297, Springer, New York, 1992. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NORTH CAROLINA CHARLOTTE, CHARLOTTE, NC 28223, USA

E-mail address, Yu. A. Godin: [email protected] E-mail address, S. Molchanov: [email protected] E-mail address, B. Vainberg: [email protected]


Spectral Theory of 1-D Schrodinger Operators with Unbounded Potentials A. Gordon, J. Holt, and S. Molchanov ABSTRACT. This paper presents the spectral theory of the Schrodinger operator with piecewise constant random and deterministic potentials that are unbounded from above and below, and in several cases strongly fluctuating. In particular, we will present models exhibiting a type of transition from discrete spectrum, to pure point spectrum, to singular continuous spectrum.

1. Introduction Many years ago two of the authors along with B. Tsagani published in the Russian journal Functional Analysis and its Applications a brief paper [6] about random and deterministic operators with unbounded potentials. We were interested mainly in bifurcations from dense point spectrum to discrete spectrum. This paper did not contain proofs, but the subject was interesting and as a result, it attracted certain interest and generated responses. One of the models from this paper (that of the sparse rectangular barriers) was studied in a more general form in the paper [3] by one of the authors. We decided to publish a complete analysis of the three remaining models from [6]. Such analysis is important due to the growing interest in spectral bifurcations and borderline examples. Some of the ideas, which we had in mind in the early 1990s, are probably still of some interest. The paper is dedicated

to the memory of Professor B. Tsagani, who proposed the return to this subject and started to work on two of the models before suddenly passing away in 2004. Model I is related to the classical criterion of Weyl (whose final version was given by A. Molchanov, see details in [2, 9]) of the discreteness of the spectrum. Specifically, if V(x) > co > -oc for all x, then E(H) is discrete for any boundary phase 00 if and only if for all h > 0 +h

lim

V+ (z) dz = oo. JXX

2 000

Mathematics Subject Classification. Primary 34L40, 47E05; Secondary 34B24, 34C10. The research of the two last authors is partially supported by NSF grant DMS-0405927. This is the final form of the paper. ©2007 American Mathematical Society 181

A. GORDON ET AL.

182

Here, as usual, the operator H = Ho : L2(0, oc) - L2(0, oo) is given by (1.2)

HO = -0" + V(x)O

(1.3)

0(0)cos0-0'(0)sin0=0

where 0 E [0, 7r). A similar result is true on the full axis, but we'll consider, in the sake of simplicity, only the half axis case. We will give examples of operators H with exotic essential spectrum and with singular continuous spectral measure. Model I presents piecewise constant nonnegative potentials increasing in mean. Model II presents piecewise constant potentials unbounded from above and below. In both cases we describe transitions from dense pure point spectrum to discrete spectrum on the entire energy axis. Finally, model III describes a class of potentials which are non-positive and tend to -oo very fast. We'll discuss the problem of the essential self-adjointness of such operators (initiated by the classical example of E. Nelson, see [11]) and the transition from discrete spectrum (unbounded from above and below) to dense pure point spectrum.

2. Model I. Potentials increasing in mean Consider a partition of Ig+ into intervals In = [n, n + 1) and define

V(x) =

r0

An

ifxE12n if x E I2n+1

for n > 0 where An > 0 and An - 00Potential V is bounded from below, but (1.1) does not hold. In particular +2 fX V+(z) dz -3 oo, while for h < 1, lim infy-00 fy +h V+(z) dz = 0. Hence E(H) is not pure discrete for any Bo E [0, ir). It means that E,,3(H) is non-empty. Theorem 2.1. With V(x) as above, EeSS(H) = {7r2k2 : k = 1, 2,

... }.

Operator H (as any 1-d Schrodinger operator on the half axis) has simple spectrum (recall that a self-adjoint operator on 7-l has simple spectrum if there exists g E 7-l such that the {E([a, /3])g : E R} is dense in 7-1). From this fact Theorem 2.1 implies

Corollary 2.2. For any 0 E [0, ir) operator H of model I has pure point spectrum with accumulation points Ak = 7r2k2.

To prove Theorem 2.1 we will first show that for each k > 1, Ak E Eess (H) using the classical Weyl's theorem for essential spectrum, i.e., A E Eess(H) < if there exists a sequence On with II Wn 11 = 1 for which On converges weakly to 0 and IIH'n - Az1i, - 0 as n - oo. To prove that for every A # Ak, A V E...(H) we'll apply the Sturm oscillation theory. The following lemma will play a central role in the current model I as well as model H.

Lemma 2.3. For A' > A > 0 let VA,A (x) be the "test potential" defined by (2.2)

VAA'(x) =

A

if x 2

(2.5)

10n(x)I < C1

I e-coVAjxj

for some co > 2 - eo, cl > 0 and co = co(Jo) uniformly in n for n
a > K(A)A60 where, uniformly in A, K(A) > K for some fixed K > 0. In the new notations (2.6) can be can be rewritten as

tan w

- - b-1+ a-1 = O ( b-1 a-1-1

1

a-1

.

More precisely, (2.7)

tanw = -w

7A

1

+

7A

1

+O

w

VA

3

A. GORDON ET AL.

184

Iterations wn = 7rn - Sn now lead to formula (2.3) for the eigenvalues. Then 'On (21)

COS VA A

On + B +

An 1

A - An

Xn+B

sin

An + B

1

cos wn + - sin wn wn

(-1)n +0(62).

nVAAn This proves (2.5).

Remark 2.4. We'll use this lemma in the proof of Theorem 2.1 in the case when B(A) = 0 and .\i = 0(1). In its full generality this lemma will be essential for model II. We also note that if .\(A) - OA where 0 E (0, 1) then the result of Lemma 2.3 is wrong: the deviation of An from 7r2n2 for n - 00A is of the order 1, not o(1). Lemma 2.5. Let An (A, A') be the eigenvalues of Lemma 2.3 and Yin (x; A, A') the corresponding eigenfunctions. For B(A) = 0 and An < A(A) uniformly in n,

f

I HOn(x; A) - 7r2n2On(x; A) 12 dx = 0(1)

as A -3 00. In fact

f1

J

I HV)n(x; A) - n27r2On(x; A) I2 dx -

2A

PROOF. Since On (x; A) is an eigenfunction for An it follows that f1

f J

1

I HOn(x; A) - n27r2On(x; A)I2 dx = (An - n27r2)2 1

J

IOn(x; A) I2 dx. 1

Brief calculations reveal that 1

I On (x; A) I2 dx

2An

1

and that (An - n27r2)2

2A,

An

A

for each fixed n as A - oc. PROOF OF THEOREM 2.1. To show that in model I for each 1, 7x212 E Ee33(H),

we define a sequence of almost eigenfunctions. Let 01(x; An, An+1) be the eigenfunctions of Lemma 2.3 corresponding to Al. By Lemma 2.5 it follows that II H'l,n1 ' = 2n - 2, n 2 2 2 l 7r qPI,nIIL2 = 0(1) as n 00. Let an an', = 2n - 4,1 an+1 = 2n + 45 and a'+1 = 2n + 2 . Define the sequence P1,n (x) by

Pl,n (x) = v(x)'bl (x - (2n + ); An, An+1) 2 where v(x) E C0 is chosen such that v(x) - 0 for I x - (2n + 2) I > 1, v(x)

Ix-(2n+2)I 0 :

lim sup 1 t->oo

Bn2,2+((t)) < 00.

7r

Then it follows (see [9,12]) that Eess (H) fl (n27r2 + (, (n + 1)27x2 - O = 0. Without loss of generality consider the solutions and 9 2,2+((t) of (2.15) such that 0(n+1)2,2_((2Mo) = a and On2,r2+((2Mo) = 0 where a, a E [(, 7r-(] and where Mo = M0(() is chosen so that An,, > n27r2 and 1/ An, < (2 for all m > 2Mo. Then on the even intervals I2n

rk(t) = rk(2n)

0k(t) = Bk(2n) + k(t - 2n), which is equivalent to (2.9). In particular, for

ki -.

n27r2 + (

and k2 =

(n + 1)27r2 -

0k2(2Mo+1)-0k1(2Mo+1) = (a-0)+7r-

2n+1 27r(n2 + n)

+O((2).

On the intervals I2n+1 the behavior of the phase is completely different. For one, there is no rotation of the phase Ok(t) for t E I2n+1. Let's consider Ok(t) mod 7r and introduce the angles -y, = tan-i (an,,), and 7r - ry,,,,. Without loss we take Mo so large that ry,,,, E (0,(2 /2) for all m > Mo (hence 7r - ry,,, E (7r - (2/2, 7r)) and in this case then for all m > Mo we have Am1 < r4/(27r(n + 1))2 + 0((7). Of course, 'Ym - 0 and 7r - ry,,, ----> 7r if m -+ oc. Using (2.11) and (2.12) it follows that both Bk2 (2Mo + 2) and 0k1 (2M0 + 2) are exponentially close to the angles 'Y2Mo+2 (k) for

k = ki = k2, i.e., O(e-2/L2Mo+2(k))

9k(2Mo + 2) = 'Y2Mo+2(k) +

and therefore 10k2(2Mo + 2) - 0k1(2Mo + 2)1

_

1

A2 Mo+1

1

2n+1

21r n(n + 1)

) +O(

S2

A2Mo+1

Since 1/ An, < const (2 + O((3) for all m > Mo we have IOk2(2Mo + 2) - 0k1(2Mo + 2)1 = 0((2).

Hence, we have shown that for fixed 1 E Z+ and arbitrarily small ( > 0 that for any given a, 0 E [0, 7r) there exists Mo (() large enough so that the phases Ok (t) satisfying (2.15) with initial conditions 0k2 (2Mo) = a, 0k1(2M0) _ satisfy the relation I0k2 (2M0 + 2) - Ok1(2Mo + 2) I =

0((2).

Define now Bk2 (2M0 + 2) = a and Ok1(2M0 + 2) _ 13. Then similar steps (using either formula (2.12) or direct integration of the Ricatti equation (2.15) with initial conditions Ok (2M0 + 2) where k = k1 i k2) give by induction that for any a, , 3 E

[(, 7r - (] and m > M0(() that Ok(2m) E [-(2/2, (2/2] (and even exponentially small neighborhood of the point ry,,,,(k) (of order exp(-2µm)) shown in Figure 1 below by shading). The last fact (together with the description of the phase on the even intervals I2,,,,, m > 0) implies that for fixed l and ( > 0 the difference e(1+1)27x2-((t) - 012,,x2+((t),

for t > 2M0(()

UNBOUNDED POTENTIALS

187

c2/2 7m. (k)

4- 0 = 7r

7r/2

- -y. (k) - (2/2

FIGURE 1. Arrows on the circle indicate the direction of the phase

Ok(t) fort E [2m+ 1,2m+2) (both phases started from the same initial value 00 = 0 for t = 2Mo(()) cannot leave the interval ((, it - (). It means that the difference above is bounded on the full half axis, i.e,. the Hamiltonian H is this model has at most finitely many discrete eigenvalues inside the interval [7x212 +C/2, 7r2(l + 1) - (/2].

Let's note that the last fact implies that n

lnIIMA ([0, 2n)) II

n

E Ak - A E Ak.

k=1

k=1

The last estimation guarantees that eigenfunctions of our Hamiltonian of model I decay super-exponentially. We can generalize model I in several directions. We will consider only one such generalization. Let 19 > 0 be fixed and consider the sequence {Bn} C [0,19]. Define (2.17)

V (X) _

JBn An

if x E I2n+1

ifxE12n

where as before An > 0 and An -3 00. In addition, let F be the set of limit points of the sequence {Bn}, i.e., r = {B : B = limk,,,c, Bnk}

Theorem 2.7. With V(x) as above, Eess(Hgo) = U°1({n27r2} ® F) for all 00 E [0, 7r). If r is discrete, i.e., card(F fl A) < oo for any A C [0,19], the spectral measure µ(Hao) of Ho. is pure point. PROOF. The proof that cc

Eess(IIBo) D U ({n27r2} ® F) n=1

is exactly the same as in Theorem 2.1. Using Lemmas 2.3 and 2.5 we construct a sequence of almost eigenfunctions cp1,n(x). They have exactly the same form as

A. GORDON ET AL.

188

before. As for the opposite inclusion, 00

Eess(Hea) C U ({n21r2I (@ r) n=1

the proof is slightly different in that on the intervals 12n the transfer matrix Mk (I2n) is not orthogonal under the modified Priifer variables (2.13) and (2.14). In this case

it is better to define the standard Priifer variables by (2.18)

(t) = r,\ (t) sin0,\ (t)

(2.19)

(t) = r,\ (t) cos Ba(t).

It is well-known that the phase Ba(t) and amplitude r,\(t) obey the equations (2.20)

0' (t) = cos2 ea (t) + (A - V(t)) sin2 ea (t)

(2.21)

ra (t) = ra (t)

si

2

(t) (1 + V (t) - A)

VA_

The transfer matrices on even and odd intervals (for Bn < A < An) have the usual forms: M,\ (12.) Ma (I2n+1)

_

cos

_

-

A - Bn n A -Bn

cosh An - A

cos A-Bn An - A )

(sinh An - A)

cosh An - A

An - A

(%/A, - A sinh

A - Bn

A - Bn)

(sin

J

and from this it follows that (2.22) MA([0, 2n]) = MA(I2n-1)MA(I2n-2) ... M. (I1)Ma(Io) 2n-2 j=o

MA(Ij+1)MA(Ij).

For n > 0 let Rn be the matrix defined by A - Bn Rn

0

0

_4A-Bn

Then it follows easily that

/ 2n-2

MA([0, 2n]) = Rn+1I 1 MA(Ij+1)MA(Ij) Ro

1

where

Ma(12n) = Rn

1

cos

-sin

A - Bn A - Bn

sin A - Bn cos

-Bn

and (2.23)

Ma(I2n+1) = Rn+2M,\(12n+1)Rn+1

These formulas admit the following representation. The appropriate phase (selected differently for different even intervals I2n, n > 0) rotates uniformly with speed vn = A - Bn on each even interval I2n after which it is subjected to the strong hyperbolic transformation M,\(12,-,), n > 1 with eigenvalues of the order exp(± An - A). Now the proof of the second part of Theorem 2.7 is similar to the proof of the corresponding part of Theorem 2.1. Assume that [A1, A21 fl


189

(F (D {n27r2 : n > 1}) = 0. Then, for sufficiently small ( > 0 and n > no(() >> 1 and condition %a(2no) = 0 for A E [Al, A2], we have for Al < A' < A < A2 that

Ba(2no + 1) - 6A, (2no + 1) E [(/2, it - (/2] The hyperbolic transformation MA(I2n0+1) will return both phases 6,\, and 6a (with exponentially high accuracy) to an angle of order O(An 1/2) and after this forever the difference between 6a, and 6a will be less than it - 2C.

Remark 2.8. If F contains an interval [a, b] then the well-known result by Gordon [4,5] and Del Rio et al. [1] gives that µ(H9°) = fit,, (HB0) only for a.e. Bo E [0, 7r) and that there exists a set 9 C [0, 7r) of Lebesgue measure 0 and of second category, such that µ(H90) = µsc(Ho0) on U°° 1({n27r2} ® [a, b]).

Remark 2.9. We have now seen in model I examples where, as a set, Ee3S(H) is discrete and unbounded from above. One might ask if it is possible to construct nonnegative potentials for which the essential spectrum is discrete and finite? The answer (see [10]) is no. In this case EeSs(H) is either empty or unbounded from above. This fact will be one of the main differences between model I and model II below.

3. Model II. Potentials unbounded from above and below. Let An, Bn > 0 An, Bn - oo where we assume Bn < A'-a0 for some 6o > 0. Define

V(x) _

Bn

An

if x E I2n_1

ifxEl2n

Then V (x) is unbounded from above and below. If An = oo for all n, then the corresponding Hamiltonian H00 '(x) = -0"(x) +V (x)O(x), 0(0) cos Oo - 0'(0) sin Oo = 0 on L2(0, oo) has the system of non-interacting potential wells of unite length with depths {-Bn} surrounded by infinitely high walls. The spectrum in this case is the set

A = {Am,,n = 7r 2m2 - Bn : m, n > 0}.

If the barriers An are finite then the operator H00 is essentially self-adjoint independently of Bn due to the following result by R. Ismagilov (see [8, 11]). Recall that a differential expression of the form L := d2/dx2 + V (X) is in the limit point case at oo if for every A there exists a solution of Lu = Au which is not square integrable at oo.

Theorem 3.1. Suppose that V E Li c and let there exist a sequence of intervals (xn, yn) for which V(x) > -(yn -xn)2 for all x c (xn, yn) and > (yn -xn)-2 = 00 . Then L is in the limit point case at oo (without any assumptions on V outside of the intervals (xn, yn)).

Therefore the spectral measure tt(dA) for H00 is unique, and the spectral properties of H00 depend mainly on the structure of the set {A n,n : n > 0, m > 1}.

Theorem 3.2. Consider the set of limit points of the set A, LA = {A : A = limk . Amk,nk}. Then Eess(HB°) = LA and if LA is discrete (i.e., LAnA is finite for any A C R) then µ(H00) = ppp(H00) for all Bo c [0,7r).

A. GORDON ET AL.

190

Remark 3.3. As in model I it follows by Gordon's theorem that if LA contains an interval [a, b] then the most we can say concerning the nature of the spectral measure µ(H60) is that µ(H60) = ppp(He0) for a.e. 0 E [0, ir) and there exists a dense 95 for which µ(H60) =

on [a, b] for all B0 E g6-

Corollary 3.4. Suppose the set A is discrete so that LA = 0. Then E(H) _ Ed(H) on (-oo, oc) for all Oo E [0, ir). While the proof of Theorem 3.2 is analogous to Theorem 2.7, there are significant differences between models I and II. For one, while the essential spectrum (if nonempty) in model I must always be unbounded from above (see [10]), whereas in model II the essential spectrum can be finite (see the example below). Secondly, the spectrum of model II can be pure discrete while in model I the spectrum must always have nonempty essential spectrum.

Example 3.5. Let Bn = ir2n4 and An = 7r2n5. Then )tnti,n = m27r2 - Bn = 7r2 (m -n 2) (m +n 2). Obviously 0 E LA since 0 = Anz n for all n > 1. Also, 0 is the

only limit point A. Therefore by Theorem 3.1 we have Ee33(H) = {0} and µ(H00) is pure point for all 00 E [0, 7).

4. Model III. Negative potentials decreasing to -oc Let V (x) = -vn for x E In - 1, n) where for all n, vn = 27rMn + 8n and Mn E Z+ and Sn E [-7r/2,7r/2]. Because V can decrease arbitrarily fast at oo it is not clear that L is in the limit point case at oo and indeed as we shall see below there are important classes of potentials V giving rise to L in the limit point and limit circle cases at oo. We will first consider a class of potentials V (x) for which L is in the limit point case at oo. Let V (x) be as above where 8n = 0 for all n. We will call this type of potential a "Nelson" potential after E. Nelson who presented results for potentials having a similar structure to the one above. Our first result is very much in the spirit of his known example. Lemma 4.1. For any sequence Mn E Z+ the operator H90 corresponding to a Nelson potential is essentially self-adjoint. PROOF. The proof is straight forward. We simply construct two solutions z/)0(x)

and 01(x) of the equation Lo = 0 satisfying 00(0) = 0 = 0'(0) and 00 '(0) = 1 = 01(0). From their structure it follows that both are not from L2 (0, oc). Therefore L is in the limit point case and hence H90 is self-adjoint. Again motivated by E. Nelson we define the class of potentials V(x) with bn = it/2 for all n. We call this class of potentials "semi-Nelson" type potentials. The next several results show that not only is there a wide class of semi-Nelson potentials

giving rise to L not in the limit point case at oo , but that even certain "small" perturbations of such potentials do not destroy this property.

Lemma 4.2. Let V(x) be as above with Sn = lr/2 for all n and define the sequence {hn} by

h0=1,

hn=

1

vnhn_1

forn>1.


191

Then there exist linearly independent solutions oo(x) and i1(x) of the equation L% = 0 in D(H) such that m

2m IV)o(x)12 dx

r.

=

f

n-1' n=1

2m +1

'm

n=1

for any m E Z+.

Hence, if {hn} E 12 then Lo = 0 has two linearly independent L2(0, oo) solutions. Thus L is in the limit circle case at oo so that H90 is non-essentially self-adjoint. As an illustrative example, let us consider the particular case when Vn have the for m vn = 27r Lnaj + it/2 where a > 0 and LxJ = max{n E Z+ : n < x}. As usual for any real number x, we can write x = LxJ + {x} where {x} is the fractional part.

Lemma 4.3. Let V (x) = -vn on [n -1, n) where vn = 27r Ln"j +7r/2. Then, if a > 1 the solutions' &o (x) and 01(x) constructed above are from L2 (0, oo). Therefore the operator H90 is non-essentially self-adjoint.

PROOF. Let's note that hen can be presented as

h 2n -

n

v_v3...v2n-1 v2v4...v2i

=exp VT' k=1

v2k-1

In (

V2k

where

\

(v2k-1)

J

27r L(2k) J + -7r/2

\

l 2k) -aJ"JJ

k "J

Now with the fact that Ln"J " n" it follows -In/2k-1)1"+O(1 ) k" )

ln(v2k_1) V2k

)

2k

a 2k

+O(k2) +O(k")

2k

+o(k2).

So then, exp

k=11n (vv2k 1)

= exp E (- 2k

k2

and therefore

h2n-K1exp(E-k) -K2exp(-aInn)

= n2

" o hen must converge and it shows that 01(x) E L2(0, oo). k=1

This shows that

Similar procedures show Oo(x) E L2(0, oo).

Small perturbations of the potential V preserve the property of non-essential self-adjointness as the next lemma shows. Lemma 4.4. Let {En} be a sequence for which en E 2irZ, and

E an En

k=0

< oo.

A. GORDON ET AL.

192

Let V(x) = -a,,, for x E [n - 1, n) and set VE(x) = -(an + en)2. Then if the operator Ho0V) _ -'O"+V' is non-essentially self-adjoint, the the operator HB0V) _ -v'" +VE'' is non-essentially self-adjoint.

Remark 4.5. The two cases considered above show that the bifurcation between the essentially self-adjoint operators and the non-essentially self-adjoint operators is quite complex. Theorem 4.6. Suppose that V (x) is a Nelson potential. Suppose the sequence {vn} satisfies the following: yn+1

lim

= 1 E 1 < o0

n-oc vn

k=1 vk

Then the spectrum E(H90) is discrete. Before proving this result we first discuss a few technical details. Consider the transfer matrix for our operators, i.e., the matrix MA(0, x) for which M A(0, x)

-

+ Vk = k2k with initial conditions 'k(O) = sin00i where 'Ok(x) satisfies k(0) = cos 00. Due to thek type of potentials we are considering the matrices have the nice form n

Mk(O,n)=Mk(n,n-1)Mk(n-1,n-2)...Mk (0,1)=fMk (j-1,j) j=1

for x = n E Z+ where cos

k2 + v3

v

(1/-,/k2 +

k2 + v

sin

Mk (j - 1,j)4'k(j - 1) and 4'k (x) = (4'k(x), Yk(x))T. Let

[/k2+v32 0

0 y

k2 + v2j

and

Oj (k)

cos

k2 + v

sin

k2 -+V32

- sin k2 + v cos k2 + v3

Then Mk(j - 1, j) has the decomposition

Mk(j - 1,j) = I371(k)Oj(k)Ij(k) Let also

Dj- ,i (k) = Ij (k)Ij 11(k) _

4

0

(k2 + a?)/(k2 + aj-1) 4

0

(k2 + aj-1)/(k2 + aj)

Then we can write n+1

Mk (0, n) = In+1(k) . fi Dj,j+1(k)Oj+1(k) 11 (k). j=1


193

It is the middle matrix Mk(0,n) = 1 D,j,3+1(k)0;+1(k) that contains the information concerning the spectrum of the operator H90. We will first show that under minimal assumptions on the sequence {vn} the spectrum in the Nelson case is always discrete. As is usual one needs to show that dim(EHeo (A)

- EHeo (0)) < 00

for any A where EH00 is a spectral projection of H90. Let NS(E, Bo) be the number of zeros of the solution u(x, E) to the problem (L - E),0 = 0 on (0, s) satisfying the boundary conditions u(0) = sin 0o, u'(0) = cos 00. Let also no,) (0, A) = lim inf (NS (A, 0o) - Ns (0, 0o)) . g-100

Then (see [12]) it follows dim(EHeo (A) - EH9O (0)) < neo (0, A) + 1.

At this point, we can introduce the Pruffer variables (2.18) and (2.19). Recall though that now u(x, A) = 0 if and only if BA(x) = 0 (mod ir). Therefore NL(A, 00) = I Ba(L)

Therefore we seek estimation of 0,N (L) - 00(L)

7

Using the Pruffer relationship the transfer matrix relationship for m < n becomes [Pa (n) cos 0,\ (n)]

= MA(m' n) [pa (m) c s 0), (m)]

where for N > 1 the transfer matrix MA (m, m + N) can be presented as above in the form N

Mk(m, m + N) = I-+N+1 ' fl Dm+a,m+,j+1(k)Om+.j (k) . Im+1(k) j=1

= Im+N+lMk(m,m+N)Im+l(k). This expression can be replaced by the equivalent expression:

[pa(L) in0a(L)]

- Ma(m L) [pa (m)in °A(m)]

where N (4.1)

Mk (m, m + N) = Ian,+N+1(k) fl Dm+a,m+.j+1(k)Om+.j (k) . In,,+1(k). j=1

It is this form that we prefer and this form of the transfer matrix will be used in the proof of Theorem 4.6.

PROOF OF THEOREM 4.6. Using the transfer matrix Mk (m, L) we can track the behavior of the phases BA(x) and Bo(x) for the solutions 0,\(x) and o(x). In particular, we will show that the relative differences (i.e., modulo ir) can in fact, never exceed 7r, i.e., (BA(x) - 9o(x)) mod 7r < ir. Now since vn+l/vn -i 1 we can write

yn vn

= 1 + En

A. GORDON ET AL.

194

where ,, ----> 0 as n - oo. Now choose m so large that A

k>

0 for the case A0 < 0 follows similarly. We take no E Z+ with no >> A0. Just as before in the Nelson case above we may assume that 01\0 (no) = 0 = 00 (no) - 00 and as in (4.1) the transfer matrix has the form k+1

MA(no, no + k) = Ino+k+l - II Dno+jOno+j ' Ino+1 j=1

where a

_ Ino+1(A)

+ ano+1

0 -A+ a2no+1

0

This matrix has the attracting eigenvector [0 1] T. Since A0 is fixed, the matrix Ino+l (A0) has an eigenvalue asymptotic to 27rMn as n -p oo, and it means that for given fixed 00, the matrix Ino+l maps the angle 00 E [-7r/2,7r/2) into Bo (A) = tan-1( A + ano+1 tan Bo)

which is strongly attracted to ±ir/2 for any 00 0. It is clear that with 00 = 0 there is no deviation from this angle after the mapping Ino+1 so that 00(A) = 0. The rotation Ono+1(A) then maps 00 (A) to the angle 01(.1) = 27rMno+1 + ano+1 +

A

2ano+1

+ rno+1(A)

where rno+1(A) = O(an+1) for all A in a bounded region. Let A0

ano+1 - - 4ano+1

- rno+1(A0) 2

Then Ao

61(Ao)

= 4ano+1

+

rno+l(Ao)

2

and

B1 (0) _ -

Ao

4ano+1

-

rno+l(Ao)

2

modulo 7r of course. Now the angles 01(0), 01(A0) are symmetrically located across the zero phase. The diagonal matrix Dno+1(A) maps 01(0), 01(A0) respectively to

A. GORDON ET AL.

196

0

FIGURE 2. Separation of the phases 01(A0) and 01(0)

the angles 01(A0) = tan- 1(ryno+1(Ao)tan 01(Ao))

01(0) = - tan-' (ryno+1(0) tan 01(0)) where A + an 2 +1 ryn

V \+an

Note that the angles are wider after multiplication by the diagonal matrix but still practically symmetric across the zero phase for large no since ryn(Ao) - ryn(0) as n - oo (see Figure 2). It is clear then that we can select rno+2(A) = O(a +2) and ano+2

Ao

- - 4an,,+2

- rno+2(Ao) 2

so that the rotation matrix Ono+2 (') takes 01(0), 01(A0) to 02(0), 02 (Ao) given by

02 (0) = -tan-1 (ryno+1(0) tan(

(

-

1 A0 4ano+1 J A0

1 0200) = tan 1 'Yo+l(Ao) tan( \ 4ano+1 /

Ao

4ano+2

+

A0

4ano+2

_

rno+2(Ao) 2

+rno+2(Ao) 2

so that these angles are symmetric about the zero phase. The divergence of the product f _/n and the choice of dk imply that we can find some Lo large enough so that BLo (0) = -ir/2 + E/2 and BLo (A0) = 7r/2 - E/2 where E is arbitrarily small.

At this moment set Sno+Lo = 0. Then, the next rotation will not change the position of BLo (0), i.e., modulo 7r we have BLo+1(0) = BLo (0). Also like before the

diagonal matrix maps BLo+1(0) to BLo+1(0) = tan (+i (0) tan BLo+1(0)) and brief calculations give OLo+1(0) = -

7 2

+

ano+Lo

f

ano+Lo+1

E 2

+ O(E2)


197

Continuing with this pattern gives for any k > 0 Lo+k(0) =

B

- +2 ano+Lo ano+Lo+k

2

+ 0(E2 )

For Ao the procedure is the same only this time we must take into account that the rotations are not exactly multiples of 27r, but instead multiples of 27r plus some small extra turns. As we have said Lo is chosen so that 9L,, (A0) = 7r/2-E/2. Denote by a(u, v) the difference between angles u and v. Then a(9Lo (Ao), 7r/2) = E/2. The counter-clockwise rotation OLo+1(Ao) gives Ao

e

- 2ano+Lo+1 + O \ a'no+Lo+l/

a(0L0+1(Ao), 7r/2) = 2

2

1

which we choose to rewrite, using the fact that ano is large in comparison with A0, as A0

a(9Lo+l (A0)) 7r/2) = E 2

2

_+0(a

Ao + ant +Lo+1

l

/

z 1 no+Lo+l

(this will simplify the calculations that follow). Now, after applying the diagonal matrix we get (again after brief calculations and dropping the smaller order terms from here on to reduce space) 0(0L0+1(Ao), 7r/2) =

_o + an Lo+Lo+1

A0

o + azno+Lo+ z (2

0

+ az

no+Lo+l

Now the next rotation gives a(9Lo+2 (Ao), 7r/2) Ao

/\0

2 -o + ago+Lo+1

2 A0 +ano+Lo+2

_o + ana+Lo+l /\o +a2no+Lo+2

2

For brevity in notation let Ko = no +Lo. Then multiplication by the next diagonal and brief calculations give a(OL0+2 (Ao), 7r/2) =

1 2

Vo + aKo+3

(

Ao + aKo+1 ' 2

3. 20

Repeating we have in general then, for any j > 1 a(0Lo+i-1(Ao), 7r/2)

2

Ao + axo+,

(VA0 + axo+1 ' 2 - j. 2

o 2

Now combining the above results with these we see that the difference between 9Lo+i (0) and 9Lo+j(Ao) is 1

E

z 0 + aKo+1 2

z

A0 +aKo+j which simplifies 1

o +aKo+'

j

Ao

2

+

o+aK2 o+1 .E - j .

z Ao + aKo+1 Ao

z

+aKo+

E

2

Ao

2

iat this difference can be made negative and that From this expression it follows tciat this would give one rotation in the relative difference of the phase Sao - 90. From here we can repeat the above construction so that by induction there exist infinitely many rotations in the phase.

A. GORDON ET AL.

198

References R. Del Rio, N. Makarov, and B. Simon, Operators with singular continuous spectrum. II: Rank one operators, Comm. Math. Phys. 165 (1994), no. 1, 59-67. 2. I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Israel Program for Scientific Translation, Jerusalem, 1965. 1.

3. 4.

5. 6.

A. Ya. Gordon, A deterministic potential with a purely point spectrum, Mat. Zametki 48 (1990), no. 6, 38-46 (Russian); English transl., Math. Notes 48 (1990), no. 5-6, 1197-1203. , Exceptional value of the boundary phase for the Schrodinger equation on the semiaxis, Uspekhi Mat. Nauk 47 (1992), no. 1(283), 211-212 (Russian); English transl., Russian Math. Surveys 47 (1992), 260-261. , Pure point spectrum under 1-parameter perturbations and instability of Anderson localization, Comm. Math. Phys. 164 (1994), no. 3, 489-505. A. Ya. Gordon, S. A. Molchanov, and B. Tsagani, Spectral theory for one-dimensional Schrodinger operators with strongly fluctuating potentials, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 89-92 (Russian); English transl., Funct. Anal. Appl. 25 (1991), 236-238.

7.

J. Holt and S. Molchanov, On the Bohr formula for the one-dimensional Schrodinger operator with increasing potetial, Appl. Anal. 84 (2005), no. 6, 555-569.

8.

R. S. Ismagilov, The spectrum of the Sturm - Liouville equation with oscillating potential, Mat. Zametki 37 (1985), no. 6, 869-879 (Russian); English transl., Math. Notes 37 (1985), no. 5-6, 476 - 482.

B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, Transl. Math. Monogr., vol. 39, Amer. Math. Soc., Providence, RI, 1975. 10. C. R. Putnam, On the unboundedness of the essential spectrum, Amer. J. Math. 74 (1952), 9.

no. 3, 578 - 586.

11. M. Reed and B. Simon, Methods of modern mathematical physics. II: Fourier analysis, selfadjointness, Academic Press, Boston, MA, 1975. 12. J. Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Math., vol. 1258, Springer, Berlin, 1987. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NORTH CAROLINA CHARLOTTE, 376 FRETWELL BLDG., 9201 UNIVERSITY CITY BLVD., CHARLOTTE, NC 28223-0001,

USA

E-mail address, A. Gordon: [email protected] E-mail address, J. Holt: [email protected] E-mail address, S. Molchanov: [email protected]


Fermi - Dirac Generators and Tests for Randomness Alexander Gordon, Stanislav Molchanov, and Joseph Quinn ABSTRACT. A Fermi-Dirac experiment FDO(n, N) is an ordered random sampling of size n without replacement from the integers from 1 to N. A complete sampling FDO(N, N) yields a string (random permutation of size N) that can

be used to simulate Fermi-Dirac experiments for any n less than or equal to N. We refer to such strings, regardless of their origin, as Ordered FermiDirac Generators, (FDO) generators. Because of their lack of collisions, such generators cannot hope to simulate Maxwell- Boltzmann (MB) experiments (in a previous publication, the authors showed this to still be the case even if truncation is used to overcome the lack of any collisions). All pseudo-random number generators (PRNGs) are, at some level, FDO generators. They are routinely combined with some sort of projection to provide simulations of MB generators. Another approach is possible. Instead of trying to produce a good MB PRNG directly, one can focus on producing a high quality PRNG that does

a good job of simulating a complete FDO experiment, where the absence of collisions (Pauli exclusion principle) is an intrinsic property. There is a natural duality between unordered FD(n, N) experiments and Bose -Einstein (BE(N - n, n + 1)) experiments (consisting of distributing N-n indistinguishable balls to n + 1 cells). We develop the asymptotics of occupation statistics for BE experiments which have many analogues to known results for MB experiments. However, some of the results are completely new with no analogue in the MB theory. We show that one of these has considerable ability to distinguish between the performances of a fairly large number of standard PRNGs. Other tests give an indication of the degree to which algorithmic chaos has destroyed local correlations inherent in dynamical systems. Finally, we several ways by which a high quality FDO generator can be transformed into an MB generator of essentially the same quality.

1. Introduction In our previous publication, [2], we explored the fact that, at some level, all dynamical systems on a finite phase space cannot repeat values and that this can result in detectable differences between the sequences produced by a pseudo random

number generator, PRNG, and those that would be produced by a true MaxwellBoltzmann experiment. In this paper, we want to focus our attention on FermiDirac sequences (no repetitions). There are three reasons for doing this: 2000 Mathematics Subject Classification. Primary 60A99; Secondary 60P99, 60K99. This is the final form of the paper. ©2007 American Mathematical Society 199

200

A. GORDON ET AL.

(1) As already mentioned, all PRNGs produce, at some level, Fermi-Dirac sequences. We will make this clear below. (2) In some sense, Fermi-Dirac sequences and the random models associated with them are the most basic and simplest sources and might, therefore, be easier to validate or test. Furthermore, as we shall see, if one has a "good" Fermi-Dirac sequence, one can obtain from it a "good" Maxwell-Boltzmann sequence.

(3) Historically, much of modern interest in probability and notions of randomness stems from the attempts begun by Maxwell and Boltzmann, and later by Gibbs, to understand how macroscopic thermodynamic properties could result from microscopic dynamical laws governing the evolution of the molecules of a gas. The explanation offered for how a nonequilibrium system could evolve to one whose

energy distribution is, asymptotically, that of the appropriate stationary distribution requires one to view the system on two levels: a coarse-grained level, and a trajectory level. We are interested in seeing that the time average spent by the trajectory in any given coarse grained element is equal to the "size" of that element with respect to the appropriate invariant measure. Under the assumption of metric indecomposability, this will happen for almost all trajectories and, assuming we are on one of those trajectories, we can imagine choosing a AT and noting the position of the system at each AT time interval. At the level of the dynamical system, the resulting sequence of points X0, XAT, X2IT,... , XnAT,..., will never repeat itself. However, at the coarse grained level, repetitions can occur. For any finite sequence thus extracted, the coarse-grained boxes can be refined into finer grained boxes so that each XiIT will be in only one of them. Now, for that sequence, the "positions" can be labeled by a pair of integers (i, j) where the first coordinate is the label of the coarse-grained box and the second coordinate is the label of the refined box. The resulting "trajectory," at the coarse-grained level, will then be a projection of the full nonrepeating position (i, j) onto its first coordinate. Models like this will be referred to as generalized Fermi -Dirac models. But, we emphasize, at some level these are Fermi-Dirac models although this may not be explicit to the observer. We return to reason (1) above and recall some of the arguments from [2] that support this assertion. In the background of any algorithmical random number generator (PRNG), there is a discrete dynamical system, i.e., a one-to-one map-

ping F: X -> X of a suitable finite set X into itself (preserving therefore the uniform measure on X). We can restrict our mapping to one of the ergodic components X' C X. An X' C X is an ergodic component if it consists of a point of X and all of its iterates under the mapping F. If x1 E X, then X' = {x1, x2 = F(xl),... , xt = F(xt_1)}, where t is the first time that x1 = F(xt). Note, t = X'J. In many cases, X' = X = Wk, where Wk is the set of binary words of length k. In this case, IWk! = 2k. Some typical examples can be found in [5]. The mapping F may be followed by an additional mapping W: X -> Y. Then the output of the PRNG will be yl = P(x1), y2 = ( x 2 ) , . . . , , yt = o(xt), which may not be distinct. In many cases, cP is a truncation of some of the higher order bits in the full word which are often considered to be flawed. Let N = I{yl, Y2.... Yt}11 and let z1i z2, ... , zN be a labeling of the distinct values of the yi, 1 < i < t. The X' fibers over each zi will be of the form (P-1(zi) n X' and will form a partition of the ergodic component X'. If Mi = (P-1(zi) n X11, then t = EN1 Mi. A straightforward model for such a cycle might consist of distributing t balls to N boxes in some

FERMI-DIRAC GENERATORS AND TESTS FOR RANDOMNESS

201

way so that the number of balls ending up in the ith box is equal to Mi. Important examples include the lagged Fibonacci generators under modular addition or multiplication (for historical reasons, the xor lagged Fibonacci generators should also be mentioned). In these cases, for a typical cycle, the fibers over the output words have varying heights. There are any number of ways that the ordered outcomes of a typical cycle for such a generator could have been realized from a genuinely random experiment. One way is that the assignment could have resulted from a run of a true Maxwell -Boltzmann experiment in which the N outcomes are associated with the same number, N, of boxes and the t balls are placed in the boxes sequentially in such a manner that each ball has an equal probability of going into any one of the boxes. A generator F with such an ergodic component could then be defined

by taking F: {1,2,...,N} x {1,2,...,t} ----> f x {1,2,...,t} to be any one-to-one, onto extension of the map f : {(N(1),1), (N(2), 2), ... , (N(t), t)} -> {(N(1),1), (N(2), 2), ... , (N(t), t)} where f (N(t), t) = (N(1),1), for 1 < i < t, f ((N(i), i)) = (N(i + 1), i + 1), and N(i) is the label of the box that the ith ball enters. This same construction (which is, by the way, a "copy" algorithm) will work for any other way of assigning the t balls to the N boxes. When we think of a PRNG as providing uniform numbers over some range, we are considering the sequence it produces as being somehow representative or similar to that which would have been produced by a Maxwell-Boltzmann experiment. From the discussion above, it is clear that all generators can be identified with a collection of finite ordered sequences of elements from some word space. Thus, for the rest of this paper, by a simple generator, we will mean a sequence y1, y2, , yn, yi E Y, 1 1 will consist of the next k elements of the sequence following the element indexed by the seed. A multi-generator, will consist of a collection of simple generators. The length of the string, n, is the period of the simple generator. Another random way that the ordered outcomes of a generator could have been

produced would be to arrange the possible outcomes in towers of boxes labeled (i, j), 0 < i < N - 1, 1 < j < Mi. The experiment would then consist of placing t = Ei I Mi balls into the t boxes in such a way that every unoccupied box has an equal probability of receiving the next ball. Of course, such experiments could

be run with any number, n, n < t, of balls. We refer to such an experiment as an ordered generalized Fermi-Dirac experiment with parameters n, N, {Mi}, GFDO(n, N, {Mi}). If, for each i, Mi = 1, the experiment is an ordered FermiDirac experiment, FDO(n, N). Note that a GFDO(t, N, {Mi = 1}) is equivalent to an FDO(t, t) experiment. If we forget the order, the experiments are referred to as generalized Fermi-Dirac, GFD(n, N, {Mi}), and Fermi-Dirac, FD(n, N). A generator yl, ... , YN that has been produced from an FDO(N, N) experiment will be referred to as an FDO(N, N) generator or an FDO generator of order N, or, sometimes, just an FDO generator. In this paper, we will focus on Fermi-Dirac generators, yi, ... , YN, for some integer N > 1 over an appropriate symbol space and both the ordered and unordered samples that can be extracted from them. There are two kinds of samples that we will be interested in when discussing these generators: (1) the contiguous kind, already mentioned above, which consists of a block yio+1, ... , Yi0+k (for a sample of size k and seed io); and (2) the noncontiguous kind based on considering successors of intervals of integers. To illustrate, suppose we are interested in

A. GORDON ET AL.

202

a noncontiguous sample based on the interval of integers [1, m]. Define the corresponding sequence of indices i(1) = that i, 1 < i < n, such that y2 = 1, i(2) = that

index i such that y2 = 2, and so on. The noncontiguous sample then consists of the Yi(1)+1, Yi(2)+1i ... , yi(m)+1 Note that, if the generator is the result of a PRNG, F, then yi(1)+1 = F(1), ... , yi(r.)+1 = F(m). The samples we extract from Fermi-Dirac generators are naturally ordered, however, if we forget the order we are merely extracting a subset of the integers in the label space. Contiguous samples are the kind that are often extracted from a generator. Statistics based on them, therefore, are of principle interest in efforts to evaluate the "quality" or performance of a generator. Noncontiguous samples are involved in assessing generators based on dynamical systems. In order to destroy the correlations inherent in dynamical systems, even chaotic ones, the sampling interval AT must be large in comparison to the time scale on which the dynamical system evolves. As we will see, noncontiguous samples provide an indication of whether a PRNG has been built in such a way as to break the natural ("short-term") correlations inherent in dynamical systems. In order to evaluate the performance of a Fermi-Dirac generator with respect to statistics based on contiguous samples, one needs a battery of statistics whose theoretical distributions are well understood and for which relevant probabilities can be effectively approximated. For Fermi-Dirac generators, there is a duality between statistics associated with Fermi-Dirac models and corresponding occupation statistics for Bose-Einstein models. The asymptotic theory of these occupation statistics parallel the well known analogous theory for Maxwell-Boltzmann models.

In Section 2, we review the Maxwell- Boltzmann asymptotic theory for occupation statistics. In Section 3, we develop the asymptotic theory of occupation statistics associated with Fermi - Dirac/Bose -Einstein models. The main value of these asymptotic results appears to be the development of intuition into what might represent useful relationships to impose on the parameters of the models. To obtain hard estimates on tail probabilities and sometimes on the entire distribution, when needed, one can utilize a technique developed in [2] that utilizes factorial moments and classical results due to Chebyshev. In Section 4, we discuss some testing results for a number of generators based on occupation statistics for contiguous samples. We also discuss results for noncontiguous samples for several PRNGs. Finally, in Section 5, we discuss several ways that a Fermi-Dirac generator can be used to produce (with varying efficiency) a Maxwell -Boltzmann generator.

2. Review of limit theorems for MB statistics Following [2], we denote by MBO(n, N) the result of an ordered MaxwellBoltzmann experiment in which n balls are placed sequentially in N boxes, labeled

from 0 to N - 1, and the order is kept track of. Each ball has an equal probof ability of being placed in each of the boxes. Each possible outcome such an experiment has probability (1/Nn). The classical Maxwell- Boltzmann experiment, MB(n, N), consists of ignoring the order and only keeping track of the cell occupation statistics: xo, ... , x2, ... , xnr_1 where xi = #{balls in cell i}. For these, one obtains the classical multinomial probability distribution PMB(x0 = ko, ... , ZN_1 = kN-1) = n!/(ko! ... kN-1!)(1/NTh), here ko + ... + kN_1 = n. So, the elementary outcomes are the ways in which the integer n can be written as the

FERMI--DIRAC GENERATORS AND TESTS FOR RANDOMNESS

203

sum of N integers ko, . . , kN_1. From the point of view of serving as a source of uniformly distributed numbers, an MBO(n, N) is much more useful. If one starts anywhere in the outcome sequence i1, ... , in, say at the position jo, and takes the next m, m < n - jo, results, according to assumption, one will have a theoretically valid, ordered sample of size m from the numbers ranging from 0 to N - 1 under the uniform distribution. Of course, one cannot rule out that the entire sequence .

obtained by the experiment was not a wildly improbable one and might not be representative in any useful way of the "typical" MBO(n, N) experiment. However, if n is large and much larger than m, then with exception as unlikely as one might wish, one can expect the collection of all such samples to be representative of what one would get in sampling from a uniform distribution on the numbers from 0 to N - 1. In this section, we will recall some of the results from [2], with some

additional details, on the limit theorems for MB experiments. Suppose we have available an MB(n, N) experiment. Let 7ro = 7ro(n, N), 7r1 = 7r1(n, N),..., 7rN-1 = 7rN-1 (n, N) be the corresponding occupation numbers of these cells. These random variables satisfy a multinomial distribution: for any xo, z1,. .. , xN-1i EN 01 xi = n, P(7r0 = x0, ... , 7CN-1 = xN-1) = n!/(xo! . xN-1!)(1/N)n. An important fact is that for any A > 0 and independent Poissonian random variables vo, v2i ... , vN-1 with parameter A, P(7r0 = x0, ... ) 7rN-1 = xN-1) =P(v0=x0,...,FN-1=xN-1Ix0+...+xN-1=n).

In analytical applications, the optimal choice of A is A = n/N (the density of the particles). Let's also introduce the random variables 'Yo,'yi, ... depending on the parameters n, N: rys (n, N) = #{7ri = s, i = 0, ... , N - 1},

s = 0, 1, ... , n.

Of course, Yl+2-N+...+-nyn=n

(2.1)

In statistical physics, formulas (2.1) are equivalent to the conservation laws for energy and momentum. The rth-factorial moment, pr (s), for ys is p, (8)

- r!

N

n!

(N - )n-rs

r (s!)r(n - sr)!

Nn

Based on this, it follows that

[N(n ) se-n/Nl re-nr/(N(N-1))-r(r-1)/(N-(r-1))-sr(sr-1)/(n-(sr-1))+r2s/(N-1)

s!N

J

n) se-n/N1r pr(s) < [N s! l`N

e-r(r-1)/N-sr(sr-1)/n+r2s/N

The asymptotics of the pr(s) are determined by the [(N/s!)(n/N)se-n/N]r factor that appears in both the lower and upper bounds for pr(s). There are two different situations: the low density case, n > N. In the first case, yo = o(N), in the second case, -yo = O(N). The limit

A. GORDON ET AL.

204

theorem results we now formulate were used in [2], those concerning the ys statistics are essentially due to von Mises, see [9]. We begin with the low density case.

Proposition 2.1. Assume that (N/s!)(n/N)s - as, n, N -> oo. Then ear(s) (a3)r and, therefore, ys is asymptotically Poisson with parameter as.

In particular, if n2/(2N) -> a2, n, N -> oo then -y2 (n, N) has asymptotically a Poisson distribution with parameter a2, and -ys (n, N) -4 0 in probability for any s > 3. Furthermore, if we let Mn = maxie-a2,P{Mn>1}---*

1-e-a2.

Joint asymptotic distribution behavior is modeled by

Proposition 2.2. Let n3/(6N2) -> a3. Then, for fixed z and x3 > 0, we have (

P{ y2 = xz =

3a3N n

l 3a3N n y3 = x3 7

-z

e-z2/2

e

_a3 (a3)x3 3

Proposition 2.3 (Dynamical limit theorem). Let T2 = min{t : max {7ri(t, N)} = 2}, z 0, _.2/2 P( T2 >a } -,e 1

f(

.

vfN-

Thus, the random variable (T2)2/(2N) has the standard exponential limiting distribution. One can introduce the second, third, etc. collision times and find the joint limiting distribution, see [2]. Let now n >> N (the high density case) and more specifically,

()8e" N,a s>0, that is

n=N[lnN-sln(lnN)(1+o(1))]. then

Proposition 2.4. The number of s-occupancy cells -/,(n, N) has asymptotically a Poisson distribution with parameter as.

Proposition 2.5. Let To = min{t y°(t, N) = 0}. Then P{T° < t} _ e-a0, a° = Ne-tIN, or r T N1nN P To = ° N < - In a° =e -a0 . :

P{y0(t, N) = 0}

This gives, for - In a° = x, P{To < x} -> e_e

x E R;

the classical double exponential distribution.

Proposition 2.6. If m(t) = mini 0} - 1- e-a0.


205

The theory for the limiting distribution of M(t, N) = maxti oc. We will not go into details, but we summarize in broad fashion the results that can be found in [4, Chapter 2, Section 6]. We want to include them for comparison with the corresponding results for Bose-Einstein models. Proposition 2.7. The asymptotic distribution for M(t, N), t, N -> 00 depends on the behavior of the limit parameter t/(N1nN), t,N - oo as follows: (1) when

t/(N1nN) - 0, the limiting distribution of M(t,N) is concentrated in one or two points; (2) when t/(N1nN) -+ x > 0, the limiting distribution of M(t,N) is concentrated on countably many points and takes the form P{M(t, N) < r + k} exp{-Ayk+1/(1 - y)} where y is the root of the y + x(lny - y + 1) = 0 in an

interval 0 < y < 1; (3) when t/(N1nN) - oo, t,N -> oo, then M(t,N), when suitably centered and scaled has, asymptotically, the classical double exponential distribution.

3. FD(n, N)/ BE(N - n, n + 1) duality and limit theorems Each run of a true FD(n, N) experiment chooses a partition of the remaining (unchosen) N - n cells into n+1 groups so that each partition is equally likely. This implies that each run of an FD(n, N) experiment is equivalent to running a BE (N-

n, n + 1) experiment. An extensive family of tests can be based on occupation statistics for various patterns associated with FD experiments and corresponding patterns associated with their equivalent BE experiments. These are discussed in [8]. In this paper, we are going to limit ourselves to BE analogs of the statistics for MB experiments that were recalled in the previous section. Before proceeding with this agenda, however, there are a couple of technical issues that we need to discuss. We have identified an FDO-generator with a sequence yl, y2, ... , yN where yz E {0, 1,-, N-1} and each integer from 0 to N-1 occurs exactly once in the sequence. An FDO-generator could be the result of a run of an FDO(N, N) experiment and we are interested in testing/validating whether that is the case for a given FDOgenerator. If a generator is the result of an FDO(N, N) experiment, then we can use the first n, n < N, terms in the sequence, yl, Y2, ... , yn to simulate/represent the outcome of an FDO (n, N) experiment. Running such an experiment can be accomplished in two steps: (1) First running an FD(n, N) experiment (equivalent to uniformly picking a subset of the N cells of size n).

(2) Then uniformly picking a permutation on the n cells to determine the order in which they are filled.

If we use an FDO-generator to simulate an FDO(n, N) experiment, n < N, the result yl, y2i ... , yn generates both a candidate FD(n, N) experiment and an experiment in choosing a random permutation of order n. The generator can be evaluated on the basis of how well such simulations produce representative results for both types of experiments. However, we have found that most decent FDPRNGs do not produce obviously flawed results on such experiments. Of course, just because a statistic based on a single sample doesn't supply enough evidence to reject a null hypothesis, doesn't mean that, for instance, the empirical distribution of the statistic based on all possible samples or a large collection of "independent" samples might not indicate that there is a problem. This leads to a desire to repeat

A. GORDON ET AL.

206

samples. If one takes the next n elements in the sequence, y,,,+1, y2, , yen, the result should produce a valid random permutation of size n, however, it cannot be claimed to be a run of an Fl) (n, N) experiment, since the elements of the subset {yi, y2, , yn} are not available. With modification, of course, it can be consid-

ered to be an FD(n, N - n) experiment. Thus, tests based on repeated FD(n, ) experiments cannot be considered to be identically distributed, which leads to additional complications. We are not going to deal with these complications in this paper. Rather, we will focus here on some tests that avoid this problem but still illustrate some of the issues involved in building tests that are appropriate for FDOgenerators. There is one last technical issue to mention. An FDO-generator has a beginning. If we choose a seed i0 < N - n and then extract a contiguous sample of size n, Yio+1, ... , Yio+n, we have to view this as having first chosen a sample yl, ... , yio (or equivalently, R(io) = {I.... , N} - {yl, . . , yio }) and then selecting a random nonrepeating string of length n from R(io). So, if we are considering an FDPRNG to be a simulation of an FDO-experiment, we need to choose a seed to identify which FDO-experiment. Thus, a given FDPRNG can be considered to be a simulation of N different FDO-experiments. .

3.1. Limit theorems for BE experiments. By definition, this experiment consists of distributing n indistinguishable particles over N cells. Let 7ri (n), i = 0, 2, ... , N - 1, be the corresponding occupation numbers. As before, -ys (n) = #{i : 0 < i < N - 1, 7ri (n) = s }, s = 0, 1, 2. .... Of course, 7ro (n) + +7rN_1(n) = n, ryl (n) + 2ry2 (n) + = N, and there is a one-to-one = n, 'Yo (n) + 'yl (n) + correspondence between nonnegative solutions of the equation x0 + + XN_1 = n and the elementary events {7r0 = x0, ,7TN-1 = xN-1}. It is well known that the _ (N+n-1) number of such solutions is equal to n N-1 The random variables 7ri are conditionally independent and identically geometrically distributed with an arbitrary parameter p. That is P{7r = l} = qp1, l > 0, E7r = p/q = p/(1 - p). One can optimally select p from the equation E7r = n/N. So, take p so that p/(1 - p) = n/N, or (n'+n-1)

p

n

N+n

Combinatorial calculations in the BE case are simpler than in the MB case of Section 2 above. As before, we will discuss two cases: the low density case, n/N 1, and the high density case, n/N >> 1. The easiest way to establish the asymptotics of the 'f statistics is to use techniques developed in [9] to derive formulas for their factorial moments of all order,

recognize them asymptotically as those of a Poisson distribution and then use a result due to von Mises to imply that they are then asymptotically Poisson. The technique in [9] is based on inclusion exclusion methods that establish that the kthfactorial moment of -ys, µk (s) = k!S(s, k), where S(s, k) = > P{7ri1 = s, ... , 7rik _ s} and the sum is taken over all distinct subsets of {0, . . . , N - 1} of size k. We now establish an expression for µk (s) that will be specialized to handle the low and high density cases.


207

Proposition 3.1. The kth factorial moment of 78, µk(8) is given by Ilk (s) = X

(N2ns)k

(1 - 1/N)2 .. (1 - (k - 1)/N)2(1 - k/N)(1- 1/n)

(1 - (ks - 1)/n) (N + n)k(s+l) (1-1/(N + n)) ... (1- k(s + 1)/(N + n))

PROOF.

S(s, k)

-

_

-

jBE(n, N) (N)IBE(n_ks,N_k)j

N (n ( k)

-1)

k`+N

1 (N+n-1\

N-1 J (n + N - k(s + 1) - 1)!/((N - k - 1)!(n - ks)!) N! k!(N - k)! (N + n - 1)!/((N - 1)!n!) 1))

1

(N+n-1)...(N+n-k(s+1))

k!

1))

1

- k!

(N + n)k(S+1) (1 - 1/(N + n))

(1 - k(s + 1)/(N + n))

1 (N2ns)k (1-1/N)2 ... (1-(k-1)/N)2(1-k/N)(1-1/n) (1-(ks-1)/n) (N+n)k(s+1) (1-1/(N+n)) (1-k(s + 1)/(N+n))

k!

. .

Since µk(s) = k!S(s, k), the result follows.

El

3.1.1. The low density case. This first result is a specialization of Proposition 3.1 to the case when n -- 0, n, N -- oo. Proposition 3.2. Let N(n/N)s -> as > 0, n, N -- oo. Then the random variable -y, ,(n) is asymptotically Poisson with parameter as and s) µk (s) =

(N()

X

k

(1-1/N)2...(1-(k-1)/N)2(1-k/N)(1-1/n) ... (1-(ks-1)/n)

(1 + n/N)k(s+l) (1 - 1/(N + n)) . . . (1 - k(s + 1)/(N + n)) We can also consider BE-random variables such as 'y>g = #{i : 1 s} but, except for the s = 2 case, we defer that analysis to [8]. Here we deal only with obtaining explicit estimates for P{-y>2 = 0} which we have found useful below and in the testing section. Proposition 3.3. Let n2/N --3 a2 > 0, n, N --> oo. Then P{-y>2 = 0} -> e-a2 and, more explicitly: (1-1/N)(1-2/N)...(1-(n-1)/N).

O} - (1+1/N)(1+2/N)...(1+(n-1)/N)' e-n(n-1)/N e-n(n-1)/(N-A) < P{7>2 = 0} < (2) where A = maxn,N{n2/N} (n2/N -> a2). PROOF. Let's begin by considering the probability P{'y>2 = 0}. Clearly, (1)

(N)

P{-Y>2 = 0} _ (N n

1)

N(N-1)...(N-n+1) (N+n-1)...(N+1)N (1-1/N)(1-2/N) ... (1-(n-1)/N) (1+1/N)(1+2/N)...(1+(n-1)/N)

A. GORDON ET AL.

208

But, ln((1 - x)/(1 + x)) = -2x - 2x3/3 - 2x5/5 - , so -2x/(1 - x2) < ln((1 x)/(1 + x)) < -2x. It follows that for (i/N)2 < n2/N2 < A/N, where A = maxn,N{n2/N} (n2/N -' a2), e-(2 E it i/N)/(1-A/N) < P{y>2 = 0} < e-2 E it i/N = e-n(n-1)/N or

e-n(n-1)/(n'(1-A/iv)) < P{y>2 = 0} < e-n(n-1)/iv

From this, it is clear that P{y>2 = 0} -> e-12 and

e-n(n-1)/(r'-A) < P{y>z = O} < provides effective estimation in terms of the asymptotics.

It is not straightforward to introduce the "time" r>2(t) = min{t : y>2(t) > 0}, nonetheless, one can formally put

P{r>2 < aV1N_} = 1 - P{y>2(aV) = 0} - 1 -

e_a2.

Proposition 3.4. If MN,n = maxi 0.

(3.1)

We begin with the factorial moments. This result follows from Proposition 3.1 as did Proposition 3.2 with only one minor and evident modification.

Proposition 3.5. Suppose N2/n -> a > 0, n, N -+ oo. Then, for all s, the random variable ys is asymptotically Poisson with parameter a and

N) 2

( X

k

(1-1/N)2...(1-(k-1)/N)2(1-k/N)(1-1/n).. (1-(ks-1)/n) (1 + N/n)k(.+1) (1 - 1/(N + n)) ... (1 - k(s + 1)/(N + n))

Actually, a stronger result is available:

Proposition 3.6. Under condition (3.1) and fixed s > 0, the random variables -yo(n),-/j(n) , ... , ys (n) are asymptotically independent and Poissonian with the same parameter .A = a.

We will only offer a sketch of the proof of this result, which would be quite tedious to present fully. We begin by proving this result for yo(n) and yi(n). For


209

N>mo+ml,n>N,N=N-(mo+ml),n=n-2N-m1 we have P{yo(n) = m0,'Y1(ii) = ml}

N!/(mo!ml!(N - mo - m1)!) (N+ 1) (N+n-1\ n

N!/(mo!ml!(N - mo - ml)!)\N-mo m01 11) (N+n-1\ n

\

J

(N - 1)!

N!

mo!ml!(N- mo -ml)! (N- 1 -mo -ml)!

(n-N+mo-1)!

X

/

(N

1-

I\1 - N

m0!ml!

nN-(mo+m1)-1 (1 X

1

m0!ml!

(N) n

1 amOe-Ql mo!

J L

1-

N

- (N -

mo+m1)J1 N

mo)/n)N-(m0+m1)-1

nN-1

(l-

x 2

n!

(n - 2N + 2mo + m1)! (n + N - 1)! 1 )2 mo+ml - 1)2(

mo+m1

1 ml!

1

n-N+mo)

1

N-mo-m1-1 n-N+mo

eo(1/n) e-2N2/n+o(1/n) amle-a

J

To deal with the general situation involving {y, = mp, ys = ms}, p < s, s > one writes the P{yy = mp, ys = ms} as a (possibly multiple) sum of probabilities of the form N+n-1l N. \ Nv-1 P{yo = mo, ... , 'YS = ms } _ mo! ... ms!(N - mo - ... - ms)! N+n-1 N-1

with N = N-mo- -ms, n = n-N(s+1)-(m1+2m2+ +sms). Calculations similar to those above allow one to separate the summands which, except for the appropriate factor, can be seen to iteratively sum (asymptotically) to one. There is no similar result available for MB experiments. We introduce a "fictional" collision time by the formula To = min{t : min 7ri(t) > 1} = min{t : yo(t) =01. i a} = P{7ri > a, i = 0,1, ... , N - 1}

P{Xo > a}N P{Xo + ... +XN_1 = n}

P{Xo+...+XN_1 =n}

'

where the X i are i. i. d. r. v. s with values greater than or equal to a and P{ Xo = s

gpsl \Lja qpZ) =

qps-a (i.e., X

= X + a).

We will prove that, in all natural situations, the ratio of the probabilities of sums to truncated sums being equal to n tends to 1 and, therefore, the limit theorems for extremes of the i.i.diT.v.s Xi, i = 0, . . , N - 1, are the same as for the (dependent) occupation numbers 7ri, 0 < i < N - 1. For given n, N, let's consider, first, the limit theorems for the extremes of N independent geometric random variables with parameter p = p(n, N) = n/ (N + n). For this situation, the random variables of interest are just the maximum and minimum values, respectively, obtained on the N trials and we are no longer conditioning on the sum being equal to n. We will denote these maximum and minimum values by MN ,n and mn, n, respectively. This topic is a classical one (cf. [3]). However, the results for integer-valued random variables have some new features. For Mn n, there are three situations with different answers: n = o(N), n - aN, .

and n >> N, i.e., N = o(n). We begin with the case n = o(N). In this case with N, n --> oo, p = n/ (N + n) ---* 0, we have on the basis of standard calculations that e-Pk/(1_pk)N < (1

_P k)N < e_PkN

and hence the asymptotics of (1 - pk)N will be the same as those of e-PkN. Furthermore, for each integer k > 1, PM ,n < k} = (1- pk)N - e-PkN and there will be a largest value for k, K = K(n, N) such that pKN = (n/(N + n))KN > 1. To find K, we consider the inequality -K log ((N + n)/n) + log(N) > 0, which leads


211

to

K

_

log N log N + log (1 + n /N) - log n ]

_

1

1 + (log(1 + n/N)/ log N) - log n/ log N] The asymptotics of log n/ log N will essentially determine how MN',, will distribute itself as n, N - 00.

Proposition 3.8. If n = o(N), N, n --> 00, then one can find integers K = K(n, N) = [In N/ In ((N + n)/n)] such that

P{MNn>K+1}->0 and P{MNn 1. Thus, asymptotically, the only values that can carry mass are K - 1, K, K + 1. If p-0 - 00, which

is usually going to be the case, then P{MNn < K} -- 0 and, if p('-Q) , 0 then P{MNn < K+1} ---* 1, so P{MNn = K} --> 1. On the other hand, /3 may approach

1 in such a way that p(1-0) -> c E (0, 1] and then P{MNn = K} -p e-c and P{MNn = K + 1} -> 1 - e-c (this merely requires that /3 - 1 - log(1/c)/ log(N/n) since then pl-0 tie-(1-0)log((N+n)/n) - e- 109(1/c) = c). Finally, /3 may approach 0 in such a way that p-Q -+ c E [1, oo), in which case, P{MNn < K} , e-c and so P{MNn = K - 1} -> e-c and P{MNn = K} -- 1 - e_c (this requires that /3 log c/ log(N/n)). Let P1, P2 be two probability measures defined on the nonnegative integers. We will say that P1 is c-equivalent to the P2 on the set L C {0, 1, 2.... } if for each

jEL,1-E Jl and [log N/(log(N + n) - log n)] _

[logN/(log(a+1)-loga)]. For any such N, let Jo = [logN/(log(a+1)-loga)] and let /3 = log N/(log(a + 1) - log a) - [log N/(log(a + 1) - log a)]. We have that, for e-4+"N any integer z such that z + J > J1, P{MN n < z + J} ^= e Po+pNPo p = 3e e-PO c0 = where d = pa, co log(po). The assertion follows. F-1

=-

e-de-cV,

Remark 3.10. The inequality P{Mn n < z + J} < e-no+JN holds for all z such that z + J > 0. Thus, the asymptotic distribution can be used to give upper bounds on the initial segment [0, J1] but not effective lower bounds there. Proposition 3.11. For any real z and aN = N2/n -> a > 0, P1 < MNn l

NlnN aN

+ zN } )))

-> exp(-e-az)

N-.oo

PROOF. If h is an integer, h > N log N/2aN, 1 / P{M 0 and Np2k+1 < Np2ryd+1 < p(2-y-1)J _, 0 (recall that pJ+1 < 1). These observations all us to modify the two crucial points in the proof of Proposition 3.13 to complete the proof of this result.

Proposition 3.15. For any real z and aN = N2/n -f a > 0, P{MN,n
y(N log N) /aN, z < y < 1, then for any such k, pk+1 < pry(N log N)/aN =

1

G1 +aN/N

\ ry(NlogN)/aN

- e-(aNIN)1'((Nlog N)IaN) =

N-ry

, 0.

Furthermore, we have

Np2k+1 < N

2ry(N log N)/aN 1

(1+ aN/N

- Ne-(aN/N)2y(NlogN/aN) =

N'-2-y

-+ 0.

These observations allow us to modify the two crucial points in the proof of Proposition 3.13 to complete the proof of this result.

Remark 3.16. Both Proposition 3.13 and Proposition 3.14 involve parameters that depend on N, n. Depending on how n, N -+ oo, the support of the asymptotic distribution in Proposition 3.13 can vary over the entire set of positive integers

or it could oscillate between two pairs {K - 1, K} and {K, K + 1}. Actually, almost any pattern of oscillation on sequences of pairs could be achieved. Because the remainder /.3(N, n) in Proposition 3.14 can range over the interval [0, 1), the parameter d = po 0 can also vary over the interval (1/po,1]. Again, virtually any pattern could be realized.


215

The result for the random variables mN,n is straightforward.

Proposition 3.17. Let n > N - oc, then P{mN,n = 0} = 1 + o(1) if n = o(N2); P{mN,n > a} - (e-1/co )a if n/N2 -+ Co;

P{mN,n/AN > a} -- e-a if n = N2AN, AN

oo.

PROOF. For each a > 0, let Xa = a+X. Then E(Xa) = a+p/q and Var(Xa) p/q2. Since the Var(Xo) = Var(Xo), we trivially have that Var(Xo)/ Var(Xo) -+ 1. Furthermore, we still have that (n - NE(Xo))2/(Var(Xo)N) = 0. However,

(n - NE(X0a))2/(Var(Xo)N) = a/(N(1 + N/n))(N2/n)2. If n = o(N2), this does not have to converge to 0. Take n = N3/2 for example. Then, for a = 1,

a(l/N)(N2/n)2 (1/(1 + N/n)) -* 1 and + Xn,_1 = n}

1

P{Xo+'''+XN_1=n}

e

P{Xo +

This doesn't matter, however, because P{mn, n > 0} - 0, so the first part of the proposition still holds. In the other two cases it is easy to check that

_

(N1

0'

just what is needed to establish the other two results. The proposition is established.

For testing purposes, it will be convenient to have the following specialized precise result:

Proposition 3.18. P{mN,n > k} =

(1-(kN+1)/n)...(1-(kN+(N-1))/n) (1+1/n)...(1+(N-1)/n) 4. Testing

The tests we include in this section are not meant to provide any sort of validation for generators. It can be argued that validation of generators really needs to be conducted in terms of some particular application or applications, which is not our focus right now. First, let's describe the generators that are considered. Excelrand: Excel's rand() generator is widely used and well thought of. It seeds from the clock and we have not seen a description of its algorithm. In particular, we do not know how it relates to the linear congruential generator RAND, the Unix system rand O , or the MAPLE generator "rand O ."

Excelgen: Excel provides another way to generate random numbers through its Tools drop down menu under Data Analysis/Random Number Generation. It is possible to seed the uniform generator provided under this option, although it is laborious. It seems likely that this generator is rand () , but we do not know that for sure. It can be used to extract a relatively small noncontiguous sample.

Unixrand: Unix provides two standard generators that are described in the man pages. This one is evoked by the command randO and can be seeded.

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A. GORDON ET AL.

random(): The other generator provided by Unix. The algorithm is not disclosed, except to say that it uses a nonlinear additive feedback random number generator. It is considered to be a good generator. It can be seeded. GGL: is a multiplicative linear congruential PRNG that uses the algorithm X,,, = 16807X,,,_1 mod (231 - 1).

Lagged-Fibonacci generators:: Lagged-Fibonacci generators use an initial set of seed values x1i x2, ... , XP and two lags p and q, 1 < q < p, to generate successive elements xi for i > p by means of the recursion xi = xi_p o xj_q. Here, o is some modular binary operation which might be +, -, *, or (xor) (exclusive or). Following Marsaglia [5], we designate such a generator by F(p, q, o). Lagged-Fibonacci generators of maximal period are obtained when an associated polynomial is primitive over appropriate modular rings, see [1]. This is true for example of F(17, 5, +) on integers mod 2' (period is (217-1)2m_1, m < p).

Other "good" choices are F(31, 13, +), F(55, 24, +), and F(250,103, +). The initial seeds of these generators can be represented by m x p binary matrices (and to get the maximal orbit, you need this matrix to have maximal rank). The + and * versions have massive numbers of disjoint ergodic components, which has made them candidates for massively parallel applications, see [6]. Even though these generators are not FDO-generators, certain tests can be applied to them anyway. We include them in part to illustrate this and because of their importance. We will deal only with + versions. fdo-3: We have been involved in an effort to produce true random numbers from alpha decay. We used data from a prototype to produce an FDO-generator on the integers from 0 to 218 - 1. randomhalf: This is an FDO-generator on [0, 218 - 1] that was produced from the Unix generator random(). mrand: This is the MAPLE random number generator. It can be seeded. As indicated in the introduction, we are interested in conducting analyses for two different types of samples. Given a generator yl, ... , y,,, a contiguous sample of size k and seed io consists of the next k sequence elements after yio, yio+1, . . . , yio+k Unless otherwise specified, we will consider the index sums to be modulo n. A noncontiguous sample consists of the successors to the integers in an interval assuming that the generator assumes all the values in the interval. Thus to extract the noncontiguous sample associated with the integer interval [1, 2], we would determine i1, i2 so that yi1 = 1, yi2 = 2. The noncontiguous sample would then consist of yil+l,

yi2+i (and order does matter). For PRNGs, yi,+1 = F(1), Yi2+1 = F(2) if F is the underlying dynamical mapping. Noncontiguous samples give some indication of whether the supposedly chaotic system being sampled by the PRNG involves time intervals long enough to wipe out short-term correlations between the nearest phase points in the sample. The assumption that the generator sequence is a good FDO or MBO generator would certainly require this. There should be no clear correlations between successors of integer intervals and a noncontiguous sample should pass all randomness tests, those for independence between its elements in particular. If this does not turn out to be the case, then one can expect that contiguous samples which have "close" points in common will be likely to suffer other undesirable correlations. There are some generators, such as the lagged Fibonacci generators, for which these concepts are a bit ambiguous. The seed of one of these generators is a


217

TABLE 1. Tests of generators based on noncontiguous samples Sample Size Statistic p-value Pattern 400/(200 bits) S200 All ones 2-199 mrand 200/(100 bits) 10-12 1000000... S100 Unixrand 200/(100 bits) S100 10-4 No obvious random() 200/(100 bits) 10-6 Alt. with runs' S100 Excelgen 40/(20 bits) S20 2-19 All ones randomhalf 800/(400 bits) > .64 No discernible S400 fdo-3 800/(400 bits S400 No discernible -1

Generator GGL

'The pattern is alternating 0, 1 punctuated with runs of Is.

vector of numbers which has full rank (in a certain sense) and the number 1 is not going to be in the cycle at the level of the vectors. We can still look at the sequence generated by any given seed and extract noncontiguous samples, but there is no reason to believe that they will be correlated or not correlated. In fact, one can easily dream up seeds for particular lagged Fibonacci generators where there will be a strong correlation and others where there is no reason to believe there will be any obvious correlations. For example, the cycles corresponding to the seeds [0, 2, 0, 3, 2, 2, 0, 3, 2,1,1, 3, 3, 1, 3, 2,1] and [0, 2, 0, 3, 2, 2, 0, 3, 2, 1,1, 3, 3, 1, 3, 2, 0] for

F(17, 5, +) (+ mod 4) are disjoint but have significantly correlated substrings much longer than those involved in just the two seeds themselves. Since disjoint cycles for such generators are proposed as independent sources for massively parallel machines, this may be something to consider further. We will deal with the noncontiguous samples first.

4.1. Noncontiguous samples. The generators that we have extracted noncontiguous samples from and analyzed are: GGL, mrand, Excelgen, Unixrand, random(, and randomhalf. Table 1 presents the results. The statistic S,, is calculated as follows: a noncontiguous sample starting at 1 of size 2n is extracted from the generator and n successive pairs are then compared. A 1 is assigned if the first term is less than the second, a 0 is assigned otherwise.

S,, is then the number of Is. S,, should be binomial if the generator is FDO. If the generator is MBO, S,,, should still be binomial on samples for which the same value is never assigned to two consecutive integers. In an MBO experiment, the likelihood of two consecutive balls going into the same cell is vanishingly small and is easy to check for.

4.2. Contiguous samples. We looked at the performance of a number of generators on a very large number of individual occupation statistics "/s. On single sample tests, all generators appear to perform reasonably well. If one wants then to consider the empirical distributions of these statistics, one runs into issues with repeated samples. Repeat sampling of permutations is straightforward and requires no adjustments. However, once a BE sample has been produced, the next sample will be from a different phase space. Thus, the statistics one gathers are not identically distributed. Adjusting for this takes some effort and will be handled elsewhere. Still, there are two statistics that use the BE statistics derived here but are based on permutations and can therefore be repeat sampled, namely, the

A. GORDON ET AL.

218

TABLE 2. generators based on contiguous samples Generator

Sample Size Min > 0 p-value Min > 1 p-value 1008

.4226

> .30

.1736

Excelrand

1013

.4452

.011

.1737

.03

fdo-3

1043

.4497

.003

.1829

.002

randomhalf F(31,13, +)

1018

.4676

.00007

.2014

.000003

1440

.4632

.00001

.1903

.00002

.03

MN,., = maxz k} can be calculated precisely using Proposition 3.18 (which we did for k E {0,1}). At the sample sizes that we ran, MN,,, did not show anything interesting. However, the high-density mN,,n statistic shows considerable promise to actually differentiate between generators. We tested GGL, Excelrand, fdo-3, randomhalf and F(31, 13, +). The results for tests based on contiguous samples are included in Table 2 The true value for P{mlo,loo > 0} = .4060 and for P{mlo,loo > 1} = .1491. It is interesting that GGL appears to perform the best on this test among generators

that are usually thought to be better than it is. The relatively poor performance of fdo-3 is significant, since this generator was derived from alpha decay data, furthermore, the output of the device from which it was derived passed standard batteries of tests that are used to "certify" generators (including the NIST tests).

5. Transforming FDO generators to MBO generators Let's compare the entropy of both the ideal MBO(n, N) and FDO(n, N) experiments. The distribution of n distinguishable particles over N boxes contains entropy H(MBO(n, N)) = n 1092 (N).

The entropy for the corresponding FDO(n, N) experiment is slightly smaller, namely, if n/N is bounded away from 1, then

H(FDO(n, N)) = log2(N) + ln(2)

+ log2(N - n + 1)

[1+2+...+(n_1)1 J

-nlog2N(21n2)N


219

If n/N - 1, then H(FDO(N, N)) = 1092 (N!) - log2((N - n)!) = N 1092 N - N 1092 e + O(log2 N) - (N - n) loge (N - n)

+(N-n)log2e-O(log2(N-n)) From these considerations, an exhaustion argument can be constructed that shows that the per bit entropy of both experiments are asymptotically the same as N -+ oc. This indicates that if one has, for each N, an ideal FDO generator (equivalently, an ideal FDO(N, N)) then one should be able to use it to simulate an ideal MBO(n, M) experiment for some suitable range of values for n and M with as little loss of per bit entropy as desired. We will discuss three methods for simulating MBO(n, M) from appropriate FDO generators. The first method will be the most efficient in the sense that the asymptotic average per bit entropy will converge to 1. However, a price will be paid in terms of computations and memory required. Another method, called the "adding boxes" method, is attractive from some points of view, is less complicated to program, but still requires memory and sacrifices a significant portion of the per bit entropy. The third method only uses the permutation data. It is the most straightforward to implement and requires no memory, however, for fixed n, asymptotically the average per bit entropy approaches 0. Before proceeding with the first method, let's consider a specific example to illustrate some of the difficulties that will have to be dealt with.

Example 5.1. Let's take N = 10, n = 3, M = 5. We want to use an FDO(10,10) generator to simulate an MBO(3, 5) experiment. Suppose the FDO generator is yi,... , ylo, 0 < yj < 9. From yi, we can determine which of the equally likely equivalence classes {0, 1}, {2, 3}, {4, 5}, {6, 7}, {8, 9} that yl is a member of and thus determine a unique, uniformly distributed number from 0 to 4, which we denote by z1. Note, some entropy has been lost in doing this. Now, Y2 can be considered to be a value from a uniformly distributed random variable on

the N - 1 = 9 numbers from 0 to 8 (by determining its rank among the numbers {0, 1, ... , 9} - {yl}). If 0 < Y2 < 4, then Y2 does represent an equally likely selection from these five integers and we can use it to determine z2. The probability that z2 is determined on this second draw is 9. But, with probability 9, 5 < Y2 < 8 and conditioned on this, the value chosen is for a uniformly distributed distribution on four values 0 to 3, not five. We may keep this value in mind and look at y3. With probability 8, z2 will be determined, if it was not determined by the y2. But, with probability 8 , y3 will represent a value for a uniformly distributed random variable

on the three values 0 to 2. We could be forced to continue to y4 without z2 yet determined. If y4 is not between 0 and 4 (which would happen with probability 7 ), then it represents a value from a uniformly distributed random variable on the two values 0, 1. We could thus be forced to continue. On the next draw, the outcome is either between 0 and 4 or is uniquely determined, so z2 may still not be determined. If it is not, the next draw will determine z2 for certain, but we will not be able to determine z3 except in some approximate sense. Thus, an FDO(10, 10) generator can determine a perfect MBO(2, 5) experiment (although it

might need six draws to do it) but it cannot, with probability one, determine a perfect MBO(3, 5) experiment. If N > 15, however, we would always be able to

A. GORDON ET AL.

220

determine a perfect MBO(3, 5) experiment (in general, if N > nM, an FDO(N, N) generator will be able to simulate a perfect MBO(n, M) experiment).

We are going to do three things to simplify the rest of our discussion of this case: (1) we will fix n, (2) we will seek samples of size n which are uniform on

[0, M - 1] where M is a power of 2 and M > Mo = 2m0 for some positive integer mo,and (3) we will be allowing N to always be large enough so that an MBO(n, M0) experiment can always be successfully completed (or, at least, we aren't going to worry about that kind of complication). Let's first consider how we can get on average as many bits as possible out of a draw from a uniform distribution on the numbers from 0 to N -1 (this would correspond to taking the case that M = 2). If N = 2k for some positive integer k, then all the numbers from 0 to N - 1 = 2k - 1 are equally probable and so we can extract k lower bits with probability 1 (that is, the dyadic expansion for yl may be extracted). If 2k < N < 2k+1, then we + 2k1 be the nonzero have to do something different. Let N = 2k1 + 2k1-1 +

< k1 = k. As in

terms of a dyadic expansion for N. Here, 0 < kl < k2
k1 - [(k1

2k1-1 + ... + 2k1

2k1

(k1 -

N

k1_12k1-1 + ... + k12k1

[(k1 - k1

N

+ kl

-

k1-1)2k1-1-k1 +.

+ (k1 -

J kl)2k1-k1]

> k1-2= [log2N]-2>log2N-3. It follows from this that the average per bit entropy of this extraction scheme is asymptotic to 1. For a given Mo = 2m0, we need mo bits for each sampling. For a given N, we can extract bits from each draw according to the above scheme and keep going until we have at least mo bits. We then have a sample point from a uniform distribution on [0, M - 1] where M = 2', m = #bits extracted. Again, as N -> oo, such an extraction method will yield an average per bit entropy of 1. If we want to extract a sample of size n > 1, then we just continue this process until we have at least nmo bits. We then extract our sample by taking the largest

block of bits, say k bits, such that k > mo and nk < #bits extracted. Taking M=2 k, we have a sample of size n by taking the first n consecutive k-blocks. The reader should not find it difficult to see that the average per bit entropy of such a procedure will still approach 1 as N --> oo. We formalize the algorithm below. Assume that we have an FDO generator, yi,... , yN.

FERMI -DIRAC GENERATORS AND TESTS FOR RANDOMNESS

221

Algorithm 5.2. Let mo, N >> 2m° and n be positive integers. Let B be a binary array of length JBI = m(log2 N + 1). Below, for any integers z > 0 and k > 0, zk means the kth bit in the binary representation of z, and r(ye), 1 < j < N, is the rank of yj in the set {0, 1, ... , N - 1} - {y1, ... , yj-11. S +-0

j

1

L: Rf-N-j+1 1 F- max{k: Xk < Rk} (Bs, Bs+1 .... Bs+1-1) , (xo, x1, ...

s- s+l

,

x1-1)

if s > nmo then return (Bo, B1i ... , Bs-1) else

j-j+1

if j > N then stop else go to L end if

end if

The random sample is then extracted from B by extracting n blocks of size k = [s/n], where [a] denotes the integer part of a real number a. The following proposition summarizes the results above:

Proposition 5.3. Let mo, n, N >> 2"'L0 be positive integers. If Algorithm 5.2 is used to extract a sample of n uniformly distributed r.v.s on the integers 0 to M - 1, where M = 2k and k is determined as indicated, then the average per bit entropy of the method approaches 1 as N - oc.

A similar result is valid for an arbitrary positive integer M. An M - bit is an integer from 0 to M - 1. One has to modify the argument in (5.1). We expand N in terms of powers of M obtaining N = 011Mk, + 011_IMk1_1 +

+ aiMkl. A

natural modification of the bit extraction method to extract M - bits instead and a little thought leads to the expression

E(#M - bits)

= k1 alN

ki

+ k1_1

kia1Mkt +

a1-IN k`

1

+ ... + k1

a1Nkl

kia1Mk1 a1_1Mk`-1 +... + a1Mk1

a1Mk' + From this a lower bound on the average number of M - bits that can be extracted

per sample point can be established which can be used to show that the average per bit entropy again approaches 1 as N -+ oo.

Remark 5.4. The above method does allow one to extract MBO samples from an FDO(N, N) generator with practically no loss of entropy provided N is large enough and we are willing to allow the exact size of the interval [0, M - 1] being sampled to be determined by the process. However, it requires significant computations and may also involve significant memory. If we wish to fix M = Mo ahead of time and if Mn nK, the scheme is guaranteed to produce the desired sample. It requires no memory and is readily adapted to methods that involve random seeding of the FDO generator (it can be started at any point in the string without having to worry about outcomes that may have preceded the ones before that term). It can even accommodate within-sample random reseeding if a simple algorithm is employed to check that no overlaps are occurring between the permutation samples (this would require memory, however).

As an example, consider GGL as an FDO(231 - 1,231 - 1) generator. If we were interested in a uniform distribution on the integers from 0 to M - 1, where M > 221, then taking K = 10 will suffice, since 10! = 3628800 > 2097152 = 221 Now, there are [N/10] = [(231 - 1)/10] = 214748364 permutations that can be sampled from GGL, which is between 227 and 228, which is quite a respectable MBO generator size. Indeed, this may very well be the best way to convert GGL to an MBO generator that has the possibility of collisions. In [2], we projected GGL

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224

onto its most significant 21 bits and rejected that it was an MBO source based on the fact that the X statistic considered in [2] was too small, this required a sample of size 8,000,000. We have not run the experiment yet, but we conjecture that, if we use the permutations instead, the resulting generator will no longer have that problem.

References 1. R. P. Brent, On the periods of generalized Fibonacci recurrences, Math Comp. 63 (1994), no. 207, 389 - 401.

2. A. Figotin, A. Gordon, S. Molchanov, J. Quinn, and N. Stavrakas, Occupancy numbers in testing random number generators, SIAM J. Appl. Math. 62 (2002), no. 6, 1980-2011. 3. B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Co., Cambridge, MA, 1954. 4. V. F. Kolchin, B. A. Sevast'yanov, and V. P. Chistyakov, Random allocations, Scripts Series in Mathematics, V. H. Winston & Sons, Washington DC, 1978. 5. G. Marsaglia, A current view of random number generators, Computer Science and Statistics: The Interface (Atlanta, GA, 1984) (L. Billard, ed.), North-Holland, Amsterdam, 1985, pp. 310.

6. M. Mascagni, M. L. Robinson, D. V. Pryor, and S. A. Cuccaro, Parallel pseudorandom number generation using additive lagged-Fibonacci recursions. 7. V. V. Petrov, Sums of independent random variables, Ergeb. Math. Grenzgeb., vol. 82, Springer, New York -Heidelberg, 1975. 8. J. Quinn, Occupation statistics revisited: Factorial moment formulas for general patterns, in preparation. 9. R. von Mises, Mathematical theory of probability and statistics, Academic Press, New YorkLondon, 1964. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NORTH CAROLINACHARLOTTE, 376 FRETWELL BLDG., 9201 UNIVERSITY CITY BLVD., CHARLOTTE, NC 28223-0001,

USA

E-mail address, A. Gordon: [email protected] E-mail address, S. Molchanov: [email protected] E-mail address, J. Quinn: [email protected]


The Spectral Problem, Substitutions and Iterated Monodromy Rostislav Grigorchuk, Dmytro Savchuk, and Zoran Sunic ABSTRACT. We provide a self-similar measure for the self-similar group 9 act-

ing faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial z2 + i. We also provide an L-presentation for G and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of G on the orbit of a point on the boundary of the binary rooted tree.

Introduction It was observed recently that the class of self-similar groups naturally appears in mathematics. The most recent examples come from combinatorics and are related to one of the most famous combinatorial problems known as Hanoi Towers Game [14]. Slightly older examples, due first of all to Nekrashevych [19] (see also [2]), are related to holomorphic dynamics and random walks. Self-similar groups can be defined as groups generated by all the states of a (not

necessarily finite) Mealey automaton [12]. Of particular interest and importance is the case when the automaton is finite, since in that case the obtained group is finitely generated. Self-similar groups have many interesting and important properties. The class of self-similar groups contains exotic examples, such as groups of Burnside type or groups of intermediate growth, as well as familiar examples, such as free groups or free products of finite groups, that are well known and are regular objects of study in combinatorial group theory. One of the most remarkable discoveries in the recent years is the observation, due to Nekrashevych, that the so-called iterated monodromy groups (IMG), which can be related to any self-covering map, belong to the class of self-similar groups and that, in the most natural situations, there is an explicit procedure representing them by finite automata. Even in the case of quadratic maps over C one gets a rich theory with wonderful applications both to holomorphic dynamics and to group theory [4, 19]. 2000 Mathematics Subject Classification. Primary 20E08; Secondary 37F10, 60B15. The first author is partially supported by NSF grants DMS-0308985 and DMS-0456185. The third author is partially supported by NSF grant DMS-0600975. This is the final form of the paper. ©2007 American Mathematical Society 225

R. GRIGORCHUK, D. SAVCHUK, Z. SUNK

226

Already the simplest examples of quadratic polynomials, such as z2 - 1 or z2 + i, show that the corresponding groups can be quite complicated and can have extraordinary properties. The group IMG(z2 - 1) is called the Basilica group after the Julia set of z2 - 1 which (mildly) resembles the roof of the San Marco Basilica in Venice (the top part of the Julia set is the roof and the bottom part is its reflection in the water). Basilica group is torsion free, of exponential growth, amenable but not elementary (and not even subexponentially) amenable [6,16], has trivial Poisson boundary, is weakly branch, and has many other interesting properties. The main object of this article is the group IMG(z2 + i) (denoted by C in the

rest of the text), introduced in [2] and later studied by Bux and Perez [7], who proved that g has intermediate growth. This is not the first example of a selfsimilar group of intermediate growth (the first examples were constructed in [8,9]), but it is the first example of a group of intermediate growth that naturally appears in the area of applications of group theory.

We start with a quick introduction to the theory of self-similar groups and, in particular, to iterated monodromy groups. We are aiming for a self-contained treatment, which would make it possible for the reader to understand the context of the paper completely without reading other sources. In particular we give very detailed calculation of the action of g = IMG(z2 + i) on the binary rooted tree. Then we show that the group is a regular branch group, thus presenting an example of a branch group which naturally appears in holomorphic dynamics. The main body of the article is devoted to the calculation of an L-presentation for 9 (i.e., a presentation of a group by generators and relations which involves a finite set of relators and their iterations by substitution). Although it is known that Lpresentations are quite common for groups of branch type the number of examples in which explicit computation is possible is rather small. The presence of L-presentations is important from different points of view. Such presentations are at the first level of complexity after the finite presentations and quite often provide the simplest way to describe a group that is not finitely presented (G is not finitely presented [20]). Further, such presentations can be used to embed a group into a finitely presented group in a way that preserves many properties of the original group. We use the obtained L-presentation of 9 to embed CJ into a finitely presented group with 4 generators and 10 relators, which is amenable but not elementary amenable (this approach has been used for the first time in [10]).

During the last stage of the preparation of the manuscript we were informed by Bartholdi and Nekrashevych of their work [5] providing calculation of an Lpresentation for IMG(z2 + c) for every post-critically finite quadratic polynomial z2 + c. It would be interesting to compare the obtained presentations. The rest of the article deals with finding a self-similar measure on 9. The notion of a self-similar measure was introduced by Kaimanovich in [18], who extends some

ideas (in particular the idea of self-similarity of a random walk) that appeared in the work of Bartholdi and Virag [6]. The self-similar measure is closely related to the problem of computation of the spectrum of a Hecke type operator that can be related to any group acting on a rooted tree and to the problem of computation of the spectrum of the discrete

THE SPECTRAL PROBLEM, SUBSTITUTIONS AND ITERATED MONODROMY

227

Laplace operator (or, what is almost the same, the Markov operator) on the boundary Schreier graph of a group (i.e., the graph of the action of the group on the orbit

of a point of the boundary). A general approach to the spectral problem (which extends the ideas outlined in [1, 15]) based on a renormalization principle and leading to questions on amenability, multidimensional dynamics and multiparametric self-similarity of operators is described in [13]. Unfortunately, the spectral problem is not solved yet in our situation. What we are able to construct is a rational map on ]R3 whose proper invariant set (shaped as a "strange attractor") gives the spectrum of the Markov operator after intersection by a corresponding line. Here we have a situation analogous to the case of Basilica group [17]. Further efforts in the description of the shape of the attractor (and hence of the spectrum) are needed. The Schreier graph in this case, viewed through a macroscope, has a form of a dendrite and this is a reflection of a general fact relating the geometry of Schreier graphs and Julia sets proved by Nekrashevych [19].

In any case, our computations allow us to find a self-similar nondegenerate measure on g, which gives a self-similar random walk on the group. The study of asymptotic properties of such random walks is a promising direction and will be one of our subsequent subjects of investigation.

1. Iterated monodromy groups The theory of iterated monodromy groups was developed mostly by Nekrashevych. A very detailed exposition can be found in his monograph [19]. Here we give a definition and some basic properties of these groups. Consider a path connected and locally path connected topological space M.

Let M1 be an open and path connected subset of M and f : M1 --* M be a dfold covering map. Fix a base point t E M and let 7ri (M, t) be the corresponding fundamental group. The set of iterated preimages of t under f has a natural structure of a d-ary rooted tree T. Namely, each point s from this set has exactly d preimages s1, ... , sd and these preimages are declared to be adjacent to s in T.

The nth level of the tree T consists of the d' points in the set f -n(t). Note that although the intersection of f-n(t) and f-'n(t) may be nonempty for m j4 n, we formally consider the set of vertices of T to be a disjoint union of the sets f -n(t),

n>0.

There is a natural action of ir1(M, t) on the tree T. Let -y c 1r1 (M, t) be a loop based at t. For any point s of f -n (t), there is a unique preimage y[s] of -y under f n which starts at s and ends at a point s', which also belongs to f -n(t). We define an action of -y on T by setting y(s) = s'. This action induces a permutation of f -n(t) because the preimages of y-1 starting at the points of f -n(t) are defined uniquely as well. The group of all permutations of f -n(t) induced by all elements of iri (M, t) is called the nth monodromy group of f. If y(s) = s' then y(f (s)) = f (s') since f ('Y[s]) = y[f(s)}, so y acts on T by a tree automorphism. The action of 7r1(M, t) on T is not necessary faithful. Let N be the kernel of this action.

Definition 1.1. The group IMG(f) = 7ri (M, t)/N is called the iterated monodromy group of f.

It can be shown (see [19] for details) that, up to isomorphism, IMG(f) does not depend on the choice of the base point t.

R. GRIGORCHUK, D. SAVCHUK, Z. 3UNIC

228

In order to describe the automorphisms induced on T by the loops from 7r1 (M, t)

we need to come up with a "coordinate system" on T. Let X = {0, 1, ... , d - 1} be a standard alphabet of cardinality d. Then the set X* of all finite words over X also has the structure of a d-ary rooted tree, where v is adjacent to vx, for any

vEX*andxEX.

The group Aut X* of all automorphisms of X * has the structure of an infinite iterated permutational wreath product li>l Sym(d) (because Aut X * = Aut X * lx Sym(d), where Sym(d) acts naturally on X by permutations). This gives a convenient way to express automorphisms from Aut X* in the form (1.1)

9 = (910,911,

,91d-1)cr

,

where g10, 911) . , 91d-1 are automorphisms of the subtrees of X* with roots at the vertices 0, 1, ... , d - 1 (these subtrees are canonically identified with X*) induced

by g, and Qg is the permutation of X induced by g (i.e., ag(x) = g(x)-the action of g on x E X). More generally, for every u E X * we define g 1u to be the automorphism of the subtree of X * rooted at u (identified with X*) induced by g. The automorphism glti, is called the section of g at u and is uniquely determined by g(uw) = g(u)glu(w), for all w c X*. The product of automorphisms written in form (1.1) is performed in the following way. If h = (hIo, h11, ... , hld-1)0'h then 9h = (9IohIQ9(o),... , 9ld-1hL,,9(d-1))agah

By definition gh(w) = h(g(w)).

Definition 1.2. A group G < AutX* is called self-similar if glu E G for all

gEGanduEX*. A convenient way to describe a particular finitely generated self-similar group G generated by automorphisms 91, 92, ... , gn is through a, so-called, wreath recursion.

In this presentation we simply write the action of each gi in the form

9i =(wl(91,...,9n)e...,wd(91,...,9n)J0`gi, where wi, i = 1, . . . , n, are words in the free group of rank n. Another language which describes self-similar groups is the language of automaton groups (see the survey paper [12] for details).

Definition 1.3. A Mealy automaton is a tuple (Q, X, it, A), where Q is a set (a set of states), X is a finite alphabet, 7r: Q x X -> Q is a transition function and A: Q x X -* X is an output function. If the set of states Q is finite the automaton is called finite.

One can think of an automaton as a sequential machine which, at each moment

of time, is in one of its states. Given a word w E X* the automaton acts on it as follows. It "eats" the first letter x in w and depending on this letter and on the current state q it "spits out" a new letter A(q, x) E X and changes its state to 7r(q, x). The new state then handles the rest of word w in the same fashion. Thus

the map A can be extended to A: Q x X * -p X * -we just feed the automaton with letters of u c X * one by one. Each state q of the automaton defines a map, also denoted by q, from X* to itself defined by q(w) = A(q,w). In the special case when, for all q E Q, the map A(q, ) is a permutation of X the map q: X* -p X* is invertible and hence, an automorphism of the tree X*. In this case the automaton is called invertible.


229

Definition 1.4. A group of automorphisms of X* generated by all the states of an invertible automaton A is called the automaton group generated by A. The class of automaton groups coincides with the class of self-similar groups. Indeed, the action on X* of every element g of a self-similar group can be encoded by an automaton whose states are the sections of g on the words from X*, transition and output functions are derived from the representation (1.1). Namely, for each u E X*, set ir(9l u, x) = 9lux and A(9lu, x) = 9l u(x) Important subclass of automaton groups consists of groups generated by finite automata. For example, we know that for groups in this class the word problem is solvable.

A standard way to visualize automata is by so-called Moore diagrams. Such a diagram is an oriented graph where the set of vertices is Q and for every q E Q, x E X, there is an edge from q to ir(q, x) labeled by (x, A(q, x)). In case of invertible automata it is common to label states by the corresponding permutations of X and leave only the first coordinate on the edge labels. An example of a Moore diagram is presented in Figure 5. We go back now to iterated monodromy groups and construct an isomorphism A: X * --> T such that the induced action of iri (M, t) on X* becomes particularly nice (self-similar). We construct A level by level. Set A(I) = t. For each vertex v in Xn we

will construct a path l in M joining t to one of its preimages s, from f-n(t) and define A(v) = s,. Choose arbitrarily d paths lo, ... , ld_i in M connecting t to its d preimages in f -'(t) and, for x c X, define A(x) to be the end of the path l.'. Now assume we have already defined A(v) and corresponding paths l for all v E Xm, m < n and A is an isomorphism between the first n levels of X* and T such that, for all vertices v on the first n levels, A(v) is the endpoint of £,,. For any word

xuEX'+i with x E X anduEXn define l3u = luf[ A( u 'u)] (lx),

where f[! (. )] (li) is the unique preimage of the path l., under fn starting at the

vertex A(u) (composition of paths is performed from left to right, i.e., the path on the left is traversed first). Define A(xu) to be the end of the path lxu.

In order to prove that A is an isomorphism of trees we need to show that f (A(xvy)) = A(xv), for all x, y c X and v c X*. Indeed, f(lxvy) = f(lvy)f(f[A(vy)](lw)) = f(lvy)f[A(

i) (W.

By definition, f[A(v)] it (h) is a path going from A(v) to A(xv), so the end A(xvy) of is mapped to A(xv) under f. Abusing the notation, we often identify the path the trees T and X* and write v for A(v) (see Figure 1, where solid lines represent edges in the tree T and dashed lines represent paths in M).

Definition 1.5. The action of IMG(f) on X* induced by the isomorphism A is called the standard action of IMG(f). The tree isomorphism A allows us to compute iterated monodromy groups using the language of self-similar groups [19]. We provide the details here in order

to keep the paper relatively self-contained and to help the understanding of the computations that follow. Recall, that for any loop -y based at t and any u E f -n(t)


230

we denote by 'y[z,] the unique preimage of -y under f fl that starts at the point u. Similarly, f[ (lx) denotes the unique preimage of the path l., starting at u. Theorem 1.6. The standard action of IMG (f) is self-similar. More precisely, of y E IMG(f) at x E X is given by

the section 'y (1.2)

ylx =

hy[.](h(.))-1.

PROOF. Let V E XTh be an arbitrary word and suppose -y(xv) = yu, for y E X

and u E X. Then vertices v and u are connected by the path p=

y[xv]

(f[U (ly))

which goes through the vertices v - xv - yu -i u (see Figure 2, where solid curves represent paths in M and dashed lines represent paths in the tree X*)). We have

f"(p) = lxy[.]ly Thus the loop £ = l,: y[xj ly 1 based at t represents the element of IMG(f) which moves

v to u. The loop £ is independent of v (and u). Thus we have ylx = lxy[x]ly1.

We are now ready to compute the standard action of IMG(z2 + i) on {0,1}*. The only critical point of this map is z = 0, which is preperiodic: f

0 4i-+ (i-1)'-i. A(vy)

A(v)

A(xv)

FIGURE 1. Isomorphism A between T and X* v

FIGURE 2. Self-similar action of Iterated Monodromy Group


231

(b)

(a)

FIGURE 3. (a) Paths connecting t = 0 to its preimages (b) Generators of the fundamental group ir(M, t) b[o]

r< 1o]

0'11,

i

0

;[Cl,

4

and, hence, the postcritical set of f is {i, i - 1, -i}. Therefore the restriction of f on Ml =C\ {i, i - 1,-i, 0} is a 2-fold covering of M = C\ {i, i - 1, -i}. Set t = 0 E C as the base point. It has two preimages ei3"/4 and e'77r/4 which are identified with the letters 0 and 1, respectively (more precisely, we set A(0) = ei3a/4 and A(1) = ei7"/4). For the paths to and 11 connecting t to its preimages we choose the straight segments shown in Figure 3(a). The fundamental group 7r1(M, t) is generated by the 3 loops a, b, c shown in

Figure 3(b) going around i, -i and i - 1 respectively. Each of these loops has two preimages a[i], b[i] and c[i], i = 0, 1, shown in Figure 4. According to formula (1.2) and Figures 3 and 4 the sections of the generators a, b, c at 0 and 1 satisfy: alo = loa[o]ll 1 = 1,

all = lla[1]1o 1 = 1,

blo = lob[o]l0 1 = a,

bIl = llb[l]ll 1 = c,

CIO = loc[0]l0 1 = b,

CII = l1c[1]li 1 = 1,

where 1 denotes the trivial loop at t, which represents the identity element of IMG(z2 + i).

Since a permutes the elements of f -1(t), while b and c do not, we obtain the following wreath recursion for the generators of IMG(z2 + i) (1.3)

a = (1, 1) a,

b = (a, c),

c = (b, 1),

232

R. GRIGORCHUK, D. SAVCHUK, Z. SUNK 0,

FIGURE 5. Automaton generating group IMG(z2 + i)

where a is the nontrivial transposition in Sym(2).

These relations show that the set of all sections of the generators a, b, c of = IMG(z2 + i) is {1, a, b, c} and that the group 9 is generated by the states of the finite automaton shown in Figure 5. We now say a few words about the relation between the dynamics of the map z --> z2 + i and the combinatorial properties of the action of 9 on the tree T. Recall that if a group G acts on a set Y then the Schreier graph of this action is an oriented graph, whose set of vertices is Y and there is an edge from y E Y to z E Y labeled by g c G if and only if g(y) = z. It is convenient sometimes to forget about the labels and/or the orientation of the edges. Every group acting on a rooted tree acts on each level of the tree. The Schreier graphs of such actions are of particular interest, since in many situations (such as the one we are in) they can be used to find the spectrum of the Markov operator on the boundary of the tree. Recent results of Nekrashevych [19] show that the Schreier graphs of IMG(f) on the levels of the tree converge to the Julia set of the map f. Therefore the structure of the Julia set of f provides understanding of the structure of the Schreier graphs of IMG(f) (and vice versa). In our case the Julia set of z2 +i is the dendrite shown

in the top half of Figure 6. The bottom half of this figure displays the Schreier graph of 9 on level 8. The set of vertices of this graph is just f-8(0) and the vertices are connected according to the action of 9 (no loops are drawn though).

2. Branch groups Another important class of subgroups of Aut X* is the class of branch groups [3, 11]. Here we give basic definitions and prove that IMG(z2 + i) is a regular branch group. This shows that branch groups arise naturally in mathematics (not just as a way to construct groups with unusual properties). Let G be a subgroup of Aut X * . Then for any vertex v E X* one can define the subgroup of G consisting of all the elements in G fixing all words in X* that do not have v as a prefix. This subgroup of G is called the rigid stabilizer of v and it is denoted by ristG(v). Furthermore, the subgroup

ristG(n) = ( U ristG(v) vex" generated by the union of the rigid stabilizers of vertices at level n, is called the rigid stabilizer of the nth level. Since elements of rigid stabilizers of different vertices on


233

FIGURE 6. Schreier graph of 9 and Julia set of z2 + i the same level commute we have

ristG(n) _ JJ ristG(v). vex, Note that, if G acts transitively on the levels of the tree, then all rigid stabilizers of the vertices on a fixed level are conjugate and, hence, isomorphic. Definition 2.1. A group G of tree automorphisms of X* that acts transitively on the levels of X* is called a branch group if all rigid level stabilizers ristG(n), n > 0, have finite index in G. If all rigid stabilizers are nontrivial then G is called a weakly branch group.

It is often easier to prove that a given group belongs to a more narrow class of regular (weakly) branch groups. Consider a self-similar group G and its normal subgroup StG(1) consisting of all elements in G that stabilize the first level of X*. There is a natural embedding

T: StG(1)yGxGx...xG given by 9 -* (910, 911, - ,91d-1)

Definition 2.2. Let K, Ko,... , Kd-1 be subgroups of a self-similar group G acting on X*. We say that K geometrically contains Ko x . . . x Kd_1 and write

Kox...xKd-1K

R. GRIGORCHUK, D. SAVCHUK, Z. §UNIC

234

if Kox...xKd_1GW(StG(i)nK). Definition 2.3. A group G of tree automorphisms of X* that acts transitively on the levels of the tree X * is called a regular weakly branch group over its normal subgroup K if K.

If, in addition, the index of K in G is finite then G is called a regular branch group over K.

It can be shown that if G is a regular (weakly) branch group than it is a (weakly) branch group.

Definition 2.4. A self-similar group G is called self-replicating if, for every vertex u in X*, the map cpu : Gu -> G given by cou (g) = gu is onto (where C. is the stabilizer of the vertex u in G). Note that 9 = IMG(z2 + i) is self-replicating. This is clear from the equalities b = (a, c),

c = (b,1),

Consider the normal subgroup N of

aba = (c, a),

aca = (1, b).

defined by

N = ([a, b], [b, c])g.

By definition, [g, h] = g-'h-'gh and

Theorem 2.5. The group

denotes normal closure in

9.

is a regular branch group over N.

PROOF. First we observe that N has finite index in 9. Direct computation shows that a2 = b2 = c2 = (ac)4 = (ab)8 = (bc)8 = 1, so 9/N is a homomorphic image of

xD4i where C2 is the cyclic group of order 2 and D4 is the dihedral group of order 8. Further, we have [b, c] = ([a, b],1)

[c, b°'] _ ([b, c],1)

Since [c, ba] = cb[b, a]cb[b, a] = [b, a]bc[c, b] [b, a] E N we have that ([a, b], 1) and

([b, c], 1) are elements in N. The fractalness of g enables us to conjugate the sections in ([a, b], 1) and ([b, c], 1) by arbitrary elements in g without leaving N. Thus we get the inclusion N x 1 -< N. The transitivity of 9 on the first level then implies

NxN - 0),

where ' is the substitution defined on words in the free monoid over the alphabet {a, b, c} by

a->b, b -> c,

Ic - aba. In order to prove Theorem 3.1 we introduce some notation and prove a few intermediate results. The group F = (a, b, c I a2, b2, c2, (ac)4)

covers 9 (the relators of r are relators of !;). The action of 9 on the binary tree induces an action of the covering group r on the same tree, which is not faithful. Let 52 be the kernel of this action. Then, obviously, a set of generators of ci as a normal subgroup in r, together with the relators in r constitutes a presentation for 9.

The embedding y t Sym(2) induces a homomorphism ': F -> r? Sym(2) defined by

fa-->(1,1)a, b

(a, c),

cF--; (b,1).

Indeed, the relators of r are mapped to the trivial element (1, 1) of r l Sym(2): W (a2) = (1,1)a(1,1)a = (1, 1), ,p (C2)

W(b2) = (a, c)2 = (a2, c2) = (1, 1),

= (b,1)2 = (b2,1) = (1,1),

Ì`((ac)4) = ((1, b)o,)4 = (b2, b2) = (1,1).

The homomorphism W induces homomorphisms W,,: F ---> F l (la I Sym(2)) 1 Sym(2) denotes the iterated permutational wreath product) defined re-

(here

cursively by '1 = ' and

W: F

Fl (TSm(2))

41

(FlSym(2))l

\a?1

Sym(2)) =Fl I Z1Sym(2)).

If, for g E F, we have Tn(g) = (glu, u E Xn)vn, with glu E F and Un E

l2

Sym(2) we call glu the section of g at u. For every g E F denote by l(g) the length of the shortest word in a, b, c representing g in F. The following lemma shows that r possesses the so called contraction property. 1


236

Lemma 3.2. For every g E IF and u E X2 +1 l(9Iv.) < l(9)

(3.2)

2

PROOF. Observe that, because of the self-similarity, all generators satisfy inequality (3.2). Indeed,

1P2(b)=(1,1,b,1)(01), T2(c) = (a, c,1, 1), where, for ease of notation, the vertices on the second level are renamed by using the identifications 00 H 0, 01 H 1, 10 H 2, and 11 H 3. All pairwise products of generators also satisfy inequality (3.2):

'P2(a)=(1,1,1,1)(02)(13),

T2(a2) = 1,

T2(b2) = 1,

T2(C2) = 1,

'y2(ab) = (T1(c), Ti(a))a = (b,1,1, 1)(0213), 'y2(ba) = (T1(a), T1(c))a = (1,1, b, 1)(0312), (3.3)

T2(ac) = (W1(1), IF1(b))a = (1, 1, a, c)(02)(13),

T2(ca) = (T1(b), T1(1))a = (a, c,1,1)(02)(13), 'y2(bc) _ (T1(ab), `yi(c)) _ (c, a, b, 1)(01), T2(cb) = (T1(ba), Ti(c)) = (a, c, b, 1)(01).

Any word w in a, b, c of length n can be split into a product of at most (n + 1)/2 products of pairs of generators (if the length of w is odd one can pair the last letter in w with 1). Therefore the sections of w on the vertices of the second level are

products of at most (n + 1)/2 letters. Thus the inequality (3.2) holds for w as well.

Define an increasing sequence of subgroups of r by Stn = ker T,,,.

Lemma 3.3. The kernel St of the canonical epimorphism r - g satisfies

St= U Qn n>1

PROOF. Let h be a word in a, b, c of length at most 2n + 1 representing the trivial element in !9. Then, since for any words u, v E X (3.4)

hlu,, = hluly,

by Lemma 3.2 we obtain that all sections of h have length at most 1 on the 2(n+1)th level. Therefore they must be trivial, because h acts trivially on the tree. Hence, h E 522(n+l).

The last lemma reduces the problem of finding generators for St to finding generators for Stn. We start from Sti = ker T and, based on it, derive generators for Stn.

Let H = Str(l) be the stabilizer of the first level of the tree in r. Lemma 3.4. The group H has the following presentation

H=(0,6,'Y,P '32=b2= Y2=P2=(PS)2=1), where )3=b, 5 =c, -y = aba, p=aca.


237

PROOF. The index of H in IF is 2 and the coset representatives are {1, a}. The Reidemeister -Schreier procedure gives the above presentation.

Obviously, each SZ7z is a subgroup of H. Therefore one can restrict IQ to H. Since H stabilizes the first level one can think of W as a homomorphism H --+ F x F. This map is given by (a, c),

ry=aba-> (c, a),

bcam (b, 1), p=aca-+(1,b), which mimics the corresponding embedding of the generators b, aba, c, aca of StG(1)

into G x G. Define the following words in F: U1 = (ba)8,

U2 = [c, ab] 2,

U3 = [c, bab] 2,

U4 = [c, ababa]2,

U5 = [c, ababab]2,

U6 = [c, bababab12.

Lemma 3.5. 521 = (U1, U2, U3, U4, U5,

U6)r.

PROOF. In order to find a generating set for 521 = ker ' we first describe AY(H). Since IF (J) = (b, 1) and W (p) _ (1, b), we get

BxB Stn X Stn. Observe that (3.7)

c1

('I'(0i(Uj))) = 1

in F (recall that, for an element h = (hJ0, hJ1) in H, cpl(h) = h1l. Indeed, since 'Pi (W(o(F))) < (a,c) = D4 it's sufficient to check only that all Ui's are trivial in D4. But this is true since all these words are squares of commutators and [D4, D4] = 7L/2Z.

Equation (3.7) for i = n + 1, together with the inductive assumption yields Stn x 1 < ql(Stn+l). Since Stn+i is normal in F conjugation by a yields 1 x h,. < iP(Stn+1). Therefore 'F(Qn+l) = Stn X Stn.

Equation (3.7) also implies that ,p(0n+1(Uj)) (0n(Uj) =

1),

p(0n+1(Uj)a) = (1, 0n(Uj)),

i.e.,

((0z (Uj) I i = 1 ... , n, j = 1, ... 6) r) = Stn X Stn


240

Therefore

Qn+1 = ker T (02 (Uj), i = 1'... n, j = 1, ... , 6) r

_ (02(Ui),i = 0,...,n,j = 1,...,6)F. Lemmas 3.6 and 3.3 prove Theorem 3.1. Since 0((ac)4) = (ba) 8 = U1i 0(a2) = b2, q(b2) = c2 and q(c2) = (aba)2 = 1 the presentation in (3.1) is slightly simplified.

Corollary 3.7. The group 9 embeds into an amenable finitely presented group of exponential growth (a, b, c, s I a2, (ac)4, [c, ab]2, [c, bab]2, [c, ababa]2, [c, ababab]2, [c, bababab]2,

as = b, bs = c, cs = aba),

which is an ascending HNN-extension of 9.

PROOF. By Theorem 3.1 and the fact that cpo(T (q(u))) = u the substitution 0 induces an injective endomorphism of g. Thus the HNN-extension construction can be applied. Since 0 is an extension of the amenable group 9 (see Section 4) by the amenable group Z generated by s, 0 is amenable. The growth of 0 is exponential because it contains a free semigroup of rank 2 (it follows from the HNN-extension construction that, for example, s and sa generate such a semigroup).

4. Self-affine measures and amenability Although the amenability of 9 follows from the intermediate growth of this group, which was established in [7], we present here a different approach based on the tools developed in [6,18]. More precisely, we construct a particular self-affine measure on 9, which proves the vanishing of the asymptotic entropy and, hence, amenability.

Let G be a self-similar group acting spherically transitively on a d-ary tree. Consider a nondegenerate probability measure y on G (the support of y generates G). Then for any x c X one can define a new probability measure ILIx on G, which is called the restriction of p on x. The details of the definition and proofs of relevant statements are given in [18] and here we only give the basic idea. We consider a right random walk gn = h1h2 ... hn on G determined by µ, i.e., {hn} is a sequence of independent variables identically distributed according to the measure µ. We consider g as embedded in 9 2 Sym(2) and keep track of the xth coordinate of the image of gn in G 2 Sym(d). Recall that hn is an automorphism of X*. For x E X, hn(x) denotes the action of hn on x. Since gn+1 Ix = gn Ix . hn+l I

(x)

the probability distribution of gn+llx is completely determined by gnl x and gn(x). Therefore the induced random walk (4.1)

(gnIx,gn(x))

on G x X is again a Markov chain. The last random walk is called a random walk with internal degrees of freedom. Since X is finite and G acts transitively on X, the subset G x {x} C G x X is recurrent with respect to (4.1). Therefore one can consider the trace of the random walk (4.1) on G x {x}, which is also a random walk. Finally, we define the measure µl x as the transition law for the last random walk on G x {x} considered as a copy of G.


241

There is a convenient way to compute ul x using the properties of the random walk (4.1). The random walk with internal degrees of freedom is governed by the matrix M = (Pxy)x,yEX

whose entries pxy are subprobability measures on G such that µ,y(h) is a transition probability of getting to the state (gh, y) from the state (g, x). With slight abuse of notation, we denote by g the 8-measure concentrated at g. Then the matrix M can be expressed as

M = >

IL (g) mg,

gEsupp µ

where Mg _ xy

gIx, 0,

y = g(x), y g(x),

The following theorem is proved in [18].

Theorem 4.1. The measure /Ix, x E X can be expressed in terms of the matrix

M as (4.2)

PIx = Axx + Mxx(1 - Mxx)

MXx,

where Mxx (resp., Mix) denotes the xth row (column) of M from which the entry pxx is removed, and MM,t is the matrix obtained from M by removing the xth row and the xth column. One can define µlu, for any w = x1x2

xn E X* by

µw = (... (AI-A-2 ... )I-.-

Definition 4.2. The nondegenerate probability measure p on a self-simila group G is called self-affine (self-similar in [18]) if there is a word w E X* such that ILIw=ae+(1-cl)fi,

where 0 < a < 1 and e is the identity element in G. For simplicity, we write a instead of ae.

Theorem 4.3 ([18]). If a self-similar group G carries a self-affine nondegenerate measure It with finite entropy, then it is amenable. In this section we construct such a measure on G. Since this measure should be nondegenerate and have finite entropy the most natural place to look for it is the space Q of positive convex linear combinations of 8-measures concentrated on the generators a, b, and c, i.e., measure p of the form

µ=xa+yb+zc, x+y+z=1, x,y,z>0. Suppose we want this measure to be self-affine with respect to x E X. By definition

this means pjx = a + (1 - a)µ or, equivalently, 11 =

ILIx - a 1 -a

R. GRIGORCHUK, D. SAVCHUK, Z. SUNIC

242

Since p(e) = 0 we get a

Thus the measure p is a fixed point of the

transformation (4.3)

1-px(e)

µI-(e)

/-tl.

(D:

which is defined in [18] and used to prove amenability of a family of groups generalizing Basilica group IMG(z2 - 1).

Let's compute plo and the corresponding transformation' in the case of g. The support of p is {a, b, c} and the corresponding matrices M9 are given by (4.4)

Ma

(0

0)

'

Mb = ( a e)

M` =

,

(bb

00)

Hence,

x M=xMa+yMb+zMc= ya+zb yc+z X

By Theorem 4.1

plo = (ya + zb) +x2(1 - yc- z)-1. Since c has order 2 in C it is easy see that in the group algebra RG

(1-yc-z)

I

=

z-1 z2-2z+1-y2 c z2-2z+1-y2 y

Therefore yx2 (z - 1)x2 µo=y a+z b+z2-2z+1-y2 c z2-2z+1-y2

and the transformation

takes the form

4)(xa+yb+zc) _

c

1 + (z - 1)x2/(22 - 2z + 1 - y2) y(z2 - 2z + 1 - y2)

z2-2z+1-y2+x22-x2 + z(z2-2z+1-y2)

a

z2-2z+1-y2+x22-x2

b

yx2

+z2-2z+1-y2+x22-x2 c. Now we are interested in a fixed point of the rational map F: ][83 -> R3 induced by 4), which maps (x, y, z) to the coefficients of 4)(xa + yb + zc). Moreover, we are searching for such a fixed point only in the invariant simplex x+y+z = 1, x, y, z > 0. Fortunately, there is such a fixed point. If C ti 0.4786202932 is the unique real root

of the polynomial Z3 - 6Z2 + 11Z - 4, then the point

(S,S2-4(

+2,-1+3(-(2)

is fixed under F, which produces a self-affine nondegenerate probabilistic measure

on G with finite support, proving amenability of G. This point is unique in the simplex of nondegenerate measures. Indeed, from the equation F1 (x, y, 1 - x - y) _ x, where F1 is the first coordinate of F, we get (4.5)

y=4(-x2+x±x x2-lOx+9).

Substitution in F2 (x, y, 1 - x - y) = y yields

(4.6) (-x4 + 12x3 - 39x2 + 40x - 12) f

x2 - 10x + 9(x3 - 7x2 + 12x - 4) = 0.


(a)

243

(b)

FIGURE 7. Uniqueness of a self-affine measure

Moving the second summand to the righthand side and squaring both sides produces the equation x(x - 1)(x3 - 6x2 + llx - 4) = 0, whose unique real root on the interval (0, 1) is C. The graphs of the two functions in (4.6) are shown in Figure 7: (a) for "plus" and (b) for "minus." The solution comes from (a), so in (4.5) "plus" should be used. It is a routine to check that indeed y = (2 - 4( + 2.

5. Spectral properties and Schur complement Let H be a Hilbert space and M be an operator on H. Let H = Ho ® H1 and there are operators A E B(Ho), D E B(H1), B: H1 -p Ho and C: Ho -+ H1, such that the matrix of M in the basis consisting of the bases of Ho and H1 takes the form:

_

M

A B

CD

The following fact is of folklore type.

Proposition 5.1. Let D be invertible. The operator M is invertible if and only if S1(M) = A - BD-1C is invertible. The matrix S1(M) is called the first Schur complement of M. PROOF. Indeed, the matrix L

=

Io -D-1C

0

D-1

is invertible. Therefore M is invertible if and only if

ML = A - BD-1C BD-1 Il 0 is invertible, which is equivalent to the nonsingularity of S1(M).

In our case the action of 9 on the boundary Xw (the set of infinite sequences over X) of the tree X* induces a unitary representation ir9(f)(x) = f (g-lx) of 9 in 1-l = B (L2 (X w )) . The Markov operator M = 3 (Ira + 1rb + 7rc) corresponding to this

unitary representation plays an important role (we do not include inverse elements because all generators are of order 2). The usual method to find the spectrum of M for a self-similar group G is to approximate M with finite dimensional operators


244

27.0 24.3

21.6 18.9 16.2 13.5 10.8 8.1

5.4 2.7

0.0

-1.1-0.9-0.7-0.5-0.3-0.10.1 0.3 0.5 0.7 0.9 1.1

FIGURE 8. Histogram of the spectrum on the 9th level

arising from the action of 9 on the levels of the tree X*. For more on this see [1]. For simplicity we write g for 7rg.

Let hn be the subspace of 7-1 spanned by the 2n characteristic functions f,,, v E Xn, of the cylindrical sets, corresponding to the 2' vertices of the nth level. Then 7-1n is invariant under the action of g and 7-ln C 7-1n+I. Also 7-ln can be naturally identified with L2(Xn). By 7gni (or, with a slight abuse of notation, by gn) we denote the restriction of 71g on 7-ln. Then, for n > 0,

Mn=3(an+bn+cn) are finite-dimensional operators whose spectra converge to the spectrum of M in the sense

sp(M) = U sp(Mn) n>0

Moreover, if P is the stabilizer of an infinite word from XW, then one can consider the Markov operator MG/P on the Schreier graph of G with respect to P. The following fact is observed in [1] and can be applied in the case of 9.

Theorem 5.2. If G is amenable then sp(MG/P) = SP(M)Common practice for finding the spectrum of M, initiated in [1], is to consider a pencil of operators

M(y,z,A)=a+yb+zc-A and find the set sp(y, z, A) of points (y, z, A) such that M(y, z, A) is not invertible.

Then the spectrum of M is just the intersection of sp(y, z, A) with the line y = z = 1, shrunk by a factor of 3. We take 1 as the coefficient at a to simplify the computation. Otherwise one can divide it out (we restrict our attention to the case when x, y, z are nonzero).

Let us consider the corresponding pencil M7 (y, z, A) = an + ybn + zcn - A and find its matrix in the basis { f : v E Xn}. The orthogonal subspaces H,(,Z) = span(fz,,, v (E Xn-1), i = 0, 1 span Hn and are naturally isomorphic to Hn_1. The


245

self-similar structure of G gives the following operator recursion (which coincides with the recursion (4.4)) n-1

((5.1)

an

= -

In-1

0

0

(a,,1

bn

,

- -

0)

0

Cn-1

/

c

1

\bn

n=

0

0/

In-1

for n > 0, where In_1 denotes the identity matrix of size 2n-1. The matrices a0i b0 and co are equal to the 1 x 1 matrix [1]. For any constant r, we write r instead of rln. Thus we have, 1 + zbn-1 - A ) M,, (y, z, A) = an + ybn + zcn - A = Cyan-1

ycn_1 + z - A is invertible the operator Mn(y, z, A) is invertible if and only if S1(Mn(y, z, A)) is invertible. The inverse of ycn_1 + z - A

in Rg is y

-y2+z2-2zA+A2 Cn'-1+_y 2 +

-z+A z2_ 2zA +

A2'

Hence, ycn_l+z-A is not invertible if and only if -y2+z2-2zA+A2 = (z - A - y) x (z - A + y) = 0. Denote the union of these 2 planes by Zi. Note, that Mn(y, z, A) is not necessary singular at each point of Z1. The first Schur complement of M is S,(Mn(y, z, A))

= yan-1 + zbn_1 - A - (yCn-i + z - A)-1

Y. an-1 + z bn-1 +

-z + A n-1 + -y2 + z2 - 2zA + A2 - A

y

y2 + z2 - 2zA + A2

If y = 0 we get Sl(Mn(0, z, A)) = zbn_1 + X1Z - A is not invertible if and only if

)

z

det(1/(Az)A

1/(A - z) - A

z

(

1

z)2

(1 - (A - z)(A + z))(1 - A + z)(1 + A - z)-0. -

Denote corresponding union of a hyperbola and two lines in R3 by Z2. Note that

z2nz1=0. If

0 then

S1(Mn(y, z, A)) = a,,-, + Y

z

y

bn-1 +

-y2 + z2 - 2zA + A2

Cn

i

-Aye+Az2-2zA2+A3+z-A y(-y2 + z2 - 2zA + A2)

= An-1(F(y, z, A)), where F : R3 -> R3 is the rational map defined by F: (y, z, A

z

(y

1

-y2 + z2 - 2zA + A2'

-Aye+Az2-2zA2+A3+z-Al y(-y2 + z2 - 2zA + A2)

)

Therefore the set spn(y, z, A) of points (y, z, A) where Mn(y, z, A) is not in-

vertible in this case (y

0) is a preimage under F of the corresponding set


246

FIGURE 9. Part of the spectrum of M(y, z, A) spn_1(y, z, A). To summarize, (5.2)

Z2 U F-1(spn-1(y, z, A)) C sPn(y, z, A) C Zi U Z2 U F-1(spn-1(y, z, A)).

Since Mo(y, z, A) _ (1 + y + z - A) we have spo(y, z, A) _ {(y, z, A) : 1 + y + z - A = 0}.

(5.3)

Denote this plane by P. Equations (5.2) and (5.3) show that n-1

n-1

F-n(P) U U F-'(Z2) C sPn(y, z, A) C F-n(P) U U F-2(Z1 U Z2).

(5.4)

i=0

i=0

Since Z2 consists of points with y = 0 every point (y, z, A) from F-1(Z2) must satisfy z = 0. But the preimages of all such points are empty. Hence, F-2(Z2) = 0 and Uz 'F-'(Z2) = Z2 U F-1(Z2). Denote the last subset by Z3. O

One can easily check that P C F-1(P) and, hence, F-n (P) = UZ 0 F(P).

Therefore equation (5.4) transforms to n-1

n

Z3 U U F-'(P) C spn(y, z, A) C F-"''(P) U Z3 U U F-'(P U Z1). i=0

i=0

Thus, the spectrum of the operator M on L2(Xw) satisfies CK)

0C

Z3 U U F-i(P) C sp(M(y, z, A)) = U sPn(y, z, A) C Z3 U U F-'(P U Z1). i=0

i=0

i=0

Note, that the sets A = Z3 U U° 0 F-i (P) and B = Z3 U U° 0 F-' (P U Z1) are almost invariant with respect to F, in the sense that

F-1(A)UZ2=A, F-1(B)UZ1UZ2=B, which is an analog of [17, Theorem 4.1] for the Basilica group. The preimages of the plane P under F4 and F5 are shown in Figure 9.


247

Note that there are points in the spectrum of M(y, z, A) which do not belong to any preimage of the plane P. In particular, the point (-1, 0, belongs to Z1 2) so it is not in the domain of F, but

det Mn(-2, 0, 2) = det((an_I + 1)(cn_1 + 1) - 4) = 0 since 4 is an eigenvalue of (an_1 + 1)(cn_1 + 1). However, this point could be in the closure of the union of all preimages of P.

On the other hand we can formulate a conjecture that the spectrum of M = A (a + b + c) is the intersection of the line y = z = 1 with A = Z3 U U°° 0 F_2 (P), shrunk by a factor of 3. This conjecture survives at least up to the 6th level. Note also that the map F is conjugate to a simpler map G: (y, z, A)

,y

(-2 + yA), 1 (-y + yA2 - A)

by the conjugator map C1

1

The histogram for the spectral density of the operator Mn acting on 9th level is shown in Figure 8. Further steps are required to identify the spectrum of the pencil M(y, z, A) and

_

of M more precisely. This is related to the problem of finding invariant subsets of the rational map F. Perhaps the spectrum of M is just the intersection of the "strange attractor" of F with the line y = z = 1, shrunk by factor of 3. In any case here we have one more example when the spectral problem is related to the dynamics of a multidimensional rational map. There is a hope that the methods developed for this type of transformations (see, for instance [21]) could help to handle this case.

References 1.

2. 3. 4. 5.

L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova 231 (2000), 5-45 (Russian); English transl., Proc. Steklov Inst. Math. 2000, no. 4(231), 1-41. L. Bartholdi, R. I. Grigorchuk, and V. Nekrashevych, From fractal groups to fractal sets, Fractals in Graz 2001, Trends Math., Birkhauser, Basel, 2003. L. Bartholdi, R. I. Grigorchuk, and Z. Sunik, Branch groups, Handbook of Algebra, Vol. 3, North-Holland, Amsterdam, 2003. L. Bartholdi and V. Nekrashevych, Thurston equivalence of topological polynomials, Acta Math. 197 (2006), no. 1, 1-51. , Iterated monodromy groups of quadratic polynomials. I, available at arXiv:math.GR/ 0611177.

6. 7.

8.

L. Bartholdi and B. Virag, Amenability via random walks, Duke Math. J. 130 (2005), no. 1, 39-56, available at arXiv:math.GR/0305262. K.-U. Bux and R. Perez, On the growth of iterated monodromy groups, Topological and Asymptotic Aspects of Group Theory, Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 61-76, available at arXiv:math.GR/0405456. R. I. Grigorchuk, On Burnside's problem on periodic groups, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 53-54 (Russian); English transl., Functional Anal. Appl. 14 (1980), no. 1,

41-43. 9.

, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939-985 (Russian); English transl., Math. USSR-Izv. 25 (1985), no. 2, 259-300.


248 10.

, An example of a finitely presented amenable group that does not belong to the class EG, Mat. Sb. 189 (1998), no. 1, 79-100 (Russian); English transl., Sb. Math. 189 (1998), no. 1-2, 75 - 95.

, Just infinite branch groups, New Horizons in Pro-p Groups, Progr. Math., vol. 184, Birkhauser, Boston, MA, 2000, pp. 121-179. 12. R. I. Grigorchuk, V. Nekrashevych, and V. I. Sushchanskii, Automata, dynamical systems, and groups, Tr. Mat. Inst. Steklova 231 (2000), 134-214 (Russian); English transl., Proc. Steklov Inst. Math. 2000, no. 4(231), 128-203. 13. R. I. Grigorchuk and V. Nekrashevych, Schur complement, operator algebras and self-similar groups, available at arXiv:math.GR/0612421. 14. R. I. Grigorchuk and Z. Sunik, Asymptotic aspects of Schreier graphs and Hanoi Towers groups, C. R. Math. Acad. Sci. Paris 342 (2006), no. 8, 545-550, available at arXiv:math. 11.

GR/0601592.

15. R. I. Grigorchuk and A. Zuk, On the asymptotic spectrum of random walks on infinite families of graphs, Random Walks and Discrete Potential Theory (Cortona, 1997), Sympos. Math., vol. 39, Cambridge Univ. Press, Cambridge, 1999, pp. 188-204. 16. , On a torsion-free weakly branch group defined by a three state automaton, Internat. J. Algebra Comput. 12 (2002), no. 1-2, 223-246. 17. , Spectral properties of a torsion-free weakly branch group defined by a three state automaton, Computational and Statistical Group Theory (Las Vegas, NV/Hoboken, NJ, 2001), Contemp. Math., vol. 298, Amer. Math. Soc., Providence, RI, 2002, pp. 57-82. 18. V. A. Kaimanovich, "Munchhausen trick" and amenability of self-similar groups, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 907-937. 19. V. Nekrashevych, Self-similar groups, Math. Surveys Monogr., vol. 117, Amer. Math. Soc., Providence, RI, 2005. 20. , A minimal Cantor set in the space of 3-generated groups, to appear in Geom. Dedicata. 21. N. Sibony, Dynamique des applications rationnelles de pk, Dynamique et geometrie complexes (Lyon, 1997), Panor. Syntheses, vol. 8, Soc. Math. France, Paris, 1999, pp. 97-185. DEPARTMENT OF MATHEMATICS, TEXAS A&M UNIVERSITY, COLLEGE STATION, TX 77843-

3368, USA E-mail address, R.Grigorchuk: [email protected] E-mail address, D.Savchuk: [email protected] E-mail address, Z. Sunic: [email protected]


On Scattering of Solitons for Wave Equation Coupled to a Particle Valery Imaykin, Alexander Komech, and Boris Vainberg ABSTRACT. We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has

a six dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free wave equation. It is assumed that the charge density satisfies the Wiener condition which is a version of the Fermi Golden Rule and that the total charge of the particle equals zero. The proof is based on a development of the general strategy introduced in the papers of Soffer and Weinstein, Buslaev

and Perel'man, and others: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.

1. Introduction Our paper concerns an old mathematical problem of nonlinear field-particle interaction. A charged particle radiates a field which acts on the particle, etc. This self-action is probably responsible for some crucial features of the process: asymptotically uniform motion and stability of the particle, increment of the particle's mass, etc. (see [23]). The problem has many different appearances: for a classical particle coupled to a scalar or Maxwell field, for coupled Maxwell- Schr6dinger or Maxwell-Dirac equations, for the corresponding second-quantized equations, etc. One of the main goals of a mathematical investigation of this problem is studying soliton type long time asymptotics and asymptotic stability of soliton solutions to the equations. First results in this direction have been discovered for KdV equation and other complete integrable equations. For KdV equation, any solution with sufficiently 2000 Mathematics Subject Classification. Primary 35L70, 35P25; Secondary 35Q51, 8lUxx, 37K40.

The two first authors are supported partially by Austrian Science Foundation (FWF) Project (P19138-N13), by research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).

The second author is supported partially by Max-Planck Institute of Mathematics in the Sciences (Leipzig), and Wolfgang Pauli Institute of Vienna University. The third author is supported partially by the NSF grant DMS-0405927. This is the final form of the paper. ©2007 American Mathematical Society 249

V. IMAYKIN ET AL.

250

smooth and rapidly decaying initial data converges to a finite sum of solitons moving to the right, and a dispersive wave moving to the left. A complete survey and proofs can be found in [5]. For nonintegrable equations the long time convergence of the solution to a soli-

ton part and dispersive wave was obtained first by Soffer and Weinstein in the context of U(1)-invariant Schrodinger equation, [18-20]. The extension to translation invariant equations was obtained by Buslaev and Perel'man [1, 2] for 1D Schrodinger equation, and by Miller, Pego and Weinstein for 1D modified KdV and RLW equations, [15-17]. A fundamental role in all these results play the techniques introduced originally in the Weinstein paper [27]. In [1, 2] the long time convergence is obtained for translation invariant and U(1)invariant nonlinear Schrodinger equation: for any finite-energy solution '(x, t) with initial data close to a soliton 0vo (x - v0t - a0)e'wot, the following asymptotics hold: (1.1)

' (x,t)=Ovt(x-vft-a±)eiWtt+WO(t)Of+rf(x,t),

t,±00.

Here the first term of the right-hand side is a soliton with parameters v+, a+, wf close to v0i a0, w0, the function WO(t)of is a dispersive wave which is a solution to the free Schrodinger equation, and the remainder r± (x, t) converges to zero in the global L2-norm. Recently Cuccagna extended the results to nD Schrodinger equations with n > 3, [3,4]. We consider a scalar real-valued wave field 1(x) in 1[83, coupled to a relativistic particle with position q and momentum p, governed by

*r (x, t) = AO (x, t) - p(x - q(t)),

(x, t) _ ir(x, t), (1.2)

tq()

p(t)

2,

1 + p2'

p(t) =

J

x E 1[83

0(x, t)Vp(x - q(t)) dx.

This is a Hamilton system with the Hamilton functional (1.3) 71 (0,-7r,q,p)=

2

f((x)2 + V(x)2) dx

+

l+p2.

J V, (x) p (x - q) dx +

The first two equations for the fields are equivalent to the wave equation with the source p(x - q). We have set the mechanical mass of the particle and the speed of wave propagation equal to one. The case of the point particle corresponds to

p(x) = S(x) and then the interaction term in the Hamiltonian is simply O(q). However, in this case the Hamiltonian is unbounded from below which leads to the ill-posedness of the problem, that is also known as ultraviolet divergence. Therefore we smooth the coupling by the function p(x) following the "extended electron" strategy proposed by M. Abraham for the Maxwell field. In analogy to the MaxwellLorentz equations we call p the "charge distribution." Finally, the form of the last two equations in (1.2) is determined by the choice of the relativistic kinetic energy 1 -+p2 in (1.3). Let us write the system (1.2) as

Y(t) = F(Y(t)),

(1.4)

t E I[8,

where Y(t) := (O(x, t), 7r(x, t), q(t), p(t)). Below we always deal with column vectors but often write them as row vectors. The system (1.2) is translation-invariant and admits soliton solutions (1.5)

Ya,,, (t)

(x - vt - a), 7r (x - vt - a), vt + a, pv), pv =

v VI-1

v2

ON SCATTERING OF SOLITONS FOR WAVE EQUATION COUPLED TO A PARTICLE 251

for all a, v E R3 with Iv < 1 (see (2.7) below). The states So,,,, solitary manifold (1.6)

Yo,,v(0) form the

S :_ {Sa,v : a, v E R3, v1 < 1}.

Our main result is the soliton-type asymptotics of type (1.1), (1.7)

(z/)(x,t),7r(x,t)) - (Ovf(x-v+t-a±),7rvt(x-v+t-a+))+W°(t)'I'±,

t

for solutions to (1.2) with initial data close to the solitary manifold S. Here W°(t) is the dynamical group of the free wave equation, Wt are the corresponding asymptotic scattering states, and the remainder converges to zero in the global energy norm, i.e., in the norm of the Sobolev space HI(R3) ® L2(R3), see Section 2. For the particle trajectory we prove that (1.8)

q(t) -+ vf,

q(t) - vft + af,

t -* ±oo.

The results are established under the following conditions on the charge distributions: p is a real valued function of the Sobolev class H2(R3), compactly supported, and spherically symmetric, i.e., (1.9)

p, Vp, VVp E L2(R3),

(1.10)

p(k) = (27r) -312

p(x) =0 for JxJ > RP,

o(x) = p1(JxJ) We require that all "nonzero modes" of the wave field are coupled to the particle. This is formalized by the Wiener condition

J

e'kxp(x) dx

0

for all k E R3 \ {0}.

It is an analogue of the Fermi Golden Rule: the coupling term p(x - q) is not orthogonal to the eigenfunctions e'kx of the continuous spectrum of the linear part of the equation (cf. [21, 22]). As we will see, the Wiener condition (1.10) is very essential for our asymptotic analysis. Generic examples of the coupling function p satisfying (1.9) and (1.10) are given in [13]. Moreover, we will assume that p(0) = 0, i.e., (1.11)

J

p(x) dx = 0,

that means that the total charge of the particle is zero (neutrality condition). The problem under investigation was studied earlier for the Klein - Gordon equation [11] without the neutrality condition.

Remarks 1.1. (i) The asymptotic stability of the solitary manifold S is caused by the radiation of energy to infinity which appears as the local energy decay. Similarly, the decay for the linearized autonomous dynamics also is caused by the radiation. Analytically, the decay is related to the existence of the continuous spectrum of the linearized dynamics. The latter is a characteristic feature of infinite dimensional Hamilton systems. (ii) The asymptotics (2.14) can be interpreted as the collision of the incident soliton, with a trajectory v_t+a_, with an incident wave W°(t)1Q_, which results in an outgoing soliton with a new trajectory v+t + a+, and a new outgoing wave W°(t)W+. The collision process can be represented by the diagram of Figure 1. It suggests to introduce the (nonlinear) scattering operator (1.12)

S : (v-, a-, %F-) H (v+, a+, q/4

V. IMAYKIN ET AL.

252

Wo(t)Y_

WW(t)T+

FIGURE 1. Wave-particle scattering

However, a correct definition of the operator is an open problem as well as the question on its asymptotic completeness (i.e., on its range). Our paper is organized as follows. In Section 2, we formulate the main result. In Section 3, we introduce the symplectic projection onto the solitary manifold. The linearized equation is defined and studied in Sections 4-5. In Section 6, we split the dynamics in two components: along the solitary manifold, and in transversal directions, and we justify the estimate concerning the tangential component. Section 7 concerns the time decay of the linearized dynamics. The time decay of the transversal component is established in Sections 8-11 under an assumption on the time decay of the linearized dynamics. In Section 12, we prove the main result.

2. Main results 2.1. Existence of dynamics. To formulate our results precisely, we need some definitions. We introduce a suitable phase space for the Cauchy problem corresponding to (1.2) and (1.3). Let L2 be the real Hilbert space L2(ll83) with norm and let HI be the completion of the real space Co (1[83) with the norm IV (x)I. Equivalently, using Sobolev's embedding theorem, H' = {O(x) E L6(1[83) :

V0(x)I E L2}. Let us introduce the weighted Sobolev spaces L2 and Ha with the norms 101" = I(1 + IxD

Definition 2.1.

1,

II0IIa = I(1 + IxI)'V 'I + I(1 + I xI )a0I.

(i) £ is the Hilbert space HI ® L2 ®1[83 ®1183 with the

finite norm IIYIIE = IVV)I + ICI + IqI + Ipl

for Y = (0, it, q, p).

(ii) £, is the space H,', ® L22 ®1183 ®1183 with the norm (2.1)

IIYII. = IIYIIEa = II,PII. + I7rI,,, + IqI + IpL.

(iii) F is the space H' ® L2 of fields F = (0, 7r) with the finite norm IIFILF = IV

I + it.

(iv) .F,, is the space H, ED L22 with the norm (2.2)

IFII. = IIFII., = I)IIa + IirL. Note that HI is not contained in L2 and for instance I v1 = no, see below. However, £ is the space of finite energy states (i.e., 11(Y) < no for Y E £) due

ON SCATTERING OF SOLITONS FOR WAVE EQUATION COUPLED TO A PARTICLE253

to the following estimates which are valid for arbitrary smooth b(x) vanishing at infinity (2.3)

- 8 ff d3x day

P( 2

(P,

0-TP) < 1012 + (V) (x), P(x - q)) 2

Iool2 - 2 (P, o-IP).

The Hamiltonian functional 7-l is continuous on the space E, and the lower bound in (2.3) implies that the energy (1.3) is bounded from below. Note that the latter is not true if p is delta-function. We consider the Cauchy problem for the Hamilton system (1.2) which we write as

Y(t) = F(Y(t)),

(2.4)

t E I[8

Y(O) = Yo.

:

Here Y(t) = (O(t), 7r(t), q(t), p(t)), Yo = (')o, r0, qo, po), and all derivatives are understood in the sense of distributions.

Proposition 2.2 ([13]). Let (1.9) hold. Then (i) For every Yo E E, the Cauchy problem (2.4) has a unique solution Y(t) E C(ll8, E).

(ii) For every t c ll8, the map U(t) : Yo F--) Y(t) is continuous on E. (iii) The energy is conserved, i.e., 7-l(Y(t)) = 7-l(Yo),

(2.5)

t c I[8,

and the velocity is bounded: 14(t) I < v < 1,

(2.6)

t E I[8,

with some v which depends on Yo.

2.2. Solitary manifold and main result. Let us compute the solitons (1.5). The substitution to (1.2) gives the following stationary equations, -v VV)V (y) = lrv (y),

v=

-v

P(y)

Pv

0 = - J v v (y)P(y) dy

VI-1 +pv'

Then the first two equations imply

(-0 + (v . V)2)ov(y) = -P(y), y c R'For jvj < 1 the equation (2.8) defines a unique function z/,, c H'(I[83). If v is (2.8)

given and jvj < 1, then pv can be found from the third equation of (2.7). Further, functions p and lv are even due to (1.9). Thus, V0, is odd and the last equation of (2.7) holds. Hence, the soliton solution (1.5) exists and is defined uniquely for any couple (a, v) with I v I < 1.

The function 0, can be computed by the Fourier transform. The soliton is given by the formulas

1

L' (x) _ - 47r

P(y) d3y

I'Y(y - x)II + (y - x)-J

7rv (x) _ -v V b (x),

Pv = yV.

Here we set y = 1/11 - v2 and x = xl1 + xl, where xll 11 v and xi 1 v for x c R'.

V. IMAYKIN ET AL.

254

From the neutrality condition (1.11) it follows that 7rzJ E La for a < 3 , and v E H,,', for a < 2. Thus, for the soliton solutions Ya,v we have

Ya,v EEC" a< 2. Definition 2.3. A soliton state is S(a) := ( (x -

(2.10)

b), b,pzJ), where

or := (b, v) with b E 1[83 and I v I < 1.

Obviously, the soliton solution admits the representation S(a(t)), where

a(t) = (b(t), v(t)) = (vt + a, v).

(2.11)

Definition 2.4. A solitary manifold is the set S := {S(a) : b E 1183, vJ < 1}. The main result of our paper is the following theorem.

Theorem 2.5. Let the Wiener condition (1.10) and the neutrality condition (1.11) hold. Let a = 1+5, 0 < 6 < z and Y(t) be the solution to the Cauchy problem (2.4) with the initial state Yo which is sufficiently close to the solitary manifold: (2.12) Yo = S(ao) + Zo, do := IIZoII3 +oo since the system (1.2) is time reversible.

3. Symplectic projection 3.1. Symplectic structure and Hamilton form. The system (1.2) reads as the Hamilton system

JDN(Y),

(3.1)

J :=

0

1

0

0

01

0

0

10

0

0

-1

0

,

Y=(

7r, q,p) E £,

where DR is the Frechet derivative of the Hamilton functional (1.3). Let us identify the tangent space to £, at every point, with S. Consider the bilinear form 1 defined on £ by Q = f dO(x) A dir(x) dx + dq A dp, i.e., (3.2)

S2(Y1i l2) = (Y1, JY2),

Y1, Y2 E £,

where

(bl,''2) + (lrl, 72) + q, q2 +plp2 (Y1, l2) and (01, 02) = f 01(x)212 (x) dx, etc., if the integrals converge.

Definition 3.1.

(i) YI { Y2 means that YI is symplectic orthogonal to

Y2, i.e., cl(Yi, Y2) = 0.

(ii) A projection operator P: £ for YI E Ker P and Y2 E Im P.

£ is called symplectic orthogonal if YI { Y2


3.2. Symplectic projection onto solitary manifold. Let us consider the tangent space Ts(Q)S to the manifold Sat a point S(o,). The vectors Tj := aa, S(a), where a,, := ab, and 19a,+3 := av, with j = 1, 2, 3, form a basis in TS. In detail,

'Ti = Tj (v) := ab, S(a) = (-aj v (y), -aj7v (y), ej, 0) Tj+3 =Tj+3(v) :=av,S(a) = (avjOv(y),av,7rv(y),0,av pv)

(:3.3)

'

_ 1'2'3'

where y := x - b is the "moving frame coordinate," el = (1, 0, 0), etc. Let us stress that the functions Tj will be considered always as the functions of y, not of x. The formulas (2.9) and the conditions (1.9), (1.11) imply that for IvJ < 1 T j (v) E Sa,

(3.4)

a -2 and v < 1. Then (i)

(3.5)

there exists a neighborhood Oa(S) of S in Sa and a map 11: Oa(S) --+ S such that 11 is uniformly continuous on 0,,,(S) n {Y E Sa : v(Y) < v} in the metric of Sa,

IIY = Y for Y E S,

and

Y - S { TSS,

where S = IIY.

(ii) Oa(S) is invariant with respect to the translations Ta, and (3.6)

11 TaY = T,, II Y,

for Y E 0,, (S) and a E 1183

(iii) For any v < 1 there exists a v < 1 s.t. v(IIY)J < v when Iv(Y)I < v.

(iv) For any v < 1 there exists an ra(v) > 0 s.t. S(or) + Z E 0,(S) if w(S(o-)) I < v and IIZlla < ra(iJJ). For proof see [11, Lemma 3.4]. We will call II the symplectic orthogonal projection onto S.

Corollary 3.5. The condition (2.12) implies that Yo = S + Zo where S = S(ao) = IIYo, and (3.7)

IIZ01I0 0.

Remark 5.4. Lemma 5.3(ii) together with energy conservation (5.4) imply the analyticity of the resolvent (Av,v - A)-1 for Re A > 0. 0, Remark 5.5. For a soliton solution of the system (1.2) we have b = v, and hence T(t) - 0. Thus, the equation (5.1) is the linearization of the system (1.2) on a soliton solution. In fact, we do not linearize (1.2) on a soliton solution, but on a trajectory S(a(t)) with a(t) being nonlinear in t. We will show later that T(t) is quadratic in Z(t) if we choose S(a(t)) to be the symplectic orthogonal projection of Y(t). Then (5.1) is again the linearization of (1.2).

6. Symplectic decomposition of the dynamics Here we decompose the dynamics in two components: along the manifold S and

in transversal directions. The equation (4.8) is obtained without any assumption on a(t) in (4.1). We are going to choose S(a(t)) := IIY(t), but then we need to know that (6.1)

Y(t) E Qa(S),

t E R,

with some Qa (S) defined in Lemma 3.5. It is true for t = 0 by our main assumption

(2.12) with sufficiently small d,3 > 0. Then S(a(0)) = HY(0) and Z(0) = Y(0) S(a(0)) are well defined. We will prove below that (6.1) holds with a = -,3 if d,3


is sufficiently small. First, the a priori estimate (2.6) together with Lemma 3.4 iii)

imply that IIY(t) = S(a(t)) with a(t) = (b(t), v(t)), and Iv(t)I < v < 1,

(6.2)

tC

if Y(t) E O_,3(S). Denote by r_,3(v) the positive number from Lemma 3.4 iv) which

corresponds to a = -,G. Then S(a) + Z E O_p(S) if a = (b, v) with vt < v and IIZtl-0 < r_o(5). Note that (2.6) implies IIZ(0)II_a < r_,3(v) if dp is sufficiently

small. Therefore, S(a(t)) = IIY(t) and Z(t) = Y(t) - S(a(t)) are well defined for t > 0 so small that IIZ(t)II-R < r_Q(v). This is formalized by the following standard definition.

Definition 6.1. t* is the "exit time," (6.3)

t,, = sup{t > 0 :

IIZ(s)II_a < r_a(v), 0 < s < t},

Z(s) = Y(s) - S(a(s)).

One of our main goals is to prove that t* = oc if d$ is sufficiently small. This would follow if we show that IIZ(t)II_a < r_a(v)/2,

0 < t < t*.

IQ(t)j < Q:= r_p(v),

0 < t < t*.

(6.4)

Note that (6.5)

Now N(t) in (4.8) satisfies, by (4.11), the following estimate, (6.6)

IIN(t)IIa

Ca(v)jIZ(t)II? p,

0 < t < t*.

6.1. Longitudinal dynamics: Modulation equations. From now on we fix the decomposition Y(t) = S(a(t)) + Z(t) for 0 < t < t. by setting S(a(t)) _ HY(t) which is equivalent to the symplectic orthogonality condition of type (3.5),

Z(t) t T(aiti)S,

(6.7)

0 < t < t*.

This allows us to simplify drastically the asymptotic analysis of the dynamical equations (4.8) for the transversal component Z(t). As the first step, we derive the longitudinal dynamics, i.e., the "modulation equations" for the parameters a(t). Let us derive a system of ordinary differential equations for the vector a(t). For this purpose, let us write (6.7) in the form (6.8)

c1(Z(t), T; (t)) = 0,

j = 1, ... , 6, 0 < t < t,:,

where the vectors Ti (t) = Th(a(t)) span the tangent space T(a(ti)S Note that a(t) _ (b(t), v(t)), where (6.9)

Iv(t)I < v < 1,

0 < t < t*,

by Lemma 3.4(iii). It would be convenient for us to use some other parameters (c, v) instead of a = (b, v), where c(t) = b(t) - ff v(T) dT and (6.10)

c(t) = b(t) - v(t) = w(t) - v(t),

0 < t < t*.

We do not need an explicit form of the equations for (c, v) but the following statement.

V. IMAYKIN ET AL.

260

Lemma 6.2 (cf. [11, Lemma 6.2]). Let Y(t) be a solution to the Cauchy problem (2.4), and (4.1), (6.8) hold. Then (c(t),v(t)) satisfies the equation wit)) = N(a(t), Z(t)),

(6.11)

0 < t < t*,

where

N(a, Z) = (9Z11 2,3)

(6.12)

uniformly in a E { (b, v)

v < v}.

6.2. Decay for the transversal dynamics. In Section 12 we will show that our main Theorem 2.5 can be derived from the following time decay of the transversal component Z(t):

Proposition 6.3. Let all conditions of Theorem 2.5 hold. Then t. = oo, and (6.13)

IIZ(t)lI-Q
0.

We will derive (6.13) in Sections 8-11 from our equation (4.8) for the transversal component Z(t). This equation can be specified using Lemma 6.2. Indeed, the lemma implies that (6.14)

II T (t) II a < C(v) II Z(t)112 p,

0 < t < t*,

by (4.10) since w - v = c. Thus (4.8) becomes the equation

Z(t) = A(t)Z(t) + N(t),

(6.15)

0 < t < t*,

where A(t) = Av(t),w(t), and N(t) := T(t) + N(t) satisfies the estimate (6.16)

II N(t) II'3< CII Z(t) II? Q,

0 < t < t*.

In all remaining part of our paper we will analyze mainly the basic equation (6.15) to establish the decay (6.13). We are going to derive the decay using the bound (6.16) and the orthogonality condition (6.7). Let us comment on two main difficulties in proving (6.13). The difficulties are common for the problems studied in [2, 3]. First, the linear part of the equation is non-autonomous, hence we cannot apply directly known methods of scattering theory. Similarly to the approach of [2, 3], we reduce the problem to the analysis of the frozen linear equation, k (t) = Al X (t),

(6.17)

t c I[8,

where Al is the operator Av1,v1 defined in (4.9) with vi = v(ti) and a fixed tl E [0, t*). Then we estimate the error by the method of majorants. Second, even for the frozen equation (6.17), the decay of type (6.13) for all solutions does not hold without the orthogonality condition of type (6.7). Namely, by (5.7) the equation (6.17) admits the secular solutions X (t)

3

(6.18)

3

Dj [Tj (vl)t + Tj+3 (vl)]

CjTj (vl) + 1

1

which arise also by differentiation of the soliton (1.5) in the parameters a and vl in the moving coordinate y = x - v1t. Hence, we have to take into account the orthogonality condition (6.7) in order to avoid the secular solutions. For this

,'ATTERING OF SOLITONS FOR WAVE EQUATION COUPLED TO A PARTICLE 261

we will apply the corresponding symplectic orthogonal projection which "runaway solutions" (6.18).

.mark 6.4. The solution (6.18) lies in the tangent space TS(0.1)S with o1 = ) (for an arbitrary b1 E R) that suggests an unstable character of the nonlinear rAcs along the solitary manifold (cf. Remark 4.1(iii)).

)efinition 6.5.

(i) Denote by II,,, Ivl < 1, the symplectic orthogonal

f ion of E onto the tangent space TS(Q)S, and Pv = I - Hv. ii) Denote by Zv = P E the space symplectic orthogonal to TS(Q)S with oa _ (for an arbitrary b c I[8). Note that by the linearity, i!I) r

RvZ=),Hl(v)Tj(v)fZ('n(v),Z), ZEE,

f h some smooth coefficients Ihi (v). Hence, the projector IIv, in the variable

x - b, does not depend on b, and this explains the choice of the subindex in If, and Pv. Now we have the symplectic orthogonal decomposition iti.20)

E,3 = Ts(Q) S + Zv,

or = (b, v),

and the symplectic orthogonality (6.7) can be written in the following equivalent forms, (6.21)

HV(t) Z(t) = 0,

PP(t)Z(t) = Z(t),

0 < t < t*.

Remark 6.6. The tangent space Ts(a)S is invariant under the operator Av,v by Lemma 5.3(i), hence the space Zv is also invariant by (5.5): A,,,,,Z E Zv for sufficiently smooth Z E Zv.

In the next section we prove the following proposition which is one of the main ingredients for proving (6.13). Let us consider the Cauchy problem for the equation

(6.17) with A = Av,v for a fixed v, IvI < 1. Recall that the parameter 3 = 1 + 6, 0 < 6 < z is also fixed. Proposition 6.7. Let the Wiener condition (1.10) and the neutrality condition (1.11) hold, Ivi i < v < 1, and X0 E E. Then (i) The equation (6.17), with AI = A,,,,,, admits the unique solution eA'tX0 X (t) E C(R, E) with the initial condition X (O) = Xo. (ii) For Xo c Zvl f1 Ea, the solution X (t) has the following decay, (6.22)

IleAIIXoll-a 0. In detail,

II(y) + v VW(y) - An(y) = -'o(y) (7.3)

AT(y)

- m2'I'(y) + v VH(y) + Q . Vp(y) - AII(y) = -no(y) AQ= -Qo -(V '(y), p(y)) + (VO.(y), Q . Vp(y)) - AP = -Po

yc

I[83.

We express the field x in terms of Q from the first two equations (applying first Fourier transform), and substitute to the second two equations. Then we obtain a closed system of linear algebraic equations for the vector components (Q(A), P(A)). The system reads (7.4)

M(A)

C=

(),

ReA > 0,

where the matrix M(A) can be computed explicitly. In detail it is given in [11, formulas (13.18) - (13.20) with m = 0]. Further, PO' = Po + (D ('o, no).

Lemma 7.1 ([11]). The matrix M(A) is invertible for ReA > 0, the matrix function M(A) (respectively, M-1(A)) admits an analytic (respectively meromorphic) continuation to the entire complex plane C. The vector components are given by the Fourier integral (7.5)

MQ(t))

= 2, f

e'wtM-1(iw

+ 0)

(QOI0)

dw


if it converges in the sense of distributions.

Proposition 7.2. The matrix M-1(iw) is smooth in w E R \ 0. The proof is based on the following

Lemma 7.3. Let p satisfy the Wiener condition (1.10), and vl < 1. Then the limit matrix M(iw + 0) is invertible for w 4 0, w E R. Remark 7.4. The proof of Lemma 7.3 is the unique point in the paper where the Wiener condition is indispensable. The Proposition 7.2 alone is not sufficient for the proof of the convergence and decay of the integral. Namely, we need an additional information about a regularity of the matrix M-1(iw) at its singular point w = 0 and some bounds at Iwi --> oo. We will analyze all the points separately. 1. First we study the asymptotic behavior of M-1(A) at infinity. Let us recall

that M-1(A) was originally defined for ReA > 0, but it admits a meromorphic continuation to the entire complex plane C (see Lemma 7.1). The following proposition is a very particular case of a general fundamental theorem about the bound for the truncated resolvent on the continuous spectrum. The bound plays a crucial role in the study of the long-time asymptotics of general linear hyperbolic PDEs, [26].

Proposition 7.5. There exist a matrix Ro and a matrix-function Rl (w), such that

M-1(iw) = Ro + Rl (w), where, for every k = 0, 1, 2, ... , IaaR1(w)I
1, w E I[8,

Iwl > 1 w E R.

II. Second, we consider the point w = 0 which is most singular. The point is an isolated pole of a finite degree by Lemma 7.1, hence the Laurent expansion holds, n

M-1(iw) = E Lkw-k-1 + h(w),

(7.7)

Iwl < so,

k=0

where Lk are 6 x 6 complex matrices, so > 0, and h(w) is an analytic matrix-valued function for complex w with IwI < so. The rest of the proof of Proposition 6.7 is similar to that of [11].

8. Frozen form of transversal dynamics Now let us fix an arbitrary t1 E [0, t*), and rewrite the equation (6.15) in a "frozen form" (8.1)

Z(t) = A1Z(t) + (A(t) - A1)Z(t) + N(t),

0 < t < t*,

where Al = Av(t1),v(t1) and

([w(t)_v(ti)].V A(t) -A, =

0

0 0

0

0

0

0

0 0

0 0

(o(

Bv(t) 0

V. IMAYKIN ET AL.

264

The next trick is important since it allows us to kill the "bad terms" [w(t) -v(tl)] V in the operator A(t) - A1.

Definition 8.1. Let us change the variables (y, t) H (yi, t) = (y + d1(t), t) where t

d1(t) :=

(8.2)

j(w(s) - v(tl)) ds,

0

t

t1.

1

Next define (8.3)

ZI (t) = (T i (yl, t),111(yi, t), Q(t), P(t)) := (W (y, t), H(y, t), Q(t), P(t)) = (`W (yl - di(t), t), H(yi - di(t), t), Q(t), P(t))-

Then we obtain the final form of the "frozen equation" for the transversal dynamics (8.4)

.j(t) = A1Z1(t) + B1(t)Z1(t) + N1(t),

0 < t < t1,

where N1(t) = N(t) expressed in terms of y = yl - d1(t), and 0

B1(t) =

0

0

0

0

0

0

0

0

0

Bv(t1)

l

0 (V(Ltict) -'vct1)), Vp) At the end of this section, we will derive appropriate bounds for the "remainder terms" B1(t)Z1(t) and N1(t) in (8.4). First, note that we have by Lemma 6.2, 0

0

t1

t

(8.5)

Bv(t) - Bv(t1)1 < I

itt l

v(s) V Bv(s) dsl < C

itt

IIZ(S) 112 a

ds.

Similarly,

I

(8.6)

t1

1(V(0"(t)- v(t1)),Vp)I 0 if dp is small enough.

12. Soliton asymptotics Here we prove our main Theorem 2.5 under the assumption that the decay (6.13) holds. First we will prove the asymptotics (2.13) for the vector components, and afterward the asymptotics (2.14) for the fields.

Asymptotics for the vector components. From (4.3) we have 4 = b + Q,

0 Z2 Q). Thus,

and from (6.15), (6.16), (4.9) it follows that Q =

O(IIZI12 a). 4 = b + Q = v(t) + c(t) + Recall that 3 = 1 + 6, 0 0.

Therefore, c(t) = c+ + O(t-(1+26)) and v(t) = v+ + O(t-(I+26)), t , oo. Since JPJ < IIZII_,3, the estimate (6.13), and (12.2), (12.1) imply that

4(t) = v+ + O(t-p).

(12.3)

Similarly, ( 12.4)

b(t)

c(t) +

f

v(s) ds = vt + a+

hence the second part of (2.13) follows:

q(t) = b(t) + Q(t) = v+t + a+ + 0(t-21), since Q(t) = O(t-a) by (6.13). (12.5)

Asymptotics for the fields. We apply the approach developed in [9], see also [6, 8,10,12]. For the field part of the solution, F(t) _ ((x, t), 7r(x, t)) let us define the accompanying soliton field as Fv(t) (t) = (,b (t) (x - q(t)), 7rv(t) (x - q(t))), where we define now v(t) = 4(t), cf. (12.1). Then for the difference Z(t) = F(t) - Fv(t)(t) we obtain easily the equation [12, eq. (2.5)],

Z(t) = AZ(t)

-

- V Fv(t) (t),

A(O, 7r) = (7r, LO).

Then (12.6)

Z(t) = W°(t)Z(0) - f

W°(t - s) [v(s)

VVFv(s)(s)] ds.

V. IMAYKIN ET AL.

270

To obtain the asymptotics (2.14) it suffices to prove that Z(t) = W°(t)IF + + r+(t) with some T+ E F and 11r+(t)JIT = O(t-a). This is equivalent to

W°(-t)Z(t) = `y+ + r+(t),

(12.7)

where jIr+(t)jH.F = O(t-b) since W°(t) is a unitary group in the Sobolev space F by the energy conservation for the free wave equation. Finally, (12.7) holds since (12.6) implies that (12.8)

W°(-t)Z(t) = Z(O) + J W°(-s)R(s) ds, t

R(s) = i(s) V,F,(s)(s),

0

where the integral in the right-hand side of (12.8) converges in the Hilbert space

F with the rate O(t-a). The latter holds since jjW°(-s)R(s)jjf = 0(s-,) by the unitarity of W°(-s) and the decay rate JMR(s)jj,r = O(s-a) which follows from the asymptotics for the vector components. More precisely, differentiating the

modulation equations (6.11) we obtain an estimate for v(t) = q(t) providing the mentioned decay rate of R(s). Acknowledgments. The authors thank V. Buslaev for detailed and numerous lectures on his results and fruitful discussions.

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14. A. Komech and H. Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field, Nonlin. Analysis 33 (1998), no. 1, 13-24. 15. J. Miller and M. Weinstein, Asymptotic stability of solitary waves for the regularized long-wave equation, Comm. Pure Appl. Math. 49 (1996), no. 4, 399-441.


16. R. L. Pego and M. I. Weinstein, On asymptotic stability of solitary waves, Phys. Lett. A 162 (1992), no. 3, 263-268. 17. , Asymptotic stability of solitary waves, Comm. Math. Phys. 164 (1994), no. 2, 305349.

18. A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable systems, Integrable Systems and Applications (ile d'Oleron, 1988) (M. Balabane, Lochak P., and C. Sulem, eds.), Lecture Notes in Physics, vol. 342, Springer, Berlin, 1989, pp. 312-327. 19. , Multichannel nonlinear scattering in nonintegrable systems, Comm. Math. Phys. 133 (1990), no. 1, 119-146. 20. , Multichannel nonlinear scattering and stability. II: The case of anisotropic potential and data, J. Differential Equations 98 (1992), no. 2, 376-390. 21. , Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), no. 1, 9-74. 22. , Selection of the ground state for nonlinear Schrodinger equations, Rev. Math. Phys. 16 (2004), no. 8, 977-1071. 23. H. Spohn, Dynamics of charged particles and their radiation field, Cambridge Univ. Press, Cambridge, 2004.

24. B. Vainberg, Behavior of the solution of the Cauchy problem for a hyperbolic equation as t -* oo, Mat. Sb. (N.S.) 78 (120) (1969), 542-578 (Russian); English transl., Math. USSRSb. 7 (1969), no. 4, 533-568. 25. , The short-wave asymptotic behavior of solutions of stationary problems and the asymptotic behavior as t -* oo of solutions of nonstationary problems, Uspehi Mat. Nauk 30 (1975), no. 2(182), 3-55 (Russian); English transi., Russian Math. Surveys 30 (1975), no. 2, 1-58. 26. , Asymptotic methods in equations of mathematical physics, Gordon & Breach Publishers, New York, 1989.

27. M. I. Weinstein, Modulational stability of ground states of nonlinear Schrodinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472-491. ZENTRUM MATHEMATIK, TECHNISCHEN UNIVERSITAT MUNCHEN, 85748 GARCHING, GERMANY

Current address: Fakultat fdr Mathematik, Universitat Wien, Nordbergstraiie 15, 1090 Wien, Austria E-mail address: [email protected] FAKULTAT FUR MATHEMATIK, UNIVERSITAT WIEN, NORDBERGSTRASSE 15, 1090 WIEN, AusTRIA

Current address: Department of Mechanics and Mathematics, Moscow State University, Moscow 119992, GSP-2 Russia E-mail address: alexander.komechmunivie.ac.at

DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NORTH CAROLINACHARLOTTE, 376 FRETWELL BLDG., 9201 UNIVERSITY CITY BLVD., CHARLOTTE, NC 28223-0001,

USA

E-mail address: brvainbememail.uncc.edu


Purely Absolutely Continuous Spectrum for some Random Jacobi Matrices Uri Kaluzhny and Yoram Last ABSTRACT. We consider random Jacobi matrices of the form (J.-) (n) = an (w)u(n + 1) + bn (w)u(n) + an-1(w)u(n - 1)

on 22(N), where an(w) = an + an(w), bn(w) = bn + /3n(w), {an} and {bn} are sequences of bounded variation obeying an --s 1 and bn - 0, and {an(w)} and {/3n(w)} are sequences of independent random variables on a probability space (0, dP(w)) obeying

r

/3n2 (w)) dP(w) < oc

n=1 Jll

and fo an (w) dP(w) = ff On (w) dP(w) = 0 for each n. We further assume that

there exists Co > 0 such that 1/Co < an(w) < Co and -Co < bn(w) < Co for every n and P a.e. w. We prove that, for P a.e. w, J,,, has purely absolutely continuous spectrum on (-2,2).

1. Introduction In this paper we study self-adjoint Jacobi matrices of the form

J({an}°n°----1 {bn}nom--1)

-

bl

a1

0

a1

b2

a2

0

a2

b3

...

where aj > 0 and bj E JR. We consider only such matrices whose entries are bounded, so they define bounded self-adjoint operators on t2(N). We say that a sequence {cn}°°_1 C JR is of bounded variation if r° 1 cn+1 - cn I < oo. We denote by Jo the free Laplacian on £2(N), which is the Jacobi matrix of the form (1.1) with an = 1 and bn = 0. Our main result in this paper is the following: 2000 Mathematics Subject Classification. Primary: 81Q10; Secondary: 47B15, 82B44. This research was supported in part by The Israel Science Foundation (Grant No. 188/02) and in part by Grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. This is the final form of the paper. ©2007 Uri Kaluzhny and Yoram Last 273

U. KALUZHNY AND Y. LAST

274

Theorem 1.1. Let (SZ, dP(w)) be a probability space and let Jw = J({an (w)}°n°--{bn(w)rn 1),

where an(w) = an + an(w) and bn(w) = bn + Qn(w). Assume that (i) {an}°°_1 and {bn}°n°--1 are real-valued sequences of bounded variation obey-

ing limn_,,, an = 1 and limn_,,,. bn = 0. (ii) {an(w)}°°_1 and { 3 ( w ) } . 1 are sequences of real-valued independent ran-

dom variables on (Il, dP(w)) obeying

fn(w)dP(w) =

J

3n(w) dP(w) = 0

for each n and f (ate (w) + an(w)) dP(w) < 00. n=1

There exists a constant Co > 0 such that 1/Co < an(w) < Co and -Co < bn(w) < Co for every n and P a.e. w. Then, for P a.e. w, J4, has purely absolutely continuous spectrum on (-2, 2) with essential support (-2, 2). ( iii)

Remarks. (1) Saying that the essential support of the absolutely continuous spectrum of Jw is (-2, 2) means that the absolutely continuous part of the spectral measure of J4, is supported on (-2, 2) and gives positive weight to any subset of (-2, 2) that has positive Lebesgue measure. (2) Our proof actually shows something a bit stronger (see Remark (3) to Theorem 2.1 below), namely, that for P a.e. fixed w, the purity of the absolutely continuous spectrum in (-2,2) will be stable under changing any finite number of entries in J,. Theorem 1.1 is essentially an extension of a result of Kiselev - Last - Simon [7, Theorem 8.1], who obtained this theorem for the special case an = 1, bn = 0, an(w) = 0 (also see [4, 8,14] for related earlier results). Part of the extension is in considering the nontrivial off-diagonal part, rather than just an(w) = 1 for all n. While this extension is fairly straight forward, it does add some technical complexity to the problem (partly due to the fact that one-step transfer matrices depend on pairs of neighboring an's and are thus not independent of each other). The more important extension in Theorem 1.1 is in adding the decaying perturbation of bounded variation J({an - 1}°°_1i {bn},0 1).

We note that the Jacobi matrix J({an},n 1, {bn}°°_1), where {CLn}°°_1 and {bn}- 1 are sequences of bounded variation obeying limn- an = 1 and limn. bn = 0, is well-known to have purely absolutely continuous spectrum on (-2,2) with essential support (-2, 2). That is, adding the decaying perturbation of bounded variation J({an - 1111, {bn}°_1) to the free Laplacian Jo doesn't change its absolutely continuous spectrum. This fact is the discrete version of Weidmann's theorem [17] (see, e.g., [16] for a proof). One may thus be tempted to think that such a perturbation of bounded variation may never be important for absolutely continuous spectrum. However, one of us have recently constructed [11] an example of a Jacobi matrix J({an}°0_1, {bn}°0_1) with an = 1, bn = bn+bn, so that limn-,, bn = limn-0,, bn = 0, {bn}°n°--1 is of bounded variation, J({an}n' 1, {bn}°°_1) (like

A. C. SPECTRUM FOR RANDOM JACOBI MATRICES

275

J({an},` 1, {bn},0 1)) has purely absolutely continuous spectrum on (-2, 2) with essential support (-2,2), but J({an}O°_1i {bn,}°_1) has empty absolutely continuous spectrum. In particular, adding a decaying perturbation of bounded variation to a Jacobi matrix can fully "destroy" its absolutely continuous spectrum (even though such a perturbation does not change the absolutely continuous spectrum when added to the free Laplacian JO). Theorem 1.1 is connected with a recent result of Breuer-Last [1] which says, roughly speaking, that absolutely continuous spectrum of Jacobi matrices which is associated with bounded generalized eigenfunctions is stable under square-summable random perturbations of the form J({an(w)}°°_1, {/3n(w)}°°_1). In particular, their result imply that, with probability one, Ju, of Theorem 1.1 has absolutely continuous spectrum with essential support (-2, 2), but it doesn't exclude by itself the possibility of embedded singular spectrum in (-2, 2). If the probability distributions of the random variables {an(w)}°0_1 and {/3n(w)}°° 1 happen to be absolutely continuous with respect to the Lebesgue measure (in fact, it suffices for only some of the distributions to be absolutely continuous, for example, absolute continuity of the distribution of /31(w) or of the distributions of any consecutive pair i3n(w), /3n+1(w) would suffice), then one can use spectral averaging (see, e.g., [15, Theorem 1.8]) to conclude almost sure purely absolutely continuous spectrum in (-2,2), namely, to recover Theorem 1.1 from the general result of [1]. The main ingredient of Theorem 1.1 which doesn't follow from the result of [1] is the purity of the absolutely continuous spectrum in (-2, 2) even in cases where all of the probability distributions of the random variables {an(w)}°0_1 and {/3n(w)}°0_1 are singular with respect to the Lebesgue measure. This ingredient is connected with a key technical difference between the analysis of the current paper and [1]. Here we develop, following the original approach of [7], estimates which are uniform in

energy over subintervals of (-2,2) and which yield bounds on certain integrated (over energies) quantities. Such an approach cannot possibly work in the more general context considered by [1], where the obtained estimates are per individual energy and uniformity over energy ranges cannot hold. The current paper and [1] can thus be viewed as complementary to each other. We note that much of the original Kiselev -Last -Simon result [7, Theorem 8.1] discussed above can be seen as a special case of the celebrated deterministic result of Deift-Killip [3], which has been considerably strengthened by Killip - Simon [6]. The Killip-Simon theorem says, among other things, that any Jacobi matrix J for which J - Jo is Hilbert -Schmidt has absolutely continuous spectrum with essential support (-2,2) (this doesn't exclude singular spectrum embedded in (-2, 2)). Thus, by spectral averaging, one can recover [7, Theorem 8.1] for the case of absolutely continuous probability distributions. While various variants and extensions of the Killip-Simon result exist (see, e.g., [9,10,13,18] and references therein), it seems that the sum-rule techniques underlying all these deterministic results cannot handle a general decaying perturbation of bounded variation, which may have an arbitrarily slow decay rate. Stanislav Molchanov has been among the most important contributors to the theory of random Schrodinger operators ever since his seminal paper with Goldsheid

and Pastur [5], which gave the first proof of Anderson localization for such an operator. It is a pleasure to dedicate this paper to him on the occasion of his 65th birthday.


276

2. Proof of Theorem 1.1 Our proof of Theorem 1.1 relies on controlling the asymptotic growth of the norms of the 2 x 2 transfer matrices associated with the problem. These are defined by

Tn,m(E,w) =Tn(E,w)Tn_1(E,w)...T,n,(E,w), where

T (E w) = ((E

-

bj(1))/aj(w) -aj-1(O)/aJ(w)) JJ

We use the following result of [12]:

Theorem 2.1. Suppose there is some m E N so that lim inf

n-oo

II Tn,m (E, w) III' dE < oo

Ja J.

for some p > 2. Then (a, b) is in the essential support of the absolutely continuous spectrum of J, and the spectrum of Ju, is purely absolutely continuous on (a, b).

Remarks. (1) This is essentially [12, Theorem 1.3]. While [12] only discusses Jacobi matrices with an = 1, the result easily extends to our more general context. (2) As noted in [12], this theorem is an extension of an idea of Carmona [2]. (3) While the fact that (a, b) is in the essential support of the absolutely continuous spectrum isn't explicitly stated in [12, Theorem 1.3], this easily follows from spectral averaging and the fact that the lim inf,,- fd IITn m (E w) I I P dE < oo condition is invariant to changing any finite number of entries in the Jacobi matrix.

To prove Theorem 1.1 we fix an e > 0 and use Theorem 2.1 on the interval A = (-2 + e, 2 - e). Since e is arbitrary, Theorem 1.1 would follow. One of the technical difficulties we need to overcome is the fact that each of the matrices Tn(E,w) depends on both an(w) and an_1(w). Thus, Tn(E,w)'s for different n's are not independent. To overcome this, we introduce the matrices 1

An(w)

= \0

an O(w)/

and define

Pn(E,w) = An+1(w)Tn(E,w)An'(w) =

((E - bn(w))/an(w) an(w)

-1/an(w) 0

J

Now, =PnPn-1...Pm =An+,(w)Tn,m(E,w)Aml(w), Pn,m(E,w) and since IIAnll < Co, IIAn1II < Co, it suffice to prove that for a.e. w E SZ,

lim inf

n--oo In, J

IIPn,m (E, W)114 dE < oo.

We choose an m so that for every n > m, (1 - e/3)/(1 + e/3) < an and Ibnl < e/3. Then for every E E A, IE - bnI < 2(1 - e/3), hence I(E - bn)/anI < 2

1

eel/9, and we can define kn E (0, 7r) by

2 cos kn = (E - bn)/an.


277

1 - E2/9, we have for every n > m and E E 0, sinkn > E/3.

Since I cos knI < (2.1)

We want to separate the deterministic and the random parts of Pn. Since

E - bn

E - bn(w)

(E - bn)(an(w) - an) + an(bn(w) - bn)

an

an (w)

anan(w)

- an(w)2coskn + an(w)

we have

On (w)

an(w)

- aw) (2CoSkn -

E - bn (w)

(

a(w)

an )

and thus

Pn(E,w) =

((E - bn(w))/an(w)

-I/an(w) 0

an (w)

an

2 cos kn - ,3n (w) /an

-1 /an

an (w)

an(w)/an

0

r(2 Cankn

L\

an (w)

-1/anl

n(o)

1

J+an \ a2 w (w) -a2n 00)].

0

To control the growth of transfer matrices, [7] uses the EFGP transform, which is connected with the following equality:

sink 2 cos k - cos k) ( 1

0 1

1

sink) (0

cosk

0) - (-sink

sink.

cosk) (1 - cos k)

We use the following modification: (0

sink) (2 cos k -1/a)

a- cos k

a

- ( cos k

sink) (0

sink cos k

0

sink )

a- cos k

Denote

C.

E_ (

0

sinkn

) - Gin - cos kn)

Bn (E) =

'

cos kn

sinkn

(-sinkn

cos kn)

kn/an 1/an so Cn ( E) -1 = ( cot 1/ sinkn 0

_

D. (E) =

-1/an 0 )

kn (2 cos an

.

Based on the previous computations, we have

P. (E, w) = D.(E) + Q. (E, w),

(2.2)

where

Dn(E) = Cn(E)-1Bn(E)Cn(E)

(2.3)

and (2.4)

Qn(E, w) _

an(w)

L-an(w)Dn(E) +

0

(a2 (w) (W) n

0)

I

We define Dij(E) - Di (E)Di_1(E) . . . Dj (E) for i > j. These matrices define the deterministic part of Pn,m (E, w) and are uniformly bounded in the following sense:

Lemma 2.2. There exists a constant CA, such that for every E E 0 and n > m, JIDn,m(E)II

CA and IIDn,m(E)-1II < CA.


278

PROOF. Since the determinant of Dn,,,.,(E) is 1, we have IIDn,-,,,,(E)II = IIDn,.m.(E)-' I

l

for every E. It is thus sufficient to consider II Dn,m(E)II. From (2.3), C,,,,., and thus, since I(Bzll = 1, we get

we have Dn, , = C 113,,(C,,C,a 11)13,,-1

111(

IIDn,mII < IICn

-f IICjCj 1111) 11CmM j=m+1

.

Since IICjCj11II = II(Cj-1 +Cj - Cj-1)Cj11II C 1 + IICj - Cj-11111C311I1,

lcoskj -coskj-1I =

2I(E-bj)/aj - (E-bj_1)Iaj-1I,

and

Icos kj + cos kj-l

Islnkj-slnkj-lI -

lSlrikj+Slrikj_llcoskj-coskj_1I,

we see that for some constant C, uniformly on 0, IICjCj11II < 1 + C(I aj

- aj-1I + Ibj - bj-1I)

Using the fact that 1 + x < ex for x > 0, we thus conclude that IIDn,m(E)II 5 IICn 1(E)IIIICm(E)II eXp(C

\

E(Ian+1 - and + Ibn+1 - bnI)

n=1

which is uniformly bounded on A. This proves the lemma.

11

PROOF OF THEOREM 1.1. If {e1, e2} is an orthonormal basis in R2, then for

any 2 x 2 matrix M, IIMII

IIMe1II + IIMe211

Thus, if we show that there exists a constant C, such that for every e E ][82 with 11 e l l= 1 and every n>

m,

f< C,

(2.5)

then it would follow that

ff II Pn,m (E, U)) 1I4 dE dP(w) < 16C, and, by Fatou's lemma, we will have f limn-,oo inf JZA lI Pn,m (E, w) ll4 dE dP(w) < 16C.

This would mean that, for P a.e. w, lim inf f I I Pn,m (E, W)114 dE < oo, n--oo

A

and, as discussed above, this would prove the theorem.

To prove the estimate (2.5), define vl(E,w) = Dn1(E)Pl_1,n(E w)e. Using equation (2.2), we get v1+1 = Dn,1+1(DI + Q1)P1-l,me = (I + Dn,1+121Dn,1)vl = (T + R.t)vl,


279

where

7Z1 = Dn,l+l QiD-l

=

al

at

Z + atADn,l+1 (0 0) Dn,t1 + 0

1

ai - al at

0

0

_1

Dn,l+1 (1 0) Dn

t.

We want to estimate the average of Ilvt+1114 = ((Z+)Z1)vt, (Z+7 i)vt)2

=

(11V1112

II7Ztv1112)2

+ 2(vi, 7Ztvt) +

= Ilvi II4 + II1Zlvl II4 + 211v11I2117Ztvl I12 + 411vi II2(vl, + 4117Z1vl

112

)Ztv1)

(vl, 7Zlvl) + 4(vl, 7Zlvt)2.

Since vi (w) is independent of at (w) and 01(w), we have -at Lt

Ilvt ll2 (vl, 7Zivi) dP(w) =

+f

dP(w)

aft d p(w) f

f

II vi

II4 dP(w)

vi lI2 (vt, Dn,t+l II

f at - at dP(w) f II 2

+

z

S ince

0) Dn t vi) dp(w)

(0

2

vt

II2 (v1, Dn,t+1 ( ) Dn l vt/) dP(w). \

1/Co < al(w) < Co, we see that at (w)

_

at (w)

Qt(w) at (w) and

- /3l(w) +O(01(w)at(w)), at

_ at2

al2

+O(ai (w)),

at

at (w)

(a w)

= tat (w) + O (ai (w)).

Thus, since at (w) and /31(w) are of zero mean, Lemma 2.2 implies that for some constant C, uniformly on A and in 1, fz Ilvlll2(vt,

)Ztvt) dP(w)I < C fn (at +/3i ) dP(w)

f

IlvzII4 dP(w).

In a similar way, one can see that for a sufficiently large constant C, fo (vt 7Zivt)2 dP(w) < C f

(at + ai ) dP(w) f Ilvt ll4 dP(w), n

z 112

fsz Ilv1112117Zivt

dP(w) < C

fn(a2

+ /312) dP(w) I II vt II4 dP(w),

f 117Z1v1II4 dP(w) < C f (a1 + 0 2

and fsz

Il1Zivt ll2 (vt, IZ1vt) dP(w)

C

f

Ilvt II4 dP(w)

f

11vt II4 dP(w).

Thus, we obtain that for some constant C, uniformly on 0 and in 1, f IIv1+1 ll4 dP(w)
0 implies that

ft

(1 + C f (al (w) +

02

a

(w)) dP(w)

/

is uniformly (in m and n) bounded from above by some constant C, and so, uniformly on A and in n,

f Ivn+1(E,

W)114 dP(w)

0. It is a linearized model for chemical kinetics [15], is equivalent to Burger's equation in hydrodynamics [6], and describes magnetic phenomena [19]. We refer the reader to [6,15,18] for more background. The long-time behavior of the parabolic Anderson problem is well-studied in the

mathematics and mathematical physics literature because it is the prime example of a model exhibiting an intermittency effect. This means, loosely speaking, that most of the total mass of the solution, (1.3)

U(t) =

u(t, z),

for t > 0,

zEZZd

is concentrated on a small number of remote islands, called the intermittent islands. A manifestation of intermittency in terms of the moments of U(t) is as follows. For

0 < p < q, the main contribution to the qth moment of U(t) comes from islands that contribute only negligibly to the pth moments. Therefore, intermittency can be defined by the requirement,

limsup (U(t)P) IlP = 0, for 0 < p < q, (U(t)e)I/e denotes expectation with respect to t . Whenever is truly random, the where parabolic Anderson model is intermittent in this sense, see [15, Theorem 3.2] and (1.4)

t_.

Section 1.2 below.

The large-time behavior of the solution u to the PAM (1.1) is determined by the spectral properties of the Anderson Hamiltonian (1.5)

fl= Ad + .

Namely, since (under natural assumptions on ) the upper part of the spectrum of ?-l in Q2(Zd) is a pure point spectrum [3,10], the solution u admits the spectral representation (1.6)

u(t, ) _

with respect to the random eigenvalues )i and the corresponding exponentially localized random eigenfunctions vi. (For simplicity we ignore the possible occurrence of a continuous central part of the spectrum.) As t increases unboundedly, only summands with larger and larger eigenvalues will contribute to (1.6), and the corresponding eigenfunctions are expected to be localized more and more far from each other. Hence, for large t, the solution u(t, ) indeed looks like a weighted superposition of high peaks concentrated on distant islands. The solution u to (1.1) describes a random particle flow in Zd in the presence of random sources (lattice sites x with fi(x) > 0) and random sinks (sites x with

THE UNIVERSALITY CLASSES IN THE PARABOLIC ANDERSON MODEL

285

fi(x) < 0).1 Two competing effects are present: the diffusion mechanism governed by the Laplacian, and the local growth governed by the potential. The diffusion tends to make the random field u(t, ) flat, whereas the random potential has a tendency to make it irregular. The solution u to (1.1) also admits a branching particle dynamics interpreta-

tion. Imagine that initially, at time t = 0, there is a single particle at the origin, and all other sites are vacant. This particle moves according to a continuoustime symmetric random walk with generator Ad. When present at site x, the particle is split into two particles with rate l;+(x) and is killed with rate l;_(x),

and _ =

where l;+ =

are independent random i.i.d. fields

(x) may attain the value oo). Every particle continues from its birth site in the same way as the parent particle, and their movements are independent. Put

fi(x) = +(x) - _(x). Then, given _ and +, the expected number of particles present at the site x at time t is equal to u(t, x). Here the expectation is taken over the particle motion and over the splitting resp. killing mechanism, but not over the random medium (k_, +). A very useful standard tool for the probabilistic investigation of (1.1) is the well-known Feynman - Kac formula for the solution u, which (after time-reversal) reads (1.7)

ll

ll

u(t, x)

(t, x) E

[0,00)

X Zd,

0

where (X (s))SE[o,oo) is continuous-time random walk on Zd with generator Ad start-

ing at x E Zd under Ex. Our main interest concerns the large-time behavior of the random field u(t, ). In particular, we consider the total mass, i.e., the random variable (1.8)

U(t) _

u(t, x) _ ]Eo [exp{ .EZd

ft

lJ

l1 (X (s)) ds }J ,

t > 0.

JJ

Note that U(t) coincides with the value i(t, 0) of the solution u to the parabolic equation (1.1) with initial datum u(0, ) = 1 instead of u(0, ) = So. One should have in mind that, because of this, our considerations below also concern the large-time asymptotics of u. We ask the following questions: (1) What is the asymptotic behavior of U(t) as t ---> oo? (2) Where does the main mass of u(t, ) stem from? What are the regions that

contribute most to U(t)? What are these regions determined by? How many of them are there and how far away are they from each other? (3) What do the typical shapes of the potential l; (-) and of the solution u(t, ) look like in these regions? We call the regions that contribute the overwhelming part to the total mass U(t) relevant islands or relevant regions. The notion of intermittency states that there does exist a small number of relevant islands which are far away from each other and carry asymptotically almost all the total mass U(t) of u(t, ). This effect may also be studied from the point of view of typical paths X (s),

s E [0, t], giving the main contribution to the expectation in the Feynman-Kac 'Sites x with e(x) _ -oo may be allowed and interpreted as ('hard') traps or obstacles, sites with £(x) E (-oo, 0) are sometimes called "soft" traps.

W. KONIG

286

formula (1.8). On the one hand, the random walker X should move quickly and as far as possible through the potential landscape to reach a region of exceptionally high potential and then stay there up to time t. This will make the integral in

the exponent on the right of (1.8) large. On the other hand, the probability to reach such a distant potential peak up to t may be rather small. Hence, the first order contribution to U(t) comes from paths that find a good compromise between the high potential values and the far distance. This contribution is given by the

height of the peak. The second order contribution to U(t) is determined by the precise manner in which the optimal walker moves within the potential peak, and this depends on the geometric properties of the potential in that peak. It is part of our study to understand the effect of intermittency for the parabolic Anderson model in great detail. We distinguish between the so-called quenched setting, where we consider u(t, ) almost surely with respect to the medium t;, and the annealed one, where we average with respect to l;. It is clear that the quantitative details of the answers to the above questions strongly depend on the distribution of the field t; (more precisely, on the upper tail of the distribution of the random variable l; (O)), and that different phenomena occur in the quenched and the annealed settings. It will turn out that there is a universal picture present in the asymptotics of the parabolic Anderson model. Inside the relevant islands, after appropriate vertical shifting and spatial rescaling, the potential t; will turn out to asymptotically approximate a universal, nonrandom shape, V, which is determined by a characteristic variational problem. The absolute height of the potential peaks and the diameter of the relevant islands are asymptotically determined by the upper tails of the random variable (0), while the number of the islands and their locations are random. Furthermore, after multiplication with an appropriate factor and rescaling, also the solution u(t, ) approaches a universal shape on these islands, namely the principal eigenfunction of the Hamiltonian Ad + V with V the above universal potential shape. Remarkably, there are only four universal classes of potential shapes for the PAM in (1.1).

1.2. Intermittency. Let u denote the solution to the equation in (1.1) with =

initial datum u(0, ) = 1 instead of u(0, ) = Jo, with an i.i.d. potential

(l;'(x))XEzd. Assume that all positive exponential moments of t;(0) are finite. Let denote probability and expectation w.r.t. l;. Prob(.) and A first, rough, mathematical approach to intermittency consists in a comparison of the growth of subsequent moments of the ergodic field u(t, ) as t -+ oo. Define Ap(t) = log(it(t, 0)P),

p E N,

and write f oc, if AP-1

(1.9)

« AP

p-1 p Note that, by Holder's inequality, always Ap_1/(p - 1) < Ap/p. If the finite

moment Lyapunov exponents

AP= l 1Ap(t), pEN,


287

exist, then the strict inequality'\p_1/(p - 1) < Ap/p implies p-intermittency. To explain the meaning of Definition 1.1, assume (1.9) for some p E N \ {1} and choose a level function 4 such that Ap_1/(p - 1) eeP(t)})

as t -> oo. Hence, by Birkhoff's ergodic theorem, for large t and large centered boxes B in Z d, u(t, x)p

IBI-1 xEB

u(t, x)P.

IBI-1 xEBf1P(t)

This means that the pth moment (u (t, 0)p) is "generated" by the high peaks of ft (t,

)

on the "thin" set P(t) and therefore indicates the presence of intermittency in the above verbal sense. Unfortunately, this approach does not reflect the geometric structure of the set IF(t). This set might consist of islands or, e.g., have a net-like structure. Theorem 1.2. If t; = (Z;(x))XEZd is a nondeterministic field of i.i.d. random variables with (et£(°)) < oc for all t > 0, then the solution u(t, ) is p-intermittent

forallpEN\{1}. This is part of [15, Theorem 3.2], where, for general homogeneous ergodic potentials t;, necessary and sufficient conditions for p-intermittency of u(t, ) have been given in spectral terms of the Hamiltonian (1.5).

1.3. Annealed second order asymptotics. Let us discuss, on a heuristic level, what the asymptotics of the moments of U(t) are determined by, and how they can be described. For simplicity we restrict ourselves to the first moment. The basic observation is that, as a consequence of the spectral representation (1.6), (1.10)

U(t)

etat(O

(in the sense of logarithmic equivalence), where At(cp) denotes the principal (i.e., largest) eigenvalue of the operator Ad + with zero boundary condition in the "macrobox" Bt = [-t, t]d n Zd. Hence, we have to understand the large-time behavior of the exponential moments of the principal eigenvalue of the Anderson Hamiltonian l in a large, time-dependent box. It turns out that the main contribution to (et)t (c)) comes from realizations of the potential having high peaks on distant islands of some radius of order a(t) that is much smaller than t. But this implies that A (Z;) is close to the principal eigenvalue of 7-l on one of these islands. Therefore, since the number of subboxes of

W. KONIG

288

Bt of radius of order a(t) grows only polynomial in t and is spatially homogeneous, we may expect that

etat(0) ti et\Ra(t)(0) for R large as t - oo. The choice of the scale function a(t) depends on asymptotic "stiffness" properties of the potential, more precisely of its tails at its essential supremum, and is determined by a large deviation principle, see (1.17) below. In Section 2 we shall see examples of potentials such that a(t) tends to 0, to oo, or stays bounded and bounded away from zero as t -- oo. In the present heuristics, we shall assume that a(t) ---> oo, which implies the necessity of a spatial rescaling. In particular, after rescaling, the main quantities and objects will be described in terms of the continuous counterparts of the discrete objects we started with, i.e., instead of the discrete Laplacian, the continuous Laplace operator appears etc. The following heuristics can also be read in the case where a(t) - 1 by keeping the discrete versions for the limiting objects. The optimal behavior of the field in the "microbox" BRa(t) is to approximate a certain (deterministic) shape (p after appropriate spatial scaling and vertical shifting. It easily follows from the Feynman-Kac formula (1.8) that eH(t)-2dt
0,

denotes the cumulant generating function of (0) (often called logarithmic moment generating function). Hence, the peaks of mainly contributing to (U(t)) have height of order H(t)/t. Together with Brownian scaling this leads to the ansatz

Zt(.) = a(t)2 [([.(t)])

(1.12)

-

H()], t

for the spatially rescaled and vertically shifted potential in the cube QR = (-R, R)d.

Now the idea is that the main contribution to (U(t)) comes from fields that are shaped in such a way that t;t :s cp in QR, for some cp: QR -* 118, which has to be chosen optimally. Observe that (1.13)

t;t ti cp in QR

in BRa(t).

Ht t) +

a(t)2 (0,('t)) Let us calculate the contribution to (U(t)) coming from such fields. Using (1.10), we obtain (1.14)

(U(t)l1{fit

cp in QR}) eH(t) expl tARa t

l

(-7()) 1

The asymptotic scaling properties of the discrete Laplacian, Ad, imply that (1.15)

ARa(t)

a(t)2

\a(t)

/)

a(t)2 it((p),


289

where AR(cp) denotes the principal eigenvalue of A + co in the cube QR with zero boundary condition, and A is the usual Laplacian. This leads to (1.16)

(U(t)ll{fit ^ W in QR})

eH(t) eXPS a(t)2 AR(,p)T Prob(t

W in QR).

In order to achieve a balance between the second and the third factor on the right, it is necessary that the logarithmic decay rate of the considered probability is t/ca(t)2. One expects to have a large deviation principle for the shifted, rescaled field, which reads (1.17)

Prob(t ,:; cp in QR) ti exp(-a(t)2IR(cc)},

where the scale a(t) has to be determined in such a way that the rate function IR is nondegenerate. Now substitute (1.17) into (1.16). Then the Laplace method tells us that the exponential asymptotics of (U(t)) is equal to the one of (U(t) ii{fit .:: cp in QR} with optimal cp. Hence, optimizing on cp and remembering that R is large, we arrive at (1.18)

(U(t))

eH(t)

exp -a(t)2 X

where the constant x is given in terms of the characteristic variational problem (1.19)

X = lim

inf

R-*oo W: QR-R [IR(cp) - A'

The first term on the right of (1.18) is determined by the absolute height of the typical realizations of the potential and the second contains information about the shape of the potential close to its maximum in spectral terms of the Anderson Hamiltonian 'H in this region. More precisely, those realizations of with t .:: cp. in QR for large R and W. a minimizer in the variational formula in (1.19) contribute most to (U(t)). In particular, the geometry of the relevant potential peaks is hidden via X in the second asymptotic term of (U(t)).

1.4. Quenched second order asymptotics. Here we explain, again on a heuristic level, the almost sure asymptotics of U(t) as t -+ oo. Because of (1.10), it suffices to study the asymptotics of the principal eigenvalue A Like for the annealed asymptotics, the main contribution to at(e) comes from islands whose radius is of a certain deterministic, time-depending order, which we denote a(t). As t -f oo, the scale function a(t) tends to zero, one, or oo, respectively, if the scale function a(t) for the moments tends to these respective values (see also (1.22) below). However, a(t) is roughly of logarithmic order in a(t) if a(t) -> oo, hence it is much smaller than a(t). The relevant islands ("microboxes") have radius Ra(t), where R is chosen large.

Let z E Bt denote the (certainly random) center of one of these islands h = z + BR&(t) meeting the two requirements (1) the potential is very large in h and (2) has an optimal shape within B. This is further explained as follows. Let ht = maxBt be the maximal potential value in the large box Bt. (Then ht is a priori random, but well approximated by deterministic asymptotics, which can be deduced from asymptotics of H(t).) Then - ht is roughly of finite order within the relevant "microbox" B. Furthermore, - ht should approximate a fixed

W. KONIG

290

deterministic shape in B. Hence, we consider the shifted and rescaled field in the box B, (1.20)

a(t)) - ht],

fitO = a(t)2

in QR = (-R, R)d.

Note that (1.21)

1

e(z + ) ti ht + a(t)2 W

tt ^ W in QR

(t)

in j3-

- z.

A crucial Borel - Cantelli argument shows that, for a given shape cp, with probability one, for any t sufficiently large, there does exist at least one box b having radius Ra(t) such that the event {et cp in QR} occurs if IR(cp) < 1, where IR is the rate function of the large deviation principle in (1.17). If IR(cp) > 1, then this happens with probability 0. For the Borel-Cantelli argument to work, one needs the scale function a(t) to be defined in terms of the annealed scale function a(t) in the following way:

a(t)

(1.22)

a(a(t))2 = dlogt,

i.e., &(t) is the inverse of the map t --> t/a(t)2, evaluated at d log t. Note that the growth of a(t) is roughly of logarithmic order of the growth of a(t), i.e., if the annealed relevant islands grow unboundedly, then the quenched relevant islands also grow unboundedly, but with much smaller velocity. Hence, with probability one, for all large t, there is at least one box h in which the potential looks like the function on the right of (1.21). The contribution to At (e) coming from one of the boxes b is equal to the associated principal eigenvalue (1.23)

AB_;, (ht + a(t)2 9(

ht + a(t)2 AR (,p),

where we recall that AR(cp) is the principal Dirichlet eigenvalue of the operator A+ in the "continuous" cube QR. Obviously, At (e) is asymptotically not smaller than the expression on the right of (1.23). In terms of the Feynman-Kac formula in (1.7), this lower estimate is obtained by inserting the indicator on the event that the random path moves quickly to the box b and stays all the time until t in that box.

It is an important technical issue to show that, asymptotically as t --. oo, At (e)

is also estimated from above by the right-hand side of (1.23), if cp is optimally chosen, i.e., if .R(cp) is optimized over all admissible cp and on R. This implies that the almost sure asymptotics of U(t) are given as (1.24)

log U(t)

At(e)

ht - a(t)2 X

,

t -' oo,

where X is given in tterms of the characteristic variational problem (1.25)

lim

inf

R-oo co: QR-'R,IR(co)

oc, where v =

(1 - -y)/(d + 2 - d-y). Now we identify the four universality classes, ordered from thick to thin upper tails of (0). For the moment, we focus on the large time behavior of the pth moment (U(t)P) for any p > 0. In all cases there is a scale function a: (0, 00) -+ (0, oo) and a number X E R such that

H(pt a(pt)-d)

-

1

1 log(U(t)P) = (X + 0(1)), as t T oo. 0, (pt) 2 pt a(pt)-d pt The scale function a describes how fast the expected total mass, which at time t = 0 is localized at the origin, spreads, in the sense that (> zEZd u(t,z)1{JzJ < Rat)}) = 0. lim liminf a(t)2 log (3.4) RToo tToo t (KzEZd u(t, z)) Moreover, in the three classes where the mass does not concentrate asymptotically in a single point, there exists R > 0 such that u(t, z)I{IzI < Ra(t)}) (3.5) liminf a(t)2 log (>ZEZd < 0. tToo t (EZEZd u(t, z)) (3.3)

(1)

-y>1,or-y=1andk*=oo.

This case is included in [16] as the upper boundary case p = oo in their notation,

see Section 2.1. Here X = 2d, the scale function a(t) = 1 is constant, and the first term on the right hand side in (3.3) dominates the sum, which diverges to infinity. The asymptotics in (3.4) can be strengthened to 1 (u(t, 0)) _ tTOCtlog

du(tz

0,

i.e., the expected total mass remains essentially in the origin and the intermittent islands are single sites, a phenomenon of complete localization. We call this the single-peak case. (2) y = 1 and ,* E (0, oo). This case, the double-exponential case, is the main objective of [16], see Section 2.1. The prime example is the double exponential distribution with parameter

p E (0, oo),

Prob{e(0) > r} = exp{-er/P}, which implies H(t) = pt log(pt) - pt+o(t). Here a(t) -+ 11V/ * E (0, oc), so that the size of the intermittent islands is constant in time. The first term on the right-hand side in (3.3) dominates the sum, which goes to infinity. Moreover, (3.6)

X =gmin : Zd-R E g2=1

1

2

E (g(x) - g(y))2 - p x,yEZZd

x^'y

g2 (x) log g2 (x)

,

xEZd

where we write x - y if x and y are neighbors. This variational problem is difficult to Analise. It has a solution, which is unique for sufficiently large values of p, and

W. KONIG

296

heuristically this minimizer represents the shape of the solution. As noted in [11], for any family of minimizers gp, as p Too, gp converges to 6o, which links to the single-peak case. Furthermore, as p 10, the minimizers gp are asymptotically given by o(1))e_ 2 d/2

(1 + uniformly on compacts and in L1(Rd). Consequently,

x=pd(1-1logp+o(1) 2

asp10.

I

El

ry=1 andtt* = 0. Potentials in this class are called almost bounded in [16] and may be seen as the degenerate case for p = 0 in their notation. This class contains both bounded and unbounded potentials, and is analyzed for the first time in [17], see Section 2.5. The scale function a(t) and hence the diameter of the intermittent islands goes to infinity and is slowly varying, in particular it is slower than any power of t. The first

term on the right hand side in (3.3) dominates the sum, which may go to infinity or zero. Moreover, (3.7)

x=

(/

min 9EH'(1

)

(

f I Vg(x) I2 dx - p

lid

J

g2 (x) log gz (x) dx

l

119112=1

This variational formula is obviously the continuous variant of (3.6), and it is much easier to solve. There is a unique minimizer, given by 9p(x) _

()d/4exp(_x2),

representing the rescaled shape of the solutionon an intermittent island. In particular, x = pd(1 - 2 log(p/7r)), which is the asymptotics of (3.6) as p 10. Hence, on the level of variational problems, (3) is the boundary case of (2) for p . 0. (4) ry (-oo, 0] such that t _ a(t)d+z Y) = H(y), ilTOO t H(a(t)d uniformly on compact sets in (0, oo). It is not too difficult to see that this assumption holds if Assumption H holds for the index ry < 1, for a defined by (1.22) and

p

H(y) = ry-1 Y'. Here a(t) - oo as t H a(t) is regularly varying with index (1 - y)/(d + 2 - dry). The potential is necessarily bounded from above. In this case, the two terms on the right hand side in (3.3) are of the same order, and (3.3) converges to zero. Moreover (3.8)

x=

inf

(l

d

119112=1

jf

IV9(x)12 dx - p d

/

/Rd

2,,

9

(x) - 9z (x)

ry -1

dx9


297

In the lower boundary case where -y = 0, the functional f g2 -Y must be replaced by the Lebesgue measure of supp(g). In this case the formula is well-known and well-understood. In particular, the minimizer exists, is unique up to spatial shifts, and has compact support. To the best of our knowledge, for ly E (0, 1), the formula in (3.8) has not been analyzed explicitly, unless in d = 1. One can also show [17] that (3.8) converges to (3.7), as would follow from interchanging the limit -y T 1 with the infimum on g. This means that, on the level of variational formulas, (3) is the boundary case of (4) for -y 11. The variational problems in (3.6), (3.7), and (3.8) encode the asymptotic shape of the rescaled and normalized solution u(t, .) in the centered ball with radius of order a(t). Informally, the main contribution to (U(t)) comes from the events that

u(t, ['a(t)1) Ju(t, ['01(t)1)112

_ 9

where g is a minimizer in the definition of X. To the best of our knowledge this heuristics has not been made rigorous in any nontrivial case so far. Note that in case (1), formally, (3.6) holds with p = oo and hence the optimal g is 10.

References 1.

2.

P. Antal, Trapping problems for the simple random walk, Thesis, ETH, Zurich, 1994. , Enlargement of obstacles for the simple random walk, Ann. Probab. 23 (1995), no. 1, 1061-1101.

3. M. Aizenman and S. Molchanov, Localization at large disorder and extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245-278. 4. M. Biskup and W. Konig, Long-time tails in the parabolic Anderson model with bounded potential, Ann. Probab. 29 (2001), no. 2, 636-682. E. Bolthausen, Localization of a two-dimensional random walk with an attractive path interaction, Ann. Probab. 22 (1994), 875-918. 6. R. A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc. 108 (1994), no. 518. 7. , Stationary parabolic Anderson model and intermittency, Probab. Theory Related Fields 102 (1995), 433-453. 8. M. Donsker and S. R. S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (1975), 525-565. 9. , On the number of distinct sites visited by a random walk, Comm. Pure Appl. Math. 32 (1979), 721-747. 10. J. Frohlich, F. Martinelli, E. Scoppola, and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), no. 1, 21-46. 5.

11. J. Gartner and F. den Hollander, Correlation structure of intermittency in the parabolic Anderson model, Probab. Theory Related Fields 114 (1999), 1-54. 12. J. Gartner and W. Konig, Moment asymptotics for the continuous parabolic Anderson model, Ann. Appl. Probab. 10 (2000), no. 3, 192-217. 13. , The parabolic Anderson model, Interacting Stochastic Systems (J.-D. Deuschel and A. Greven, eds.), Springer, 2005, 153-179. 14. J. Gartner, W. Konig, and S. Molchanov, Almost sure asymptotics for the continuous parabolic Anderson model, Probab. Theory Related Fields 118 (2000), no. 4, 547-573. 15. J. Gartner and S. A. Molchanov, Parabolic problems for the Anderson model. I: Intermittency andrelated topics, Comm. Math. Phys. 132 (1990), no. 3, 613-655. 16. , Parabolic problems for the Anderson model. II: Second-order asymptotics and structure of high peaks, Probab. Theory Related Fields 111 (1998), no. 1, 17-55.

17. R. van der Hofstad, W. Konig, and P. Morters, The universality classes in the parabolic Anderson model, Comm. Math. Phys., to appear. 18. S. Molchanov, Lectures on random media, Lectures on Probability Theory (Saint-Flour, 1992) (P. Bernard, ed.), Lectures Notes in Math., vol. 1581, Springer, New York, 1994, pp. 242-411.

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19. S. Molchanov and A. Ruzmaikin, Lyapunov exponents and distributions of magnetic fields in dynamo models, The Dynkin Festschrift: Markov Processes and their Applications (M. Freidlin, ed.), Birkhauser, 1994, pp. 287-306. 20. A.-S. Sznitman, Brownian motion obstacles and random media, Springer Monogr. Math., Springer, Berlin, 1998. MATHEMATISCHES INSTITUT, UNIVERSITAT LEIPZIG, AUGUSTUSPLATZ 10/11, 04109 LEIPZIG, GERMANY



An Inverse Problem for Gibbs Fields Leonid Koralov ABSTRACT. According to classical results of Minlos and Ruelle, given a poten-

tial of pair interaction that is regular and stable, and a small value of activity, one can define the corresponding Gibbs field (a measure on the space of configurations of points in Rd or Zd). In this paper we consider a problem that may be viewed as a converse to the result of Minlos and Ruelle. Namely, we show that for a sufficiently small constant fil and a sufficiently small function p2(x), x E Zd, there are a pair potential and a value of activity, such that pl is the density and p2 is the pair correlation function of the corresponding Gibbs field.

1. Introduction Let us consider a translation invariant measure y on the space of particle configurations on the lattice Zd. For a given configuration each site can be occupied by one particle or be empty. An m-point correlation function p.m(xl, ... , Xm) is the probability of finding m different particles at positions x1,. .. , xn e Zd. The following natural question has been extensively discussed in physical and mathematical literature: given pl(xl) - fil and p2(xl, x2) = p2(xl - x2), does there exist a measure p, for which these are the first correlation function (density) and the pair correlation function, respectively? There are well-know conditions on fil and p2 that are easy to verify and are necessary (but not sufficient) for the existence of the translation invariant measure. Let 1; (x) be a stationary random field with values 0 and 1. Its distribution can be viewed as a measure on the space of particle configurations. Let fil and p2(x) be its first two correlation functions. Then the mean of the field is El; (x) and its covariance is given by fil

- Pl

C(x) = E(S(x) - pl)(S(0) - Pl) = lp2 (x) - pi

ifx=0 otherwise.

2000 Mathematics Subject Classification. 60G55, 60G60. Key words and phrases. Gibbs field, Gibbs measure, cluster functions, pair potential, correlation functions, Ursell functions. The work of the author is partially supported by NSF Research Grant. This is the final form of the paper. ©2007 American Mathematical Society 299

300

LEONID KORALOV

It is well-known that positive-definiteness of C(x) is necessary and sufficient for the existence of some random field with such a covariance. For example, if C(x) is positive-definite, one can always take to be a Gaussian field. However, the requirement that only takes values 0 and 1 imposes additional requirements on the correlation functions. In particular, there is a simple condition on the variance of the number of particles in a given region, which follows from the fact that the number of particles (sites where fi(x) = 1) is integer. The original statement of this condition can be found the the paper by Yamada [12], and a more general version in the paper of Costin and Lebowitz [3]. In the series of papers [5-7] Lenard provided a set of relations on the functions p,,,, which are necessary and sufficient for the existence of a measure with such correlation functions. However, given pi and P2, it is not clear how to check if there are some p3, p4,... for which these relations hold. There are several recent papers which demonstrate the existence of particular types of point processes (measures on the space of particle configurations), which correspond to given pi and P2 under certain conditions on pi and p2. In particular, one dimensional point processes of renewal type are considered by Costin and Lebowitz in [3], while determinantal processes are considered by Soshnikov in [10]. In [1] Ambartzumian and Sukiasian prove the existence of a point process corresponding to a sufficiently small density and correlation function. Recently Costin and Lebowitz [3], and Caglioti et al. [2] suggested generalizations of their results. In [11] Stillinger and Torquato consider fields over a space with finitely many points. Besides, for the lattice model, they discuss possible existence of a pair potential for a given density and correlation function using cluster expansion without addressing the issue of convergence.

In this paper we show that if pi and p2 are small (in a certain sense), there exists a measure on the space of configurations for which pi is the density and P2 is the pair correlation function. Moreover, this measure is the Gibbs measure corresponding to some pair potential and some value of activity. In a sense, this is the converse of the classical statement that a given potential of pair interaction and a sufficiently small value of activity determine a translation invariant Gibbs measure on the space of particle configurations in Zd (or Rd) and the sequence of infinite volume correlation functions.

2. Gibbs random fields Let us remind the reader of the construction of the Gibbs random field corresponding to a given potential of pair interaction and a given value of activity. Let 4)(x), x E Z' be a real-valued function (potential of pair interaction), and let U(xl,... , xn) = I1 0 is called the activity. The inverse temperature, which is the factor usually present in front of the function U, is set to be equal to one (or,

AN INVERSE PROBLEM FOR GIBBS FIELDS

301

equivalently, incorporated into the function U). The total mass of the measure is the grand partition function °O

n

e-U(xl,...,x')

-(A z, 4)) (x1....,x.)EAn

n=0

The m-point correlation function is defined as the probability of finding m different particles at positions x1i ... , xm c A, 00

m+n

PA (xl, ... , xm) = E (A, z n=0

n!

e-U(xl,...,x,,,,yi,...,y,..)

Y,

(yi,....yn )EA

The corresponding measure on the space of all configurations of particles on the set A (Gibbs measure) will be denoted by µA. Given another set A0 C A, we can consider the measure uA obtained as a restriction of the measure µA to the set of particle configurations on A0. Given a potential of pair interaction 4D(x), we define g(x) = e-41(x) - 1, x E Zd. We shall make the following standard assumptions:

g(x)>-a>-1

(2.1)

g(O) = -1;

(2.2)

for x

0.

g(x) = g(-x) for all x;

lg(x) I < c < oc. x#0

Clearly, any function g(x) which satisfies (2.1) - (2.2) defines a potential of pair interaction via 4)(x)

ln(g(x) + 1).

It is well-known ([8, 9]) that when A -> 7Ld in a suitable manner (for example, A = [-k, k]d and k -> oo) the following two limits exist for sufficiently small z: UZd on the space of all configurations on (a) There is a probability measure Zd, such that

µno -' µno

(2.3)

as A -> 7Gd

for any finite set A0 C Zd. (b) All the correlation functions converge to the infinite volume correlation functions. Namely, (2.4)

A

P

as A _ 7Gd.

The infinite volume correlation functions are the probabilities with respect to the measure pzd of finding m different particles at positions x1, ... , xm E Zd. To make these statements precise we formulate them as a lemma.

Lemma 2.1 ([8, 9]). Assuming that (2.1) and (2.2) hold, there is a positive z = z(a, c), such that (2.3) and (2.4) hold for all 0 < z < z when A = [-k, k]d and

k->oo. Thus, a pair potential defines a sequence of infinite volume correlation functions for sufficiently small values of activity. Note that pm (x1, . . . , xm) = 0 if xi = xj for i j, since two distinct particles can not occupy the same position. Also note that all the correlation functions are translation invariant, Pm(x1) .... xm)

=Pm(0,x2-x1,...,xm -x1).

LEONID KORALOV

302

Thus, pi is a constant, P2 can be considered as a function of one variable, etc. Let p,,,, be the function of m - 1 variables, such that (2.5)

pm(xl,...,xm) = pm(x2 -X1) ...,xm -x1).

3. Formulation of the result The main result of this paper is the following theorem.

Theorem 3.1. Let 0 < r < 1 be a constant. Given any sufficiently small constant pl and any function p2(x), such that p2(0) = 0 and Ex#o Ip2(x) - pit rp2, there are a potential fi(x), which satisfies (2.1) -(2.2), and a value of activity z, such that pl and p2 (x) are the first and the second correlation functions respectively for the system defined by (z, oD).

Remark. As will be seen from the proof of the theorem, the pair potential and the activity corresponding to given pl and p2(x) are unique, if we restrict and z. The method of the proof consideration to sufficiently small values of allows one to explore the properties of the pair potential based on the properties of the correlation function. The main idea of the proof is the following. Assuming that a pair potential and a value of the activity exist, we express the correlation functions (or, rather, the cluster functions, which are closely related to the correlation functions) in terms of the pair potential and the activity. This relationship can be viewed as an equation for unknown 4) and z. Then we use the contracting mapping principle to demonstrate that this equation has a solution. New estimates on the Ursell functions (see below) are needed to prove that the right hand side of the equation on (P and z is indeed a contraction.

4. Cluster functions and Ursell functions In this section we state a useful expression for cluster functions in terms of the pair potential. The cluster functions are closely related to the correlation functions.

The reader is referred to [9, Chapter 4] for a more detailed exposition. We also formulate a lemma on the growth of Ursell functions. Its proof can be found in [4]. Let A be the complex vector space of sequences Vb, w = (0m(x1, ... , xm))m>0

such that, for each m > 1, V),,,, is a bounded function on Zmd, and '0 is a complex number. It will be convenient to represent a finite sequence (x1,.. . , x,,,.) by a single letter X = (x1, ... , x.,,,,). We shall write

'O(X) = wm(xl,...,xm) Let now ?il, 02 E A. We define `Vl * 02 (X)

_

01(Y)02(X\Y), YCX

where the summation is over all subsequences Y of X and X \Y is the subsequence of X obtained by striking out the elements of Y in X. Let A+ be the subspace of A formed by the elements V) such that V5o = 0. Let 1 be the unit element of A (10 = 1, 1m - 0 for m > 1).


303

We define the mapping r of A+ onto 1 + A+ : *

*

+ ...

3

2!

The mapping r has an inverse r-1 on 1 + A+: 2

3

cp(X,.) correspondIt is easy to see that r (X) is the sum of the products cp(X1) ing to all the partitions of X into subsequences X1i ... , X,.. If cp E A+ and V) = Pcp, the first few components of 0 are

00=1;

"/,

w2(xl,x2) _ Let be a pair correlation function which satisfies (2.1) - (2.2), and let z < z(a, c). Note that the sequence of correlation functions p = (p .)m, o (with po = 1) is an element of 1 + A+. 01(x1)

_

1(x1);

Definition 4.1. The cluster functions w.m (xl,... , xm ), m > 1 are defined by

w = r-lp. Thus, w2(xl,x2) = P2(xl,x2) - Pl(xl)Pl(x2),

wl(x1) = Pl(xl);

or, equivalently, w1 = P1;

W2(x) = P2(X) - Pl

where Wm are defined as in (2.5).

Let b E 1 + A+ be defined by ,,/'o

= 1;

0m (xl, ... , x,,,) = e -U(xi,...,xm)

Define also

cp = r-lo. Definition 4.2. The functions V),,,, and cp,,,, are called Boltzmann factors and Ursell functions, respectively.

Lemma 4.3 ([9]). The cluster functions can be expressed in terms of the Ursell functions as follows

rz L nl 00

wm(xl,...,xm) = Zm

n=0

n

m+n(x1i...,xm,y1,...) Yn).

yi,...,y,EZd

We next state a lemma which provides a bound on the growth of Ursell functions

in terms of the function g(x) = exp(--P (x)) - 1. Lemma 4.4 ([4]). Suppose that functions gl (x) and 92 (x) satisfy (2.2) with c
0, k = 1, 2 be the corresponding Ursell functions. Then there exist constants ql and q2 such that

0i+n(0, yl, ... , yn) < n!q +1,

k = 1, 2,

yi,...,y,,. EZd

I(Pl+n (0, yl, ... , yn) - 1+n (0, y1, ... , yn) I < n!q2+l y,,...,ynEZd

1: Igl (x) - g2 (x) I. x#0

LEONID KORALOV

304

5. Proof of the main result Let us recast the main theorem in terms of the cluster functions and examine a system of equations, which relates the first two cluster functions with the pair potential and the activity. Theorem 3.1 can clearly be reformulated as follows

Proposition 5.1. Let 0 < r < 1 be a constant. Given any sufficiently small constant wl and any function 02(x), such that 02(0) = -wi and E.#0 1w2(x)I < rwi, there are a potential (D (x), which satisfies (2.1) -(2.2), and a value of activity z, such that 01 and 02 (x) are the first and the second cluster functions respectively for the system defined by (z, D). Consider the power expansions for w1 and w2, which are provided by Lemma 4.3.

Let us single out the first term in both expansions. Note the translation invariance of the functions w,,,, and corn and the fact that cp(xl, x2) = g(xl - x2). zn 1 (5.1) wl = z + z2 00 'Pl+n(O, Y1.... yn),

E

n=1

n

i

yi,...,ynEZd

00

(5.2)

zn

w2(x) = z29(x) + z3 E nl n=1

Let

o2+n(O, x, Y 1 ,-

.

,

yn).

V1....,ynEZd

0o zn -

A(z, g) = n=1

n!

Wl+n(O) yl, .... yn), Y1,....YnEZd

01) zn-

B(z, 9)(x) _ n=1

nI.

cP2+n(O, x, yl.... , yn)yi,...,ynEZd

Thus equations (5.1) and (5.2) can be rewritten as follows (5.3) z = w1 - z2A(z, g) (5.4)

g = z2 - zB(z, 9)

Instead of looking at (5.3) - (5.4) as a formula defining 01 and w2 by a given pair potential and the activity, we can instead consider the functions 01 and 02 fixed, and g and z unknown. Thus, Proposition 5.1 follows from the following.

Proposition 5.2. If wl and w2 satisfy the assumptions of Proposition 5.1, then the system (5.3) - (5.4) has a solution (z, g), such that the function g satisfies (2.1) -(2.2) and z < z(a, c). Now we concentrate on the proof of Proposition 5.2. We shall need the following notations. Let C be the space of functions g, which satisfy (2.2) with some c < oo. Let 11911 = Ex00 19(x) I. This is not a norm, since g is not a linear space, however d(91, 92) = 1191 - 9211 is a metric on the space C. Let 9, be the set of elements of 9 for which 1I9II < c. Note that if c < 1 then all elements of C, satisfy (2.1) with a = c. We also define J1.1,12 = [alzo, a2zo1. Let D = Izp ,a2 x Cc. Note that if c < 1 then (z, g) E D implies that z < z(c, c) if zo is sufficiently small. Thus, the infinite volume correlation functions and cluster functions are correctly defined for (z, g) E D if zo is sufficiently small.


305

Let us define an operator Q on the space of pairs (z, g) E D by Q(z, g) = (z', g'), where

z' = wl - z2A(z, g),

(5.5)

- zB(z,g)(x)

g'(x) _ -z2

(5.6)

for x

0;

g'(0)

We shall prove the following lemma.

Lemma 5.3. Let 0 < r < 1 be a constant. There exist positive constants al < 1, a2 > 1, and c < 1 such that the equation (z, g) = Q(z, g) has a solution (z, g) E D for all sufficiently small z0 if 01 = z0, 02 (0) = -zo, and E.#o 102 (x) I < rzo . Before we prove this lemma, let us verify that it implies Proposition 5.2. Let 0 < r < 1 be fixed and let wl be sufficiently small for the statement of Lemma 5.3 to be valid. Let w2 be such that 02(0) = -wi and Ex#o I w2 (x) I < rw2. Let (z, g) be the solution of (z, g) = Q(z, g), whose existence is guaranteed by Lemma 5.3. Let i and w2 be the first two cluster functions corresponding to the pair (z, g). Note that wl and i J satisfy the same equation z = wl - z2A(z, g);

Z=01 _Z2 A (z, g).

Therefore, 01 = 0'. The functions 02 and w2 also satisfy the same equation

g(x) =

wz2

- zB(z,g)(x);

Thus, w2 (x) = o2 '(x) for x

g(x)=w (2

-zB(z,g)(x);

for x

0.

0. The fact that w2 (0) = w2 (0) follows from

w2(0) _ -w1 = -wit = w2(0). Thus it remains to prove Lemma 5.3. The proof will be based on the fact that for small zo the operator Q : D -- D is a contraction in an appropriate metric. Define hjzl - z2

dzo(zl,z2) =

zo

The value of the constant h will be specified later. Now the metric on D is given by

p((zi, 91), (z2, 92)) = dZ0 (zl, z2) + d(gi, 92)

Lemma 5.3 clearly follows from the contracting mapping principle and the following lemma.

Lemma 5.4. Let 0 < r < 1 be a constant. There exist positive constants a1 < 1, a2 > 1, and c < 1 such that for all sufficiently small zo the operator Q acts from the domain D into itself and is uniformly contracting in the metric p for some value of h > 0, provided that wl = z0, W2(0) = -zo, and E,.#o Iw2(x)1 < rzo.

PROOF. Take c = (r + 2)/3, a1 = 2r3(r + 1), a2 = 2. We shall need certain estimates on the values of A(z, g) and B(z, g) for (z, g) E D. Namely, there exist

LEONID KORALOV

306

universal constants u1, ... , u6, such that for sufficiently small zo we have sup

(5.7)

(z,g)ED

sup

(5.8)

u1,

I A(z, 9) I

I B(z, 9) (x) I G U2,

(z,9) ED x#0

sup

(5.9)

(zi,9),(z2,9)ED

(5.10)

sup

IA(z1, 9) - A(z2, 9) I

u3Iz1 - z21,

IA(z, 91) - A(z, 92) I

u4d(91, 92),

(z,91), (z,92) ED

sup

(5.11)

E I B(zl, 9)(x) - B(z2, 9)(x)1 _< u5Iz1 - z2

(zi,9),(z2,9)ED x00

sup

(5.12)

IB(z,91)(x)-B(z,92)(x)I

u6d(91,92)

(z,9i),(z,92)ED 2540

These estimates immediately follow from Lemma 4.4 above. The fact that QD C_ D is guaranteed by the inequalities (5.13)

zo + (a2zo)2u1 < a2zo

(5.14)

zo - (a2zo)2u1 > alzo rzo2

(5.15)

+ a2z0u2 < C.

(a1z0)2

It is clear that (5.13) - (5.15) hold for sufficiently small zo. Let us now demonstrate that for some h and for all sufficiently small zo we have (5.16)

p(Q(zl, 91), Q(z2, 9z)) < 1 P((z1, 91), (z2, 92))

if (zl, 91), (z2, 92) E D.

First, taking (5.7), (5.9), and (5.10) into account, we note that (zz

dzo

A(zl, 91), zz2A(z2, 92))

< dzo (ziA(zl, 91), zzA(zl, 91)) + dzo (z2A(z1, 91), zzA(zz, 91)) + dzo (zz2A (z2, 91), z2 2A (z2, 92))

< ulhl zi - z221 + u3h(a2 zo)2Iz1 - z21 + u4h(a2zo)2d(91, 92) zo

zo

zo

If h is fixed, the right hand side of this inequality can be estimated from above, for all sufficiently small zo, by s (dzo (zl, z2) + d(91, 92))

Similarly,

E Iz1B(z1,91)(x) - z2B(z2, 92)(x) x 0

o be a sequence of positive integers and P = (PT)T>o a sequence of partitions of X. The elements of PT are called "clusters" of rank r. We say that (X, P, n) is a hierarchical structure if the following hold: (1) no = 1 and every Q E Po has exactly one element.

(2) For r > 1, every Q E Pr is a disjoint union of nr clusters in Pr_1. (3) Given x, y E X, there is a cluster Q of some rank containing both x and y. Let us state some immediate consequences of this definition. Every cluster of rank r > 0 has size Nr := f3 =o ns. Given x E X and r > 0, there is a unique cluster of rank r containing x. We denote this cluster by Q,(x). The map d(x, y) := min{r : y E Q,(x)}, is a metric on X and Qr(x) = {y : d(x, y) < r}. Note that Qr(x) = Qr(y) whenever d(x, y) < r. Given an integer n > 2, a hierarchical structure is called homogeneous 2000 Mathematics Subject Classification. 47B80, 47A55, 93A13. This work was supported by FQRNT and ISM grants. This is the final form of the paper. ©2007 American Mathematical Society 309

E. KRITCHEVSKI

310

Q3(x1)

Q2(x3)=Q2(x5)

Q2(X9) Q1 (x8)

Q1(X3) X1

x2,x3,x4,x5,x6rx7,x8,X9,x10,Xll,Xl2,Xl3,XI4)Xl5i...

Qo(x1)

here d(xl, xl) = 0, d(xl, x3) = 1, d(xl, x6) = 2, d(xl, x8) = 3, etc. FIGURE 1. A hierarchical structure

of degree n if n, = n for all r > 1. Figure 1 illustrates the first three levels of hierarchy in a hierarchical structure with nl = 3, n2 = 2, n3 = 3. The free Laplacian on the hierarchical structure (X, P, n) is defined as follows. For each r > 0, let Er : 12(X) -* 12(X) be the averaging operator

(Er)(x)

Nr

V) (y)

d(x,y)i be a sequence of positive numbers such that E°O 1 pr = 1. In the sequel we set po := 0 and r

Ar:=Eps, r=0,1,...,oo. S=O

The hierarchical Laplacian A on l2 (X) is defined by 00

A:_>prEr. T=O

Clearly, A is a bounded self-adjoint operator and 0 < A < 1. A hierarchical model is a hierarchical structure (X, P, n) together with the hierarchical Laplacian A. The basic spectral theorem for A is the following.

Theorem 1.1.

(1) The spectrum of A is given by U (A) = {Ar :r=0 ,--- ,0 0 1 -

Each Ar, r < oo, is an eigenvalue of A of infinite multiplicity. The point A,,. = 1 is not an eigenvalue. (2) Er - Er+1 is the orthogonal projection onto the eigenspace of Ar and 00

0 = E Ar(Er - Er+l). r=0 (3)

For every x E X, the spectral measure for bx and A is given by

r1

1

lb

,\r

where S(Ar) stands for the Dirac unit mass at Ar. Note that u does not depend on x.

HIERARCHICAL ANDERSON MODEL

311

A proof can be found in [5] or [8]. We now define the hierarchical Anderson model associated to (X, P, n) and the hierarchical Laplacian A. Consider the probability space (1k, F, 2) where SZ := I[8X, F is the usual Borel a-algebra in S2, and P is a given probability measure on (Il, F). For w E SZ, we set

E w(x)(6

VW

')bx.

xEX

V,J is a self-adjoint (possibly unbounded) multiplication operator on 12(X). Let

H, :=A+V, wEQ. The family of self-adjoint operators {H }WEO indexed by the events of the probability space (S2, F, IP) is called the hierarchical Anderson model. The main problem

is to understand the spectral properties of H,. The first natural question is to describe the spectrum o-(H,) of H, as a set. Note that, unlike the spectrum of the standard discrete Laplacian on Zd, the spectrum a(A) of the hierarchical Laplacian is a disconnected set. Lemma 1.2. Suppose that the random variables {w(x) : x E X} are i.i.d. with distribution v, i.e., P = ®xEX v. Let S be the support of v and ch(S) the convex hull of S. Then, for P-a. e. w E S2,

c (o,(0) + ch(S)) n ([0,1] + S). Sc In particular, if S is connected, then o,(H,) = a(0) + S, for IP-a.e. W E Q. The proof of Lemma 1.2 (given in the appendix) is a straight forward adaptation

of the proof of the corresponding well-known result for the standard Anderson model on Z' , where the spectrum is almost surely equal to [0, 4d] + supp(v) (cf., for example, [3, Proposition V.3.5]). We denote by Qac(H,) the absolutely continuous part of the spectrum of H4, and by o'cont(H,) the continuous part. In [8], Molchanov proved the following localization result:

Theorem 1.3. Suppose that (1.1)

prr1+E < oo, 1: r=1

for some e > 0. If the random variables {w(x) : x E X} are i.i.d. with a Cauchy distribution, 1 h de, 7f (e - a)2 + h2 for some parameters a E IIB, h > 0, then 0 for P-a.e. W.

ko,,h(e)de :=

A proof will be given in Section 3. A generalization of Molchanov's theorem will be stated and proved in Section 4. Molchanov's theorem is remarkable for several reasons. First of all, it is a result valid for any disorder. Given a coupling constant c > 0, the random operator O + cVW can be rewritten as O + ExEX cw(x)(8x )bx. If w(x) has a Cauchy distribution ka,h, then cw(x) has a Cauchy distribution kca,ch. Hence, under the hypotheses of the theorem, A + cVu, has pure point spectrum for IP-a.e. W. The next observation is that the condition (1.1) is very weak, it particular it does not at all depend on nr. To appreciate the generality of (1.1), one has to look at the density I

E. KRITCHEVSKI

312

of states and at the spectral dimension of A. Let xo E X be given and consider the increasing sequence of clusters Qr(xo), r > 0. Let PT be the orthogonal projection onto the Nr-dimensional subspace 12(Qr(xo)) :_ {0 E 12(X) : V)(x) = 0 for x V Qr(xo)}. Let e1T) < e2r)
YEX (Sx I OMV) = 1 yields that A generates a random walk on

X. We recall that the random walk on Z' generated by the standard discrete Laplacian is recurrent if d = 1, 2 and transient if d > 2. The corresponding result for the hierarchical Laplacian is:

Proposition 1.5. Consider a homogeneous hierarchical structure of degree n > 2. Suppose that there exist constants C1 > 0, C2 > 0 and p > 1 such that CiP_r

< Pr C

C2P-r

for r big enough. Then: (1) The spectral dimension of this model is log n d(n, p) = 2 log p

Hence 0 1 with d = d(n, p). Set Pr = Cp T, where C is a normalization constant making E°O 1 Pr = 1. Then pr satisfies (1.1). Therefore, we can construct homogeneous hierarchical Anderson models of arbitrary spectral dimension for which Theorem 1.3 holds. Since Cauchy random variables play a very special role in the theory of random Schrodinger operators, it is natural to ask whether one can extend Theorem 1.3 to distributions other than Cauchy. A partial answer to this question is that one can


313

prove localization for very general distributions of w(x), at any disorder, provided that one imposes stronger decay conditions on pr than (1.1). We will prove the following new result:

Theorem 1.6. Assume that there exists a sequence Ur > 0 with E°° 1 ur I < 00 and 00

EPrur Nr < 00. r=1

Then:

(1) For allwES2, Qac(Hw)res. (2) If P is conditionally a.c. then ocont(H,) = 0 for P-a.e. w. A proof will be given in Section 3.

Remark 1.7. The condition (1.2) is more demanding than (1.1). The required decay of pr imposes an upper bound on the spectral dimension of A. Theorem 1.6 and Proposition 1.5 allow us to construct hierarchical models with spectral di-

mension d < 4 that exhibit localization at arbitrary disorder. If (X, P, n) is a homogeneous hierarchical structure of degree n > 2 and Pr = Cp-r with p > f, then the hypothesis (1.2) is fulfilled for ur = r1+'. Given 0 < d < 4 one can adjust p> to make d(n, p) = d. If pr = Cr-2-En-r/2 then the model has spectral dimension d = 4 and (1.2) is verified for ur = r1+''2 One can also construct trivial models with d = 0 by taking pr to decrease faster than p-r for any p. We emphasize that homogeneity of the hierarchical structure is not required for Theorem 1.6. Remark 1.8. Theorem 1.6 is an improvement of the main localization theorem

of [5]. In [5], instead of (1.2), it is assumed that Y_1 prNr-iur-iur < 00, which imposes the spectral dimension to be < 2.

2. Localization criterion In this section we will formulate and prove a sufficient condition for HW to have Qac (H,,,) = 0 for all w E Q, and a sufficient condition for H, to have a,o,,t (HH,) = 0 for P-a.e. w. Both Theorems 1.3 and 1.6 will follow from this localization criterion and will be proved in the next section. Consider the truncated hierarchical Laplacian r

6'r

L p.,

Eg,

r > 0,

s=0

and set

H,,,,r:=V,,+Or,

r>0.

Fix w c Q. Note that for any Qr E Pr, the subspace l2(Qr) is invariant for HH,,,r. This is the main reason for working with the truncated hierarchical Laplacian. Let a(w, Qr) be the set of the eigenvalues of the restricted operator Hw,r r 12(Qr) and o,,, :_ U u(w, Qr) where the union is over all clusters of all ranks. Clearly, o is a countable subset of R, and hence of zero Lebesgue measure. For z c C\a,, r > 0, and x, y c X, we set Gw,r(x, y; z) := (Sx I (Hw,r - z)-1bv)-

E. KRITCHEVSKI

314

For z E (C\ow, r > 0 and t E X, let gw,r(t; z) be the average of cluster Qr(t), i.e., gw,r(t; z) :=

1

Nr d(t',t) 0 be given. Then r (2.5)

Gw,r(x, y; z) = Gw,o(x, y; z) - E p.,Ns-lgw,s-l (x; z)gw,s(y; z), s=d(x,y)

and r

(2.6)

gw,r(x; z) = N Gw,o(x, x; z) 11 (1 - p.s'Yw(Q3(x); z)). s=1


315

PROOF. The formula (2.5) holds for r = 0 since po = 0. For s > 1, the resolvent identity yields

(H,,J,8 - z)-'6y - (Hw,s-1

- z)-lsy = -(Hw s-1 -

z)-1ps Es(Hw,s

Observe that E, (H,,, - z) l6y = gw,s(y;z)1Qs(y). Taking (5

- z)-1Sy.

) in the above

equation yields (2.7)

Gw,s(x, y; z) - Gw,s-1(x, y; z) = -psgw,s(y; z)(6x I (Hw,s-1 - z)-l1Qe(y)).

Note that by (2.1), Ns-lgw,s-1(x; z),

(6. I (HL,,.-l - z)-l1Qs(y)) =

0,

if d(x, y) < s, if d(x, y) > s.

The formula (2.5) follows after adding (2.7) for s = 1, 2, ... , r. The proof of (2.6) is similar. The resolvent identity yields (6x

1 (Hw,r - z)-11Q. (x)) = (1 - pr'yw(Qr(x); z)) (6x I (Hw,r-l - z)-11Q,-1(x)),

and (2.6) follows after iterating the above formula.

Proposition 2.3. Let ur > 0 be a sequence with >°O 1 ur 1 < oo. Let w E Q and x E X be fixed. Then for m-a. e. e E I[8, there exists a finite constant C. such that 1/2

Ceus,

Igw,s(y;e)I d(x,y)<s

for alls>0. PROOF. Since l2(Qr(x)) is an Nr-dimensional invariant subspace for Hw,r and since II1Q,r(x)II2 = v, we have by Lemma A.1(see appendix) that for Mr > 0,

m({e c R \ o,. : II (H.,,

- e)-11Q,.(x) II2 > Mr})
o

R (e - \)2

dµ (A) iu

(e - A)2 + E2

.

Since for any z c C\R,

lim II (H,,,, - z)-1 - (H,,, -

z)-1II = 0

r_0o

we have that the weak-* limit limr_,w measure for H,,,,r and Sx. Therefore

I

dlL-x (A)

JII$ (e -

\)2

equals µ', where u',, is the spectral

dµ-'r(A) dµw'r(A) a>o rl oc Jj (e - \)2 + E2 < >o,P 1 Ja (e - A)2 + E2

I

-x'

- sup fR sup

(e

r>1 yEX

I

(')

- SUP (Hw,r -

e)-16x 112

(x, y; e) I2 < 00.

In the final equality we used the fact that {b, : y E X j is an orthonormal basis for l2(X). Since m(au,) = 0 and since the bound (2.11) holds for m-a.e. e E R\o-, we have that for every fixed w E Q, Gw,x(e) < oo for m-a.e. e E R. This proves part (1). Part (2) follows from the fact that Gw,.T (e) < cc for P ® m-a.e. (w, e) E S2 x R and the Simon -Wolff criterion. Part (3) is an immediate consequence of part (2) and assumption (2.2).

3. Proofs of localization theorems In this section we show how to apply the localization criterion (Theorem 2.1) to prove Theorems 1.6 and 1.3. PROOF OF THEOREM 1.6. Since rye,, (Qs (x); e) = >d(. y)<s g,,,,s (y; e), Cauchy-

Schwarz inequality and Proposition 2.3 yield that for m-a.e. e E IR, (3.1)

I'Y,,,(Q.(x);e)I

Hence Es>o pslry,,,(Qs(x); e) I < oo and the result follows from parts (1) and (2) of Theorem 2.1.

The proof of Molchanov's Theorem 1.3 requires more work. Let us introduce some notation. It is convenient to form the complex number v = a + ih E C+ and to denote by k the Cauchy distribution ka,h. If v is a Borel probability measure on the Riemann sphere S := C U fool, and T: S -> S is a Borel measurable map, then Tµ will denote the induced Borel probability measure on S: (Tv)(B)

vr\T-1(B)\

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318

for Borel sets B C S. If n > 2 is a integer, and v is a Borel probability measure on S, we let A,v :_ -r(v * v), n times

where T(z) = z/n, and * is the usual convolution of measures. Hence, if Y1, ... Yn are i.i.d. random variables on S with distribution v, then Anv is the distribution of (Y1 +- +Yn)/n. The following proposition, whose proof is left an an easy exercise for the reader, summarizes some of the well-known classical facts about Cauchy distributions.

Proposition 3.1. Cauchy distributions have the following properties: (1) If v1, v1 E C+, then kv1 * kv2 = kv1+v2

(2) If 'r(z) _ -2 or if T(z) = (az+b)/(cz+d), where a, b, c, d E R, ad-bc > 0, then Tkv = krvi

for allvEC+. (3) For all v E C+ and integers n > 2, Ankv = k,,.

We will use fractional linear transformations of the special form z

1 + pz' where p > 0. Note that Tp+p, = Tp o Tp,. The transformations 7-p are important in the hierarchical Anderson model because of the following recursive relation, proved by Molchanov in [8].

Proposition 3.2. Let w E SZ and z E C \ Qw be given. Then for all x E X, (3.2)

yw(Qo(x);z)

1

w(x) -z

Moreover, for all r > 1 and Qr E Prr, we have (3.3)

7w(Qr_i; z))'

yw(Qr; z) = Tpr 1 n r Qr-1 CQr

where the sum contains nr terms, one for each rank r - 1 subcluster Qr_I of Qr. PROOF. The formula (3.2) is clear since Gw,o(x, y; z) := (w(x) - z)1(Sx Sy). For r > 1, the resolvent identity yields

(Hw,r - z)-11QT - (Hw,r-1 - z)-11Qr = -pryw(Qr; z)(Hw,r-1 - z)-11Qr. Taking (1Q,. I ) in the above equation yields

Nr-1yw(Qr-1; z) = -pryw(Qr; z) E Nr-1yw(Qr-1; z).

Nryw(Qr; Z) -

Q;._1CQr

Q!1_1CQr

The formula (3.3) follows after dividing by Nr in the above equation and then solving for yw(Qr; z).


319

PROOF OF THEOREM 1.3. Since for every fixed w, m(ow) = 0, we have that for P ®m-a.e. (w, e) E S1 x R, e V a,,,. Hence, there is a Borel measurable set E C 1[8

with the properties (1) m(II8\E) = 0, and (2) for every fixed e c E, there is a set 1 e E J with IP(S2e) = 1 and such that

E lie, and hence for IP-a.e. w E 11, all the eE representation formulas in Propositions 2.2 and 3.2 are valid. It follows from (3.2) and from Proposition 3.1 that {yw(Qo(x); e) : x E X} are i.i.d. Cauchy random where vo(e) _ -(e - a + ih)-1. Moreover, (3.3) variables with distribution and Proposition 3.1 yield that for r > 1, {ryw,,(Qr; e) : Qr E Pr} are i.i.d. Cauchy random variables with distribution where vr(e) = TA,.vo(e). Since Ar -> Aoo = 1

as r - oo, the closure of the orbit

V(e) := U{vr(e)}, r>O

is equal to V(e) U {rivo(e)}, a compact set in C. Since supr>olvr(e)I < oo, there is a constant K(e) < oo, such that P(Iyw(Qr; e) I > u) < K(e) U

for all real u > 0, integer r > 0 and Qr E Pr. Let us now fix x E X and take Ur = r1+E. By the Borel-Cantelli lemma, for P-a.e. w E ie, there exists a finite constant Lw (e) such that (3.4)

I '/w (Qr (x); e) I < L. (e) Ur,

for all r > 0. Part (3) of Theorem 2.1 yields that 14,cont = 0 for IP-a.e. w c Q. Since x E X is arbitrary, the result follows. Note that, in the proof given above, (3.4) is a significant improvement of (3.1) is no longer present. since the factor

4. A generalization of Molchanov's theorem In this section we generalize Theorem 1.3 to convex combinations of Cauchy distributions. Let M denote the set of Borel probability measures on C+. Given v c M and a Borel set B C 1I8, we set

k is a Borel probability measure on I[8 and for every bounded Borel function f:JR-'C,

J

fdk Idv(v).

We call k a mixed Cauchy distribution. Note that the usual Cauchy distribution ka+ih is a special case of the above general defintion: ka+ih = k where v = b(a+ih). Let S be the set of mixed Cauchy distributions k,,, such that the distance from I[8 to the support of v is strictly positive.

E. KRITCHEVSKI

320

Theorem 4.1. Assume that there exists a sequence ur > 0 with E0 1 ur 1 < oe and 00

E prur < 00. T=1

If the random variables {w(x) : x E X} are i.i.d. with a mixed Cauchy distribution kV E S, then o0Ont(H,) = 0 for F-a.e. w.

Remark 4.2. Theorem 4.1 is valid at arbitrary disorder. Indeed, if w(x) has a distribution k E S, and c > 0, then cw has again a distribution of the form kv' E S. Precisely, v' = Tv, where T(z) = cz.

Remark 4.3. If f is any compactly supported continuous probability density on R, and e > 0, then there is a mixed Cauchy distribution kV E S s.t. supl f (e) - dkv/del < E. eex

To see this, recall that the harmonic function F(v) = ff f (e) dk (e), v E C+, converges uniformly to f as sv 10, and note that the integral can be approximated by finite Riemann sums. The proof of Theorem 4.1 parallels that of Molchanov's Theorem 1.3. First, we need a generalization of Proposition 3.1 to the case of mixed Cauchy distributions.

Proposition 4.4. Mixed Cauchy distributions have the following properties: (1) If vi, v1 E M, then kV1 *kV2 = kv, *V2

(2) If r(z) = -z or if r(z) _ (az+b)/(cz+d), where a, b, c, d E IR, ad-bc > 0, then

Tkv=krv,

for alluEM. (3) For all v E M and integers n > 2, Ankv = kAnv

The proof is a straightforward consequence of the definitions together with Proposition 3.1 and Fubini's theorem. PROOF OF THEOREM 1.3. Let E be the set of full Lebesgue measure as in the

beginning of the proof of Theorem 1.3, and let e E E be fixed. It follows from (3.2) and from Proposition 4.4 that {'y, (Qo(x); e) : x E X} are i.i.d. mixed Cauchy random variables with distribution kvo, where vo = You, To(z) = -(e - z)-1. Since the distance from the support of v to IR is strictly positive, there is a closed disk Do C C+ such that supp(vo) c Do. The relation (3.3) and Proposition 4.4 yield that for r > 1, {'y,,,(Qr;e) : Qr E Pr} are i.i.d. mixed Cauchy random variables with distribution kv,., where yr = -r,, A., yr_1.

For r > 0, let D. := TArpo. Since Tar is a fractional linear transformation taking C+ onto itself, Dr is again a closed disk in C+. We claim that for all r > 0, (4.1)

supp(er) C Dr.


321

The proof is by induction. By construction of Do, (4.1) holds for r = 0. Let r > 1 and suppose (4.1) holds for r-1. Since supp(vr_1) C Dr-1i and Dr_1 is convex, we have that supp(An,.vr_1) C Dr-1. Hence support(rrPrAn,, yr-1) C YP,, Dr-1 = Dr, and (4.1) holds for r, proving the claim. Let D be the closure of Ur>0 Dr. D is a compact subset of C+ and for all r > 0, supp(er) C D. It follows that there is a constant K(e) < oc, such that K(e) I > u) 0, integer r > 0 and Qr E Pr. The rest of the proof is the same as in Theorem 1.3, with the more general sequence ur instead of r 1+E.

El

Appendix A PROOF OF LEMMA 1.2. Let e c S. Given a cluster Qr or rank r and given e > 0, consider the event B(Qr, e) :_ {w : max lw(x) - eI < e}. XEQ,

Since v((e-E, e+E)) > 0 and the random variables {w(x) : x c Qr} are independent, we have that IP (B(Qr, E)) = (v ((e - E, e + E)) ) Nr > 0.

According to the hierarchical structure, X is a countable union of disjoint clusters of rank r: X = Uj001 QrJ. Let Br,E := limsupj B(Qr,j, E). Since the events {B(Qr,j, E), j = 1, ... , oo} are independent and E001F(B(Qr,j, E)) = oo, the second Borel-Cantelli Lemma yields that F(Br,E) = 1. Letting B:= n,0 Br,l/(r+1), we also have P(B) = 1. Hence, for w E B, there is a sequence (Qr,j,, : r > 0) of clusters of increasing rank with the property that

(Al) .Q xIw(x) - eI < r + 1' r

for all r > 0. Let A E a (A). Then A= as for some s E {1, ... , oo}. For r > 0, we let qr = min(r, s). It is easy to see that for each r _> 0, there exists a unit eigenvector L'r of A corresponding to the eigenvalue Aq,,, and such that 0r(x) = 0 for x Qr,j,,. Indeed, take a unit vector /'r constant on every rank qr subcluster of Qr,j,,, zero outside Qr,jr and such that Eq,.+i&r = 0. Then lim I I A7/Jr - As br I I = 0,

r-oo and, by (A.1),

lim II H,,O, - (As + e)2/JrII = 0, r-oo

for w c B. Hence, Or is a Weil sequence for the operator H, and As + e E o7(H,,,). We have shown that for IP-a.e. w E Q, Q(A) + S C o ,(H,). The inclusion o(H,,,) C (o() +ch(S)) n ([0, 1] + S) follows from general facts about the spectrum of a sum of two self-adjoint operators.

Lemma A.1. Let A be a Hermitian N x N matrix and v E CN. Then for all

M>0

m({e : II(A - e)-1vlI' > M})
0}

-yEr

be equipped with the natural Frechet topology. Then

U: Om - I,(((c*)n, em) is a topological isomorphism, where F((C*)n, £,,,) is the space of all analytic sections over (C*)n of the bundle Em, equipped with the topology of uniform convergence on compacta. 2It is sometimes also called the Gel'fand transform, due to Gel'fand's work [12].

P. KUCHMENT

328

(4) Under the transform U, the operator

P: H2(Rn) - L2(Rn) becomes the operator of multiplication by a holomorphic Fredholm morphism P(z) between the fiber bundles £2 and £o. Here P(z) acts on the fiber of £,,,, over the point z E T as the restriction to this fiber of the operator P acting between H2(K) and L2 (K).

Let us now mention another common way of looking at P(z). If z = exp ik, then commuting with the exponent exp ik x one reduce the bundle £,,,, to the trivial

one with the fiber H'n(Tn), where as before Tn = I8n/I'. On the other hand, the operator P(z) takes the form P(x, D + k) acting between Sobolev spaces on the torus Tn. In other words, the options are either dealing with the restriction of a fixed operator to an analytically "rotating" subspace, or with a polynomial family of operators between fixed spaces.

We will need to see how the structure of the Floquet solutions (see Definition 2.1), and in general, the structure of functions of Floquet type (2.2) reacts to the Floquet transform. For instance, in the constant coefficient case, where the role of the Floquet solutions is played by the exponential polynomials

pjxj,

e

jI 0, then the intersection of Fp with a sufficiently small neighborhood

of -i° is a (smooth) analytic manifold that coincides with the set of zeros of the function A(ik). Analogously to the Floquet surface -D = 4bP, we define the surface (2.12)

' := p(-i8) = {z I z = (exp 1;1i ... exp) t; E °},

and the tubular domain V := p(T),

(2.13)

where T was defined in (2.11). The results of Lemmas 2.10 and 2.11 can be restated in terms of these new objects:

Lemma 2.12. Let A0 > 0. Then

(1) 4 nV=T. If A0 > 0, then (2) The intersection of 4D with a sufficiently small neighborhood of XF is a (smooth) connected analytic manifold. (3) The intersections of 4D with neighborhoods of the tube V form a basis of neighborhoods of I in 4D. (4) For a sufficiently small neighborhood DE of T in D there exists an analytic H2(Tn) such that for any z E 4DE the function of x function p :

uz(x) = zxp(z,x) is a nonzero Bloch solution of the equation Pu = 0.

3. Representation of solutions by hyperfunctions and distributions The main result of this paper (Theorem 3.1 below) is analogous to the results of [2, 3] that characterize the classes of solutions of the Helmholtz equation that can be represented by means of distributions or hyperfunctions on S (see also the introduction to our paper). In order to state it, we need to introduce a new object. Let us denote by h(w), w E Sn-1 the indicator function of the convex domain G introduced in the previous section. Namely,

h(w) := sup(w t;),

(3.1)

BEG

is the inner product in R. The next main theorem will E; 1 be stated in terms of this function. where w

Theorem 3.1. Suppose that A0 > 0. (1) Let u be a solution of the equation Pu = 0 in 1Pn satisfying for some N the estimate (3.2)

Iu(x)1 < C(1 + Ixl)Neh(x/IxUIxI.

Then u can be represented as

= (A(0,uc(x)), where u£ is the analytic positive Bloch solution corresponding to E (see Lemma 2.10), and u(t;) is a distribution on E. The converse statement is also the function u(x) in (3.3) is a solution of the true: for any distribution p on equation Pu = 0 in Rn which satisfies for some N the growth condition (3.2). (3.3)

u(x)

P. KUCHMENT

332

(2) Let u be a solution of the equation Pu = 0 in ][RT satisfying for any e > 0 the estimate (3.4)

Iu(x)I 0},

where (O)m,N := Sup{IIbMIHm(K+7)(1 + I_y!)Neh(ry/i7i)iryi}. yEP

The operator P* clearly maps continuously W2 into Wo. It is also clear that due to (3.2), the linear functional (u, 0) :=

fn

u(x)O(x) dx

INTEGRAL REPRESENTATIONS

333

is continuous on the space Wo. Since Pu = 0, Schauder elliptic estimates together with the periodicity of the operator show that estimates similar to (3.2) hold also for the derivatives of u. One observes by a simple argument that u is a continuous functional on Wo, which annihilates the range of the dual operator P* : W2 --> Wo. Now we can apply Floquet theory arguments analogous to the ones used in [19, Section 3.2] or in [21] to obtain (3.3). However, some technical details needed in the cases considered in [19,21] and in this paper are significantly different, so we provide the details of this derivation. First of all, we need to obtain a Paley-Wiener type theorem for the Floquet transform in the spaces W,,,. Let us denote by V* the tube that consists of all points z E (C*)' such that z-1 = (zi 1, ... , zn 1) E V, where the tube V is defined in (2.13). We introduce the space A°°(V*) of holomorphic functions on the tube V * that are infinitely differentiable up to its boundary aV * . Analogously, if £ is a holomorphic Banach bundle in a neighborhood of V*, we denote by A"o (V*, E) the space of sections of £ over the (closed) tube V* that are holomorphic in the interior and infinitely differentiable up to the boundary of V*. This space is equipped with the natural Frechet space topology. The following statement is a Paley-Wiener type theorem for the transform U in the spaces W,,,,.

Lemma 3.3.

(1) The operator U : W,,,, -+ A°° (V *, £,,,,).

is a topological isomorphism.

(2) Under the transform U, the operator

P*:W2-+Wo becomes the operator P(z) of multiplication by a holomorphic Fredholm morphism between the fiber bundles £2 and go:

A-(V*,92)

'(Z)'

A- (V*, go).

Here P(z) acts on each fiber of £2 as the restriction to this fiber of the operator P* acting between H2(K) and L2(K). Before proving this lemma, we first obtain the following auxiliary statement:

Lemma 3.4. Let H be a complex Hilbert space and W(H) be the Frechet space of sequences f={ f y}, f y E H, y E F such that the semi-norm ON (f)

sup{IIf7IIH(1 + I'YI)Neh(7hI7I)I7I} ryEr

is finite for any N. Here, as before, h is the indicator function (3.1). Then a sequence f = {f.,} belongs to W(H) if and only if the function (3.6)

f (z) := E .f-7z'r ryEr

belongs to A°°(V*, H). The mapping f H f is an isomorphism of the space W(H) onto A°° (V *, H).

PROOF. Let f c W(H). We will show that the series (3.6) converges uniformly on V * as a series of H-valued functions on V*. This will imply that f is analytic in V* and continuous up to the boundary. Then we will check that the same holds for the derivatives of the series, which will imply that f E A- (V', H).

P. KUCHMENT

334

Taking into account that any z E V* can be represented as z = e-ik with Imk E G, and thus Imk y < h(y/I'yI)I'yI, we can estimate (3.7)

E rEr

If(z)II < E 'yEF

< E(1 +

y1)-n-1IIfYII(1 +

yI)n+leh('Y/I'YI)I'YI

- EF

< [(i +

_,j)-n-1

0n+1 (f) J

. yE F

Since the series E,yEF(1 + yI)-n-1 converges, this implies the analyticity in V* and continuity up to the boundary of f (z). Multiple differentiation with respect to k amounts to multiplying the coefficients of (3.6) by a polynomial with respect to -y factor. Due to the definition of the space W(H), one can get an estimate from above similar to (3.7), but with the seminorm On+d+1(f) instead of 0n+1 (f ), where d is the order of differentiation. Thus, in fact the function is infinitely smooth up

to the boundary. These estimates also prove that the mapping f E W(H) --> f E A°° (V*, H) is continuous.

Let us prove the surjectivity of this mapping. Assume that s(z) E A°°(V*, H). Let z = exp ik, then s as a function of k is periodic with respect to the reciprocal lattice P. Expanding it into the Fourier series, we get

s z-ry,

S(z) = E S_.yzry =

(3.8)

yEF

ryEF

where s_.y E H. We need to show now that {s.,} E W(H). For this purpose, we use the standard formulas for the Fourier coefficients: sry =

1n (21r)

Ls(_/3, Va E G,

where B is the first Brillouin zone, and we write z = exp ik = exp[i(/3 - ia) ], a E G. Integrating by parts 1 times with respect to 0, where 1 = ( 1 1 , . , ln) is a multiindex, we obtain analogously t

(3.9)

Sly = ((Z1yr)n

s

al ii (ei(P-ia))ei(a-ia) ry d, Va E G.

Now straightforward norm estimate in (3.9) gives l

(3.10)

IIS'YIIH < C

maxll as zE* (z) Hy_ie-a'Y

for any multi-index l and any a E G. Optimizing with respect to a E G, we get (3.11)

IIS-YIIH < CN(1 +

IyI)-Ne-h('Y/I'YI)I'YI

for any N. This means that f := {s} belongs to W(H) and by its construction f = s(z). This proves Lemma 3.4. Let us now complete the proof of Lemma 3.3. We start proving the first claim of the lemma. Let a function F(x) belong to W,,,,. Consider a sequence f = { fy} of elements of H"n(K) defined as follows:

fy(x) = F(x + y) x E K, y E r.


335

Then clearly the condition F E W,,,, is equivalent to two conditions: the first one

that f E W(Hm(K)), and second that F E Ho (][8n), i.e., that the functions fy defined on shifted copies of the fundamental domain K, fit smoothly across the boundaries. Analogously, the requirement that a section 0 belongs to A°° (V*, £,,,,) consists

of two conditions. The first one that 0 E A°°(V*, Hm(K)) and the second that it is a section of the subbundle £m C V* x Hm(K). We can notice now that the Floquet transform on W,,, is the restriction of the transform f H f of Lemma 3.4 from the larger space W (H- (K)). Thus, Lemma 3.4 claims that this transform is an isomorphism of W (Hm (K)) onto AOO (V *, Hm (K) ).

On the other hand, the second conditions: the fitting of fry across the boundaries and being a section of the subbundle £, are intertwined by the Floquet transform, according to the first statement of Lemma 2.6. This proves the first claim of the lemma. Now, the second claim of Lemma 3.3 follows from the third one of Lemma 2.6. Lemma 3.3 is proved.

Let us now return to the proof of Theorem 3.1. We remind the reader that we have a solution u with the estimate (3.2), for which we need to prove the representation (3.3). Let us apply the Floquet transform U. Then the image Uu of the solution u under the Floquet transform is a continuous linear functional on AO°(V*,£o), which is in the cokernel of the operator

A- (V*, 92) "z), A- (V*, 9,,). This, indeed is a one-to-one correspondence between solutions of the required class and such functionals. Thus, we need to describe all such functionals. Let zxp(z, ) be the Bloch solution of the equation Pu = 0 introduced in Lemma 2.12. We will also employ the space C°° (iv) with the standard topology, where the smooth variety W is introduced in (2.12). Consider the mapping

t: A' (V*, So) -+ C' (fl that for a section f (z, x) E AOO (V *, £o) of the bundle £o produces

tf ('z) = (f (z-1, .), uz(')) = f f (z-1, x)uz(x) dx. '

Here z 1 = /zl-1 , ... , zn-1) As we will see soon, the following lemma will finish the proof of the theorem:

Lemma 3.5. The mapping t is a topological homomorphism and the following sequence is exact: (3.12)

A°°(V*,£2) P(zi' A°°(V*,£o) -+ C- (T) -p 0.

PROOF OF THE LEMMA. Continuity of P(z) is already established. Continuity

of t is obvious. The complex property of the sequence (3.12) (i.e., that tP(z) = 0) follows from the construction of t. Thus, the only thing that requires proof is exactness in the second and third terms. The topological homomorphism property will follow then from exactness and the open mapping theorem. So, we only need to prove that: i) any section 0 E A°° (V*, £o) such that to = 0 belongs to the range of P(z) and ii) any function f E C°° (T) is in the range of t.

336

P. KUCHMENT

Let us start with the first of these tasks. So, let 0 E A°° (V*, Eo) be such that

to = 0. Consider the inverse P-1(z) to the morphism P(z). It is defined (and hence holomorphic) in a neighborhood VE* of the tube V*, except for an analytic submanifold, whose intersection with V* is xP (see Lemma 2.12). Let us consider

the function f = P-1(z)O(z). The only thing now to prove is that this function does not have any singularities along xP. This is a local question, so let us return in a neighborhood of a point of ' to the quasi-momenta coordinates k and consider the structure of the inverse P-1(z). As it was shown in the proof of [21, Lemma 21], the inverse has the form B(k)/A(k), where B(k) is an analytic operator-valued function.

This means that f (k) = (B(k)0(k))/A(k). The condition tO = 0 guarantees that the numerator g(k) = B(k)o(k) E A°°(V*, H) vanishes on xP, where H is a Hilbert

space. Our goal is to prove that this is sufficient for its smooth divisibility (on ,9V *) by A. We recall here that A is analytic in a vicinity of 8V * and has simple zeros along IkP (Lemmas 2.8 and 2.11). We notice that it is sufficient to prove this for scalar functions, i.e., for H = C. This can be justified in many different ways. For instance, the statement is local, and locally, due to the Fredholm nature of the morphism P(z), one can project the problem onto a finite dimensional subspace, using a lemma by M. Atiyah [5] (see also [19, Lemma 1.2.11 and Theorem 1.3.9; 36, Lemma 2.1]), which will reduce it to a finite-dimensional, and thus also to scalar case. So, we will assume in this part of the proof that g E A°° is a scalar function. According to a result of [17, 27] (see also [29, 31] and [29, Chapter VI, Theorem 1.1]), it is sufficient to check the divisibility at each point of XP on the level of formal Taylor series. So, let us pick a point k of IF and introduce coordinates x E Rn'-1 in the tangent space Tk(') E il[8n. The complexification Tkc(T) of this

tangent space is a part of the tangent space to the boundary of the tube. Let us chose coordinates y E R'-1 in Tk (W) fl ll that correspond to the coordinates x in Tk (IF). An extra coordinate tin Tk (W) fl 1R.' is required to obtain the whole tangent

space Tk(9V*). Let us denote by g(x, y, t) and A(x, y, t) the formal Taylor series of g and A at the point k. Then we know that g(x, 0, 0) = 0 and A(x, 0, 0) = 0 (formal power series versions of vanishing of functions g and A on '). Recall that g(x, y, t) is the series for a CR-function g on the boundary (since g is the boundary value

of an analytic function). This means that g(x, y, t) satisfies Cauchy-Riemann conditions with respect to the variable z = x + iy E C'i-1. Then uniqueness of analytic continuation4 claims that g(x, 0, 0) = 0 for all x implies g(x, y, 0) = 0 for all (x, y). The same is true for k, due to analyticity of A. Now, in coordinates z = x + iy, t we are dealing with the formal series g(z, t) and A(z, t), both of which vanish at t = 0 and such that A has zero of first order at t = 0. Then, vanishing of g(z, 0) guarantees divisibility in formal series of g by A. As it was explained above, this implies smooth divisibility of g by A and thus finishes the proof of exactness in the second term of the sequence (3.12). Let us now prove the exactness in the third term of the sequence. First of all,

we notice that the vector-function uZ, defined on ' only, can be extended to an analytic vector-function (which we will denote the same way) on VE* for some small epsilon. Indeed, as it is shown in [21], VE* is a Stein manifold. Then, according to the Corollary 1 from the Bishop's Theorem 3.3 in [36] (see the original theorem in [7]), 4The uniqueness of analytic continuation in this power series setting is straightforward to derive algebraically directly from the Cauchy-Riemann conditions for power series.


337

the restriction mapping to an analytic subset of a Stein variety is surjective. Thus, the required extension of u,z exists. Let also v(z) be a holomorphic family such that tv(z)lw, = 1 (it is not hard to prove the existence of such a family). Consider a function O(z) E C°°(T). Notice that the domain V* is strictly pseudo-convex and the complexifications of the tangent spaces to the submanifold' C 19V* are parts of the tangent spaces to aV*. Thus, and OV* satisfy the conditions of [9] needed for xP to be an A°° interpolation variety, and hence the restriction mapping A°° (V *) C°° (W) is surjective. Hence, there exists a function 0 E A°° (V *) such

that 0j,y = 0. Now taking f = V)(z)v(z) E A°°(V*,E0) guarantees that if = 0. This finishes the proof of the lemma.

It is easy now to finish the proof of the theorem. Indeed, after the Floquet transform solution u becomes a continuous linear functional on A°°(V*,Ej) that annihilates the range of the operator of multiplication by P(z). Lemma 3.5 implies that such a functional can be pushed down to the space C°° (IF). Any such functional is a distribution p. Hence, the action (u, 0) of the functional u on a function 0 E Wo

can be obtained as

(u,O) = (p(z),t(z00)) Applying now the explicit formulas for the transforms U and t, one arrives to the representation (3.3). Indeed, (3.13)

t(um) (z) =

-ry

fK(x)zu(x +) dx =

JR

(x)u(x) dx.

In this calculation we used the property of the Bloch solutions uz(x+'Y) = zryuz(x). Therefore,

(u, 0) = ((µ(z), uz), 0), which concludes the proof of the theorem.

Acknowledgments. The author expresses his gratitude to Y. Pinchover, the co-author of the previous papers [21, 22], with whom this manuscript has been discussed on numerous occasions and who has made many suggestions that have improved the text, and to A. Tumanov for helpful information on pick sets results.

References S. Agmon, On positive solutions of elliptic equations with periodic coefficients in lRn, spectral results and extensions to elliptic operators on Riemannian manifolds, Differential Equations (Birmingham, AL, 1983) (I. W. Knowles and R. T. Lewis, eds.), North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 7-17. 2. , A representation theorem for solutions of the Helmholtz equation and resolvent estimates for the Laplacian, Analysis, et cetera (P. H. Rabinowitz and E. Zehnder, eds.), Academic Press, Boston, MA, 1990, pp. 39-76. , Representation theorems for solutions of the Helmholtz equation on lRn, Differential 3. Operators and Spectral Theory (V. Buslaev, M. Solomyak, and D. Yafaev, eds.), Amer. Math. Soc. Transl. Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, pp. 27-43. 1.

4.

N. W. Ashcroft and N. D. Mermin, Solid state physics, Holt, Rinehart and Winston, New

5.

York -London, 1976. M. F. Atiyah, K-theory, W. A. Benjamin, New York-Amsterdam, 1967.

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338 6.

D. Battig, D. Knorrer, and E. Trubowitz, A directional compactification of the complex Fermi surface, Compositio Math. 79 (1991), no. 2, 205-229.

7.

E. Bishop, Analytic functions with values in a Frechet space, Pacific J. Math. 12 (1962),

1177-1192. J. Chaumat and A.-M. Chollet, Ensembles pics pour A' (D), Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, 171-200. , Caracterisation et proprietes des ensembles localement pics de A°° (D), Duke Math. 9. J. 47 (1980), no. 4, 763-787. 10. M. S. P. Eastham, The spectral theory of periodic differential equations, Scottish Acad. Press Ltd., Edinburgh-London, 1973. 11. L. Ehrenpreis, Fourier analysis in several complex variables, Pure Appl. Math., vol. 17, Wiley, 8.

New York, 1970.

12. I. M. Gel'fand, Expansion in eigenfunctions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR 73 (1950), 1117-1120 (Russian). 13. D. Gieseker, H. Knorrer, and E. Trubowitz, The geometry of algebraic Fermi curves, Perspect. Math., vol. 14, Academic Press, Boston, MA, 1992. 14. M. Hashizume, A. Kowata, K. Minemura, and Okamoto K., An integral representation of an eigenfunction of the Laplacian on the Euclidean space, Hiroshima Math. J. 2 (1972), 535545.

15. S. Helgason, Eigenspaces of the Laplacian; integral representations and irreducibility, J. Functional Analysis 17 (1974), 328-353. , Groups and geometric analysis: Integral geometry, invariant differential operators, 16. and spherical functions, Pure Appl. Math., vol. 113, Academic Press, Orlando, FL, 1984. 17. L. Hormander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555-568. 18. H. Knorrer and E. Trubowitz, A directional compactification of the complex Bloch variety, Comment. Math. Hely. 65 (1990), no. 1, 114-149.

19. P Kuchment, Floquet theory for partial differential equations, Oper. Theory Adv. Appl., 20.

vol. 60, Birkhauser, Basel, 1993. , On some spectral problems of mathematical physics, Partial Differential Equations

and Inverse Problems (C. Conca, R. Manasevich, G. Uhlmann, and M. S. Vogelius, eds.), Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 241-276. 21. P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal. 181 (2001), no. 2, 402-446. , Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds, Trans. Amer. Math. Soc., to appear. 23. P. Kuchment and B. Vainberg, On embedded eigenvalues of perturbed periodic Schrodinger operators, Spectral and Scattering Theory (Newark, DE, 1997), Plenum, New York, 1998, 22.

pp. 67-75. , On absence of embedded eigenvalues for Schrodinger operators withperturbed periodic potentials, Comm. Partial Differential Equations 25 (2000), no. 9-10, 1809-1826. , On the structure of eigenfunctions corresponding to embedded eigenvalues of locally 25. perturbed periodic graph operators, Comm. Math. Phys. 268 (2006), no. 3, 673-686. 26. V. Ya. Lin and Y. Pinchover, Manifolds with group actions and elliptic operators, Mem. Amer. Math. Soc. 540 (1994). 27. S. Lojasiewicz, Sur le probleme de la division, Studia Math. 8 (1959), 87-136. 28. B. Malgrange, Division des distributions, exposes 21-25, Seminaire Schwartz, Vol. 4, 195924.

1960.

, Ideals of differentiable functions, Tata Inst. Fund. Res. Studies in Math, vol. 3, Tata Inst. Fund. Res., Bombay; Oxford Univ. Press, London, 1967. 30. M. Morimoto, Analytic functionals on the sphere and their Fourier-Borel transformations, 29.

Complex Analysis (Warsaw, 1979), Banach Center Publ., vol. 11, PWN, Warsaw, 1983, pp. 223 - 250.

31. V. P. Palamodov, Structure of polynomial ideals and their quotients in spaces of infinitely differentiable functions, Dokl. Akad. Nauk SSSR 141 (1961), 1302-1305 (Russian). 32. V. Palamodov, Linear differential equations with constant coefficients, Grundlehren Math. 33.

Wiss., vol. 168, Springer, New York, 1970. , Harmonic synthesis of solutions of elliptic equations with periodic coefficients, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 3, 751-768.


339

34. R. G. Pinsky, Second order elliptic operators with periodic coefficients: criticality theory, perturbations, and positive harmonic functions, J. Funct. Anal. 129 (1995), no. 1, 80-107. 35. M. Reed and S. Simon, Methods of modern mathematical physics. IV: Analysis of operators, Academic Press, New York-London, 1978. 36. M. G. Zaidenberg, S. G. Krein, P. Kuchment, and A. A. Pankov, Banach bundles and linear operators, Uspehi Mat. Nauk 30, no. 5(185), 101-157 (Russian); English transl., Russian Math. Surveys 30 (1975), no. 5, 115-175. DEPARTMENT OF MATHEMATICS, TEXAS A&M UNIVERSITY, COLLEGE STATION, TX 77843-

3368, USA E-mail address: [email protected]


Inverse Spectral Problems for Schrodinger Operators with Energy Depending Potentials A. Laptev, R. Shterenberg, and V. Sukhanov ABSTRACT. We study an inverse problem for a class of Schrodinger operators

with energy depending potentials. In particular, we show that introduction of the discrete spectrum generically does not lead to singularities of the corresponding soliton solutions. In our last chapter we derive some new trace formulas which could be considered as generalization of a standard trace formulas for Schrodinger operators.

1. Introduction In this paper we consider the inverse problem for the operator (1.1) -0 X(x, k) + (2ku(x) + v(x))%(x, k) = k2V) (x, k), x c R. In what follows we assume that the potential functions u and v are real-valued, smooth and exponentially decay at infinity together with all their derivatives di

d3

u(x)

dxj

,

u(x)

dx3

< Cj exp(-elxl)

for j E N and some e > 0. In (1.1) k is a spectral parameter and if for some k E C there is a L2 solution of this equation, then we say that k is an eigenvalue. Such operators are sometimes called Schrodinger operators with energy depending potentials and the scattering problem for them has been considered in the papers by Jaulent [3], Jaulent and Jean [4], Kaup [5] and also by Sattinger and Szmigielski [6, 7]. In [3, 4, 6] the authors used the Gel'fand-Levitan-Marchenko approach for the inverse problem on the line given in [2]. In fact, the inverse problem has been completely solved in [6] for the regular case (when the eigenvalues are absent) by using a so-called "vanishing lemma." If the data of the inverse problem have bound state, then the situation with solvability 2000 Mathematics Subject Classification. Primary 35P15; Secondary 35L15, 47A75, 35J10. A. Laptev thanks a partial support by the SPECT ESF European program. The work of R. Shterenberg was supported in part by NSF grant DMS-0314129. V. Sukhanov and R. Shterenberg are grateful to the Royal Institute of Technology in Stockholm for its hospitality, for financial support from Gustafsson's foundation and from the Royal Swedish Academy of Science. A support by the RFFI grant 05-01-01076 is also greatly acknowledged. This is the final form of the paper. ©2007 American Mathematical Society 341

A. LAPTEV ET AL.

342

becomes significantly more complicated. The main problem is that method for constructing the solution at the first step automatically gives a number of singularities. As it has been found in [6], for one-soliton solution these singularities generically cancel in the final formulas and for a two-soliton or more solutions problem, the cancellation of singularities has been stated as a conjecture.

The main aim of this paper is to clarify the situation where there is a finite number of eigenvalues. As it has been done in [7] we will use the Riemann - Hilbert

as a method of solving inverse problem. The matrix solution of this problem has singularities in x if the data of the inverse problem contain the discrete spectrum. However we prove (see Theorem 4.1) that these singularities generically cancel at the second step of our construction. In Section 4 we consider a class of reflectionless potentials and prove that the solutions u and v are generically smooth functions. We also consider a so-called "dressing up" transform (addition of one eigenvalue to the data of the inverse problem) and also prove that such a transform generically does not lead to singularities of the potential functions u and v. In the end of the paper we derive some new trace formulas involving the potential functions u and v. If u - 0 then these formulas coincide with the corresponding trace formulas for Schrodinger operators.

2. Properties of scattering data In this section we recall some standard fact, cf. [7]. We begin with defining the Jost solution of the equation (1.1). Let us consider its two solutions satisfying the properties (2.1) f (x, k) = e'kx (1 + 0(1)), x --> +00

g(x, k) = e-'k, (1 + 0(1)), x

(2.2)

-oo.

We also introduce 00

u(s) ds ao = lim a(x). X__00 l J Since u is a real function ao 1 = ao For a real-valued k we have two pairs of linear independent solutions: (2.3)

a(x) = exp{ i

JJJ

{.f (x, k), f (x, k)},

{g(x, k), g(x, k)}.

In particular, the functions f and f could be written as linear combinations of g and g (2.4)

f (x, k) = a(k)g(x, k) + b(k)g(x, k),

f (x, k) = a(k)g(x, k) + b(k)g(x, k). Let us define the Wronskian (2.5) (2.6)

W 0] (x, k) = p'(x, k)'(x, k) - P(x, k)O'(x, k)

Proposition 2.1. The following fundamental identity holds true (2.7)

la(k) 12 - lb(k) 12 = 1.

PROOF. The Wronskian of the functions f and f is constant with respect to x. Comparing the asymptotic behavior of W [ f, 1] at +oo and -oo we complete the proof.

El

INVERSE SPECTRAL PROBLEMS

343

In follows from (1.1) that the function f satisfies the following Volterra equation (2.8)

f o sin(k(k - y)) (ku(x) + v(x))f (y, k) dy.

f (x, k) =

Therefore f can be written as a convergent series obtained by standard iterations and this immediately implies that f is an analytic function with respect to k E C+ in the upper half plane and can be continuously extended on the real line. In particular, this also implies the analyticity of the scattering coefficients a(k) in C+ and therefore a(k) in (C_. Similarly we find that the solution g is an analytic function in C+. It is well known that if u - 0 then f (x, k) - exp(ikx) as k - oo (Imk > 0). If u 0- 0 then we obtain the following.

Proposition 2.2. For Im k > 0 functions f (x, k) and g(x, k) satisfy following asymptotic relations oo

(2.9)

f (x, k) = e'kx exp i

fu(s) ds

g(x k) = e 'kx exp{ i

(1 + o(1)),

k -* oo;

f u(s)

oo. ds (1 + o(1)), k l Similarly we find asymptotic properties of the functions a(k) and b(k).

(2.10)

Proposition 2.3. The scattering coefficients a and b satisfy the asymptotic behavior as k - 00 lim a(k) = ao 1, Imk > 0, k-.cc

lim b(k) = 0,

k-.oo

Imk = 0.

From the asymptotical behavior of f at ±oo we immediately find

Proposition 2.4. Zeros of the scattering coefficient a in C+ coincide with the eigenvalues of the operator (1.1).

Remark 2.5. Due to the assumption (1.2) the scattering coefficient a(k) is also an analytical function for Imk > -e, k 0, where e > 0 is introduced in (1.2). Point k = 0 can be a pole for function a(k) or it is regular point. Therefore it can only have a finite number of zeros in C.

3. Riemann -Hilbert problem Here we describe the inverse problem by using Riemann-Hilbert approach, see also [7]. Let us assume that the scattering coefficient a(k) has N simple zeros {xj}N 1, xj E C. Denote by 1Y(x, k) ('02(X;k)) the vector defined by %P(x,

f (x,

k) =

(g('(k))'

lp(x k) = 9(x k)/a(k)

f(x,k)

fork c C+, for k c C_.

By using the analytical properties of the functions f and g and also the properties of the scattering coefficients a and b described in the previous section, we find that the vector' is meromorphic in C± and can be continuously extended to the boundary.

A. LAPTEV ET AL.

344

Let T+ (x, k) and T- (x, k) be the limits of the function W (x, k) as k approaches the real axis from upper and lower complex half plane respectively. One finds that the this vector satisfies following relation

'+(x, k) - G(k)T-(x, k),

(3.1)

G(k) -

(3.2)

r(k)

1

G(k) 1 - r(k) ) 12

'

where

r(k) _ -a(te),

(3.3)

k E R.

It follows from Proposition 2.1 that if k E R \ {0}, then I r(k) < 1. Moreover, if we denote by Q3 = (o ), then using (2.9) and (2.10) we find °1

W (x, k) = exp{ia3kx}

(3.4)

((aa()x)

(_11 (1 + o(1)),

ask - oo.

For the discrete spectrum we obtain (3.5)

reSk=xj 452(x, k) = Cij ),(x, xj),

reSk=x 01(x, k) = Cj02(X, xj), Cj O,j = 1,...,N.

Therefore meromorthic vector function T (x, k) is the solution of the Riemann -

Hilbert problem: (3.1), (3.4), (3.5). In order to exclude the direct appearance of the function a(x) from it we consider its matrix version which is slightly different. Namely, let us introduce a meromorthic in (C+ and C_ matrix {e11(x,k)

(3.6)

tt12(x,k)

521(x, k)

S22 (x,

k))

'

such that

E+ = G(k)E-,

(3.7)

= exp{iQ3kx} (1 (3.8)

10)

reSk=x e21 (x, k) = Cje11(x, xj),

_ resk=x; ell(x, k) = Cj 61 (X, xj), t

k e R,

(1 + o(1))

k -+ oo.

,

reSk=xj 62(X, k) = Cje12(x, Xj),

_

reSk=k. 512(x, k) = CjS22(x, xj), CC

CC

Cj

0,j=1,...,N.

As before we denote by E± (x, k), k E R, the limits in k of E- (x, k) from the upper and the lower complex half plane. The asymptotic formulas for the functions ejs (x, k), j, s = 1, 2, as k - oo, are given by

js(x, k) = exp{(-1)j-1ikx} 16js +

k

+ O(k-2) I.

The vector ' and the matrix E are related as follows (3.9)

q, (x, k) =

(2(x,k)01 (X

k)

)

- a(x ) (S21(x, k)) + (a(x)) 1(522(x, k)/


345

Let us now substitute the functions 01(x, k), 02(x, k) into the equation (1.1). Since they are solutions of this equation, then comparing the terms of order k°, we obtain (3 . 10)

(a(

())

+ 2i (Wll(x)) ' + 2i ((a(x)) -2 ) ' P12(x) + 2i(a(x)) 2(cp12(x))' = v(x),

(3 . 11)

( a (x))

1

i

-l

"

- 21 (c022(x)) '

(a (X))

- 2i ((a(x)) ) ' 021(x) 2

= V(X)Note that due to the symmetry of the Riemann-Hilbert problem, the equation -

2i(a(x))2(p21(x))'

(3.11) is complex conjugated to the equation (3.10). Note also that the function

m(x, k) = det 8(x, k) = e11 (x, k)e22(x, k) - 62(x, is analytic in Cf and solves the trivial Riemann-Hilbert problem (3.12)

m+(x, k) = m (x, k),

k E R,

m(x, k) -> 1,

k)

k ---> oo.

Thus m(x, k) _- 1 and therefore comparing the terms of order k-1 and in view of (3.12) we derive (3.13)

W11 (x, k) + IP22 (x, k) = 0.

Using (3.10), (3.11) and (3.13) we obtain that the function t(x) := (a(x))2 satisfies the Riccati equation (3.14)

(t(x))' + 2icp21(x)(t(x))2 + 2icp12(x) = 0,

t(x) -* 1,

as x ---> +00.

Solving this equation leads to finding the potential u

i t'(x)

U(X) = 2 t(x)

The latter formula is not very explicit and relies on the solution of the equation (3.14). Therefore we use a different approach (see [6]). Obviously f (x, 0) = f (x, 0) and thus a(x)0jj (x, 0) + (a(x)) -10±2 (x, 0) = a(x)O21(x, 0) + (a(x)) -1' 22 (x, 0).

Thus (3.15)

t(x) = (a(x))2 = exp{ -2i

l

Joo

u(s) ds }

22(x, 0) - 112(x, 0)

= Sli(x,0) - 521(x,0) Remark 3.1. We emphasize that lt(x) I = 1 and thus potential u is regular. JJ

Remark 3.2. Assume now that the inverse problem does not contain eigenvalues.rIf reflection coefficient r(k) satisfies the conditions which are usually imposed for the Schrodinger operator then the similar arguments as in the Schrodinger operator case show that the Riemann -Hilbert problem can be solved and thus potentials u and v can be uniquely reconstructed. We do not go into details. The complete proof via Gel'fand - Levitan equations can be found in [6]. The close result in terms of Riemann-Hilbert problem has been obtained in [7].

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346

4. Class of reflectionless potentials In [6] a particular example of one-soliton solution was constructed. It was pointed out that the determinant of the system which appears when solving Riemann-Hilbert problem (3.7)-(3.8) (or the corresponding Gel'fand-Levitan problem) has zeros if the eigenvalues are included. It has been proved in [6] that if there is only one soliton solution then this singularity cancels after substituting it into (3.9). In the present section we show that this cancellation is generic. Note also that, in general, one cannot expect the global solvability. The difficulties appearing here are similar to the difficulties when considering the Schrodinger operator with complex potential. We prove that generically we obtain smooth reflectionless potentials but if we choose some "wrong" coefficients, then potential functions u and v can be singular. Already the case of one-soliton potential shows that, in general, the set of "wrong" coefficients although "small" but its structure is sufficiently complicated.

Assume that r(k) = 0 and the set of simple eigenvalues {xj}, Im(xj) > 0, j = 1,. .. , N, is given. Since the potentials u and v are real functions we obtain (4.1)

Ll = eikx (1 + N hj (x) j=1

62 _ -

k - xj

wj W

eikx N j=1

k - x3

N (4.2)

621 = e-ikx

N Ti- (x)

E j=1 kxj

S22 = e-ikx

1 +

hj(x)

E j=1 k - xj

(wl, ... , wN)t. Let B = B(x) be the block

We define h :_ (hl,... , hN)t, w (2N x 2N)-matrix such that

Bll =B22 4.3

Ajs = -

B12 =B21 =J;

A,

_, xj-xs

J=

diag{C,-.le-2ixjx}.

It follows from (3.8) that (4.4)

B

(..)

= ().

By using the properties (4.3), we find that the determinant d(x) := det B(x) is real-valued. Zeros of d define points of singularity of vector-valued functions h and w. We are going to show that these singularities generically cancel. Let us define (4.5)

p=d(

W3

Note that functions d hj and d wj, j = 1, ... , N, do not have singularities. It follows from (3.15), (4.1) - (4.2) that a2 = d + p (4.6)

d+p

The function a is regular and thus the potential u is regular. We are going to prove the regularity of 01(x, k), 02 (x, k). It follows from (3.9) that it is sufficient to prove regularity of the vector-valued function s := a2h + w.


347

Theorem 4.1. Let d(xo) = 0. Assume that (1) d'(xo) # 0. (2) dimkerB(xo) = 1. (3) p(xo) 0. Then (sd)(xo) = 0 and thus s and, correspondingly, 01 and 02 are regular functions at x0. PROOF. From (4.4) and the equality d(xo) = 0 we obtain that vector ( dw- ) (xo) belongs to the kernel of B(xo). It follows from the structure of B (see (4.3)) that the vector (ah) (xo) also belongs to the kernel of B(xo) (recall that d is real-valued). Due to the condition (2) there exists a constant or such that (dw-) (xo)

- Q (dh) (x0)

Note that if ( dw- ) (x0) = 0 then theorem is proved. Otherwise, it follows from (4.7) that (dh)(xo) = a(dw)(xo) = Ja12(dh)(xo) and thus, Jul = 1.

(4.8)

Now, the direct calculations show that (4.5) - (4.8), the equality d(xo) = 0 and the condition (3) imply (ds)(xo) = 0.

Indeed,

(ds)(xo) = a2(xo)(dh)(xo) + (dw)(xo) = (p(xo)p(xo)a + 1) (dw)(xo)

_

(/(dwj)(xo)/_(du)(xo)/\ (dj)(xo)/-a->

0, +1

(dw)(xo)/7)(dw)(xo)

- 0.

Now it follows from (1) that s is a regular function at x0.

5. "Dressing up" process Now we proceed to the general case. Let u and v be regular potentials corresponding to some scattering data with eigenvalues xj, j = 1, ... , N and coefficient j, s = 1, 2, solve the Riemann-Hilbert problem of reflection r(k). Let matrix (3.7) - (3.8). We would like to consider the possibility to add one more eigenvalue. So, let xo E (C+, xo 0 xj, j = 1, . . . , N. We are going to construct new potentials

u, v by extended initial data. Let {t;js}, j, s = 1, 2, solve the Riemann-Hilbert problem (3.7) - (3.8) with additional condition

(5.)

reSk-xo S21(x, k) = Co. i (x, xo), C

tt

reSk-srp 511(x, k) = ?7o-& (x, x0),

reSk-xo 62 (x, k) = 00612 (x, x0), CC

resk=xo 12(x, k) = C062 (x, Ko Co # 0.

A. LAPTEV ET AL.

348

The solution can be written in the following form. (5111

(5.2)

tt21J

(1 +

k - xo

wz

Sttiff

(5.3)

+

h1

\ 21) (k - xo

+

h2

k

- xo

+ 22)

-xo - xo + k w2 \ k w1

+ (112) (1 + h2 + k - KO 22) (

wl k

)

-xo)

hl

k -x

are analytic in C_ we get two additional

Since X11, S12 are analytic in C+ and

equations

ii(xo)hi + 12(x0)'1 = 0,

(5.4)

z1(xo)hz +e22(xo)w2 = 0.

Similarly to (3.15) we have

uds} _ x22(0) - 12(0) J 1(0) - 21 (0)

ix2 := exp( -2i l

(5.5)

The potential u is always regular. Note that a2 = 6-2. Recall also that (5.6)

e1162 - S12e21 = 1,

and due to the fact that both potentials u and v are real (5.7)

12(k) = 61(k)

e11(k) = e22(k),

Using (5.4) we express hi and W2 via w1 and h2 correspondingly. Substituting it into

(5.2), (5.3) and taking into account (5.6), (5.7) we arrive after simple calculations at the equalities SCC11

(5.8)

12S21

Xl _ eii (Z1) + k - Xo (eii(xo) (S22)

-

+

Xz

k-xo (e,1(XO)

x

(5.9)

( S22)

Here, Xi =

=

(S22) + k

- Siz(xo) (S21)) G61

-

(S262

xO

(S11(xo)

2)

-

S12(xo) (S21) )

Substituting it into (5.1) we obtain the

X2 =

system (5.10)

B (X1) = X2

Ck2(xo)

where b is 2 x 2-matrix with the following entries

(5.11)

12

12(x0) (zz))'

Bii = B22 == 1 - Co (S12(k)e11(xo) - Si2(xo)e11(k) k - xo )

t

C

B12 = B21 = CO IS11(xo)I2 - IS12(xo)I2

xO -

xO


349

Denote d := det B. Obviously, d is real-valued. In general, d must have zeros and thus X1, X2 are singular. But the theorem below shows that generically the vector-valued function (the solutions of the equation) (5.12)

i(x, k)

(x, k) _

2(xk)

_ a(x) 51(l(xk)\ (a(x)) 1(x, k)+ ,

1

2(x, k) 522(x, k)

will be regular and so will be the potential Define (5.13)

s(x):= d(x)(a2(x)X1(x) + )6 W),

and (5.14)

p(x) := d(x)(Sli(x, 0) - 21(x, 0))

Theorem 5.1. Let d(xo) = 0. Assume that (1) d'(xo) 5 0. (2) dim ker B(xo) = 1. (3) p(xo) 0. Then and thus i is regular at point x0. PROOF. It follows from (5.8), (5.9), and (5.12) that to prove regularity of Qr at point x0 it is sufficient to show that the function a2(x)X1(x) + x2 (x) is regular at xo. Due to the condition (1) of the theorem the result will follow if we prove the equality s(xo) = 0. The same arguments as in the proof of Theorem 4.1 show that conditions (1), (2) together with the equality d(xo) = 0 imply the identity (5.15)

(d)(xo) = a(dXi)(xo)

for some constant Q such that (5.16)

Substituting (5.8), (5.9), (5.15) into (5.5) and taking into account (5.16) and condition (3) of the theorem we obtain that (5.17)

a2 (xa)

12(x,0)))(xo)

=

(d(x)(ei(x, 0) - 21 (X, 0)) (xo) From here and (5.13), (5.15) we obtain the result.

6. Trace formulas Let us rewrite the equation (1.1) - f,,",,(x, k) + (2ku(x) + v(x)) f (x, k) = k2 f (x, k)

in the terms of function h(x)

f = exp (ikx - i

. u(t) dt + J

f.

h(t) dt I .

Then (6.1)

u2 - h2 - 2ikh + 2iuh + v + iu' - h' = 0.

A. LAPTEV ET AL.

350

Due to asymptotic properties of the function f (x, k) function h(x, k) has the following asymptotic expansion as k - oo 00

h(x)

' h

= j=1

kj.

Substituting it to the equation (6.1) we have u2 + v - 2ih1 + iu' = 0,

l-1

- E hphl-P - 2ih1+1 + 2iuhi - h'', = 0,

1 > 0.

P=1

In particular, we obtain h1 =

-1 (u2+v+iu')

h2 = 2i (u3 + vu) + uu + 4 (v' + iu"), h3 = 1 (2ih2 21

(u4

2i

- h2 - h2)

+ vu2 + 4 (u2

+ v)2 - h'2) + 4 (vu)' + 12

(u3)' - 8i (2uu" + (u')2).

On the other hand, due to analytical properties of the function a(k) it follows that a(k)

=

k

u(x) dx +

- _l exp (i

In la( k l2

27ri

ds)

Let us consider asymptotic expansion of a(k) as k - oc

In a(k) = i

J

u(x) dx +

00

aj

kj.

A simple calculation shows that n

o0

Im(k)

aj = 1

l=1

27ri

f In la(s)12sj-1 ds.

Note that for Im k > 0 h(x) dx = i .

u(x) dx - In a(k).

W

fW

From (6.2) and (6.3) it follows that 00 00

hj (x) dx =

2i

> Im(k

7 1=1

1 "

+ 2-7ri

W

f

In ja(s)12sj 1 ds.

.


351

These relations are trace formulas for equation (1.1). In particular, first three trace formulas have the following form cc

f f

°°

00

(u2

2i 1

2i

+ v) dx = 2i

(us + vu) dx = i

O

2i J In la(s) 12 ds, 00 00 Im(k2) + 2I flna(s)2sds, Im(ki) +

1=1

°°

l =1

f 000 i

(u4+vu2+((u2+v)2+(u')2))

dx

n

=

3

Im(ki) +

i l=1

27ri

J- W In

ds.

References 1. P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121- 25. 2. L. D. Faddeev, The inverse problem in the quantum theory of scattering, Uspehi Mat. Nauk 14 (1959), no. 4(88), 57-119 (Russian). 3. M. Jaulent, On an inverse scattering problem with an energy dependent potential, Ann. Inst. H. Poincare Sect. A 17 (1972), 363-378. 4. M. Jaulent and C. Jean, The inverse problem for the one-dimensional Schrodinger operator with an energy dependent potential. I, Ann. Inst. H. Poincare Sect. A 25 (1976), no. 2, 105118; II, 119-137. 5. D. Kaup, A higher-order water-wave equation and the method of solving it, Progr. Theoret. Phys. 54 (1975), no. 2, 396-408. 6. D. Sattinger and J. Sznigielski, Energy dependent scattering theory, Differential Integral Equations 8 (1995), no. 5, 945-959. 7. , A Riemann-Hilbert problem for an energy dependent Schrodinger operator, Inverse Problems 12 (1996), no. 6, 1003-1025. DEPARTMENT OF MATHEMATICS, ROYAL INSTITUTE OF TECHNOLOGY, 10044 STOCKHOLM,

SWEDEN

E-mail address: laptevOmath.kth.se DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WISCONSIN, MADISON, 480 LINCOLN DRIVE,

MADISON, WI 53706, USA

E-mail address: shterenb®math.wisc.edu DEPARTMENT OF MATHEMATICAL PHYSICS, INSTITUTE OF PHYSICS, ST PETERSBURG UNIVERSITY, ULYANOV STR. 1, ST PETERSBURG PETRODWORETZ, 198904 RUSSIA

E-mail address: vvsukhanov®mai1.ru


Theory of Point Processes and Some Basic Notions in Energy Level Statistics Nariyuki Minami ABSTRACT. Although it is not explicitly stated in physics literature, energy level statistics is based upon the hypothesis that the spectrum of a quantum Hamiltonian, after suitably normalized (unfolded), and observed in thermodynamic or semiclassical limit, looks like a typical realization of a stationary point process. In this survey article, we present a purely phenomenological view of energy level statistics based on the theory of point processes.

1. Introduction Energy level statistics is a way to observe the discrete spectra of quantum Hamiltonians, in which one is interested in statistical fluctuation of energy levels (eigenvalues) looked at as a whole, rather than the location of individual levels nor the asymptotic or "average" distribution of levels. It originates in the study by nuclear physicists in early 60s ([33]), in which the excitation spectra of heavy nuclei were modeled by the spectra of random matrices. This led to an extensive study by Dyson, Mehta et al. of random matrices. Their theory was also applied to fluctuation of spectra of small metallic particles ([10]). Later, Berry and Tabor ([3]) proposed to apply energy level statistics to semi-classical Hamiltonians in order to distinguish between regular and irregular spectra, a classification of discrete spectra of quantum Hamiltonians made by Percival ([32]) according to the property of the underlying classical Hamiltonian dynamics describing the system being integrable or non-integrable. They argued that the spectrum of a quantum Hamiltonian which are obtained through quantization of a classical completely integrable Hamiltonian looks, after suitably normalized, like a typical realization of the Poisson point process. (This assertion is now called the "Berry- Tabor conjecture" [17].) They also conjectured that if a classical Hamiltonian dynamical system has chaotic behavior, then the spectrum of its quantization has fluctuation property similar to random matrices. This conjecture was later confirmed numerically ([5]), but it is yet to be clarified mathematically. Fluctuation of spectra was investigated 2000 Mathematics Subject Classification. Primary :81Q50; Secondary 82B44, 60G55, 60H25.

Key words and phrases. level statistics, stationary point process, Palm measure, PalmKhinchin formula, unfolding, level clustering, quantum chaos. This is the final form of the paper. ©2007 American Mathematical Society 353

N. MINAMI

354

also for one-dimensional random Schrodinger operators ([28, 29, 34, 35] ), and for multidimensional Anderson tight binding models ([23]). In connection to Ander-

son transition, it is believed that the spectrum of random Schrodinger operator, considered on a large finite domain, looks like Poisson point process in the energy region where Anderson localization takes place, while in the energy region where delocalization of wave functions is expected, it looks like the spectrum of a random matrix. In order to give a meaning to the vague notion of "fluctuation of levels," suppose we are given a (random) Hamiltonian H with discrete spectrum

... < En < En+1 < ... < En+m < ...

.

Among various statistics of the spectrum, one is often interested in spacings between

consecutive levels, Si = Ei+i - Ei, and wishes to know their statistical characteristics. (see e.g. §1.4 of Mehta [20].) But the following questions immediately arise: (1) The "density" of levels is in general different in different energy regions. For example, if {Ei} are eigenvalues of the Dirichlet Laplacian -AD on a bounded domain Q C 1[8d, the cumulative distribution function

N(E) :=

1{Ey<E}

i

has the asymptotic behavior N(E) - N(E) := CEd/2 (E

oo) with a positive constant C. Hence for d > 2, levels are distributed more densely for higher energy region, while for d = 1, they are distributed more sparsely. Thus in general, Si's cannot have the same statistical nature. (2) When one is observing the spectrum in the high energy region, the exact numbering of levels is practically meaningless. (3) Even if the density of levels is uniform over the entire energy axis, it is not obvious if Si's have the same probability distribution so that one can discuss the "typical" level spacing. In physics literature, the first obstacle is overcome by "unfolding" of the spectrum. In the case of Dirichlet Laplacian for example, this amounts to considering the sequence {ai}i, where Ai := N(Ei), instead of {Ei}. By doing so, we obtain a sequence which has asymptotically uniform distribution. In fact, as L -i oo, 1: 1{A 0) h\,0 h

exists and satisfies A < m. A is called the intensity of the stationary point process N.

For the proofs of limit theorems of this kind, the following lemma is repeatedly used:

Lemma 2.4. For any function g(x) defined on [0, oo) which satisfies the conditions

(1) lim.\o g(x) = g(0) = 0; (2) (sub-additivity) g(x + y) < g(x) + g(y) (x, y > 0), the limit 1

lim x g(x)o x

POINT PROCESSES AND LEVEL STATISTICS

357

For example, Proposition 2.3 is easily proved if you consider the function O(x) := TP(N(0, x] > 0),

which obviously satisfies the conditions of the lemma and O(x) < mx. Hence the desired limit A exists and we have A < m.

Proposition 2.5. If the stationary point process N, is simple, then A = M. From a given stationary point process Nu,(dx) _ Y. mn(w)6x,n(,) (dx), nEZ

we obtain a new point process NW(dx)

6,(,)(dx), nEZ

which is stationary and simple. Obviously N and N* have the same intensity A, and by Proposition 2.5, A is equal to the mean density of N*.

Proposition 2.6. Under the assumption m < oo, a stationary point process N is simple if and only if it satisfies the condition

P(N(0, h] > 2) = o(h)

(h \ 0).

As an important example of a point process, let us introduce the stationary Poisson point process:

Definition 2.7. A point process N is called the stationary Poisson point process with parameter c > 0, if it satisfies the following two conditions: (1) For any finite family A1,..., An of disjoint Borel subsets of ][8, N(A1),... , N(An) are independent random variables; (2) If A is a bounded Borel subset of ][8 with Lebesgue measure JAI, then N(A) obeys the Poisson distribution with mean cJAJ, namely P(N(A) = k) = -cJA1 (clA Dk

k = 0, 1, 2.....

It is easy to see that a stationary Poisson point process N is "stationary" in the sense of Definition 2.1. Moreover since

P(N(0, h] > 2) = 1 - e-°h - ch e-'h = 0(h2)

as h \ 0, we see that N is simple by Proposition 2.6, which has been not quite obvious from Definition 2.7. Of course, we have A = m = c.

Proposition 2.8. For k = 1, 2, ... , the limits Irk := limo P(N(0, h] = k I N(0, h] > 0)

exist, and satisfy > k>1 Irk = 1.

Intuitively, irk is the conditional probability that a point has multiplicity k, given that the origin is occupied by that point. When the stationary point process N is simple, Propositions 2.3 and 2.6 show that .7rk = 0 for k > 2.

N. MINAMI

358

Proposition 2.9. For each x > 0 and k = 0, 1 , 2, ... , the limit Qk (x) := lim P(N(O, x] < k I N(-h, 0] > 0) exists, and is right-continuous and non-decreasing as a function of x.

Now we can state the celebrated Palm-Khinchin formula:

Theorem 2.10 (the Palm - Khinchin formula). If N is a simple stationary point process (so that A = m) such that P(N(-oo, 0] = N(0, oo) = oo) = 1,

(2.2)

then one has

F.

(2.3)

k=0,1,2,...,

where we have set qk (x)

Qk (x) - Qk-1(x) = Iim IP(N(0, x] = k lN(-h, 0] > 0),

with the convention Q_I(x) - 0. Moreover fork > 1, k-1 (2.4)

Rk(x) := 1 - E qj (x) = 1 - Qk-1(x) j=0

is a probability distribution function on (0, oo) with mean

fxdxRk(x) =

(2.5)

A

Note that if we let x = 0 in (2.3), we get AJ

qk (u) du = 1,

0

P(N(O, x] > k) = A fox qk (u) du.

We omit the proof of this theorem as well as of Propositions 2.5-2.9, leaving

them to [7, 8]. We only note that the fact Rk(oo) = 1 is a consequence of the condition (2.2). In fact, we can write intuitively

Rk (x) = 1 - P(N(O, x] < k - 1 N({0}) > 0) = P(N(O, x] > k I N({0}) > 0).

If we let x - oo, we should have Rk(oo) = P(N(0, oo) > kj N({0}) > 0). According to (2.2), this probability should be equal to 1 for all k.


359

2.2. Factorial moment measures of a point process. Given a (not necessarily stationary) point process Nu,(dx) on I[8, we define its kth factorial which is a random measure on I[8k, by letting (N,, x . . x N,) (B n IIBk),

N[k](B)

(2.8)

where B runs through 13(I[8 k), the totality of k-dimensional Borel sets, and we have set

I[8k :_ {(x1,

... xk) c Rk I xi

xj (i

j)}.

When the point process N, is simple and is written in the form

N, (dx) = E d, (,) (dx),

(2.9)

nEZ

with {xn(w)} satisfying (2.1), then we have, for any intervals I17 ... , Ik, (2.10)

N[k] (Il x

x Ik) = b{ (xi1 (w), ... , xik (w))

xi, (w) are distinct and

I

xii(w) E Ij (j = 1,...,k)}. Also we define the kth factorial moment measure m[k] (dxl,... , dxk) as the expectation of N[k], namely (2.11)

m[k] (B) := IE[N[k] (B) ],

B E 13(Rk).

Suppose B E 8(R k) is of the form

B=B it x B22 x

(2.12)

where B1,

. . .

x

+ ke = k. Then it is easy to see

, Be c 13(1[8) are disjoint and k1 +

that (2.13)

N,Ik](B)

= N,,(B

1)[kl]Nu,(B2)[k21

where for integers n > 0 and j > 1, we let n[j] = (n(n - 1) ... (n - j + 1) 0

forn>1 forn=0.

As an important example, consider the case where N, is the Poisson point process with parameter c. When B E 13(I[8k) is of the form (2.12), then by (2.13) and by the definition of the Poisson point process, one has f

(2.14)

E[N[k](Bll x ... X Br)] = IE[N(B1)[kll] ...F[N(B1)[k,l] = (cIBjI)k' j=1

This can be rephrased as follows: when the one-dimensional Borel sets B1, B2, .. . Bk are such that Bi and Bj (i j) are either disjoint or identical, then k

(2.15)

EE[N[k](B1 x ... x Bk)] = ck HIBjI j=1

This can be used to prove the following

N. MINAMI

360

Proposition 2.11. If N is the stationary Poisson point process with parameter c, then for any sequence A1, A2i ... , Ak e B(R) of k Borel sets, one has k

m[kl

(A1 x ... x Ak) = E[N[k] (A1 x ... x Ak)] = ck f I Aj I. j=1

PROOF. Let {B1,. .. , B,,,} be the disjoint partition of the set Uz 1 Ai generated by {A1i ... , Ate,}, so that we can write

Ai = U Bp pEJti

with a Ji C {1, 2, ... , m}. Then we obtain

U

IE[N[k] (A1 x ... x Ak)] = IE N[k]

(BpI x ... x Bk )

)I

((Pl,---,Pk)CJI X...XJk

E

IE[N[kI (Bp1 x ...

X

Bpk)]

(P1,...epk)EJ1X...XJk k

E

ckIBp1I

- IBp2I ... IBpkI =

(p1,...,pk)EJI X... XJk

ck

HIAjI j=1

as desired.

2.3. Applications to level statistics. As we already noted, the unfolded spectrum of a quantum Hamiltonian is often implicitly assumed to be a realization {xi} of some stationary point process. In the context of quantum chaos, this identification is often something which might be called an intentional confusion. For example in [2], the authors define E(k, S), where k = 0, 1, 2, ... , and S > 0, to be "the probability to find k (unfolded) levels in a randomly chosen interval of length L." They even write it as E(k, S, x) in order to indicate that this interval is of the form (x, x+S] with x randomly chosen from (x, ci] (c > 1) according to the uniform distribution. This is actually a right way to consider the statistics of levels, because the quantal spectrum under consideration does not have the true stochasticity. (We shall return to the discussion of this situation in §4.) But immediately, they omit x "to keep the notation simple," and continue their discussion as if the spectrum is a truly random sequence of levels. In this subsection, we shall assume explicitly that the unfolded quantal spectrum under consideration is actually represented by a simple stationary point process Nu, defined on a probability space (Il, .F, F). Thus we consider E(k, S) to be the probability to find k levels in a fixed interval (x, x+S] of length S. By the stationarity, this probability does not depend on the choice of x, so we define (2.16)

E(k, S) := IE(N(0, S] = k).

Following [2] (see also [13]), we also consider F(k, S), "the probability to count k levels in an interval of length S which starts at an arbitrary quantal level xi," and P(k, S), "the probability to obtain exactly k levels in an interval of length S beginning at an arbitrary quantal level xi and ending at the quantal level xi+k+l " F(k, S) should be interpreted as the conditional probability to count k levels in the interval (0, S], given that the origin 0 is occupied by a level. Since the event


361

{N({0}) = 1} has probability zero, we are led to define this conditional probability as (2.17)

F(k, S) := lim lP(N(0, S] = k I N(-h, 0] > 0),

or F(k, S) := qk(S) in the notation of Theorem 2.10. To give a meaning to P(k, S), we interpret P(k, S) dS as the conditional probability, given that the origin is occupied by a level, say xi, to find xi+k+l between S and S + dS. This leads us to consider

P(k, S) dS = lim F(N(0, S] < k I N(-h, 0] > 0)

S + dS] < k I N(-h, 0] > 0). In view of this, we define

P(k, S) dS :_ -dQk(S).

(2.18)

When k = 0, we shall write E(S), F(S), and P(S) respectively for E(0, S), F(0, S), and P(0, S). E(S) is often called the "gap probability." P(S), when it exists, is the probability density of the spacing to the level which lies next to the one occupying the origin, because F(S) is the probability that this spacing is greater than S. We now apply the Palm-Khinchin formula, to obtain E(k, S) = P(N(0, S] < k) - IP(N(0, S] < k - 1) =A

s{qk(x) -qk-1(x)}dx

=AJ

{F(k, x) - F(k - 1, x)} dx,

for k > 1, and

E(S) = E(0, S) = A

J/

F(0, x) dx

J is

for k = 0, where A is the intensity of the point process N. Summing over k, we get k

S

j=0

or

+

F(k, S)

(2.19)

A dS

k

E(j, S),

which is formula (13) of [2]. Recall that F(k, S) is right-continuous in S, so that differentiation from the right is possible. If both sides of (2.19) are again differentiabl in S, we can write (2.20)

P(k, S) _

-

d dS

1 d2

k

E F(i, S) i=O

1 d2 A d52

k

E(j, S) i=0 j=0

k

= a Q2 E(k - i + 1)E(i, S). i=0

i

N. MINAMI

362

In particular, letting k = 0, (2.21)

P(S)

a dS2

E(S).

The following relations are also valid: 00

1: E(k, S) = 1;

(2.22)

k=0 00

E F(k, S) = 1;

(2.23)

k=0

I

(2.24)

00

P(k, S) dS = 1

;

00

F(k,S)dS= -1;

(2.25)

r

(2.26)

00

SP(k, S) dS =

kA

1

(2.22) and (2.24) are obvious from the definitions of E(k, S) and P(k, S) dS. (2.25) and (2.26) follow from (2.6) and (2.5) respectively. (2.23) can be verified in the following way. Since F(k, S) = Qk(S) - Qk_1(S) and Q_1(S) = 0, we have 00

1: F(k, S)

k

Qk (S) = SUP

k=0

k>O

Qk

(S)

On the other hand, by the definition of Qk(S) (Proposition 2.9) and IP(N(-h, 0] > 0) - Ah, we can write

Qk(S) = lim

I P(N(0, S] < k, N(-h, 0] > 0).

Now it is not difficult to show that IP(N(0, S] < k, N(-h, 0] > 0), as a function of h > 0, satisfies the condition of Lemma 2.4. (The proof of Proposition 2.9 depends on this fact.) Hence J-IE(N(0, S] < k, N(-h, 0] > 0). h>0 Ah Qk(S) = sup Interchanging supk and suph>o, we see 00

k=0

F(k, S) = sup sup 1lP(N(0, S] < k, N(-h, 0] > 0) h>O k Ah

= sup 1 IE(N(-h, 0] > 0) = 1, h>O Ah

by Proposition 2.3. Let us now look for a representation of the number variance E2(S) in terms of F(k, S) or P(k, S). E2(S) is defined as the variance of the random variable N(0, S], i.e., (2.27)

E2(S) := ]E [N(0, S]2]

- (.\S)2.


363

We can compute, using Theorem 2.10 and AS = E [N(0, S]], 00

IE[N(0,S]2] -AS=

k(k - 1)TP(N(0, S] = k) k=2

k(k -1){IP(N(0, S] > k - 1) - TP(N(0, S) > k)} k-2 00

= 2 E kP(N(0, S] > k) k=1 S 00

kqk (u) du.

= 2A 1k=1

Thus we obtain E2(S) = AS - (AS)2 + 2A or

_ AS + 2A

Formally, this can be rewritten as (2.28)

E2 (S)

j

f

/

s/0 kF(k, u) I du,

I

k=1

\

kF(k, u) - Au I du.

I

k=1

= AS+2AJs (S-u)I EP(k,u) -A I du. 0

k=0

Here (2.29)

Ec" P(k, u)

= P2 (U)

k=0 and Y2(S) := p2(u) - A,

(2.30)

which are in general singular, are often called two level correlation and two level cluster function respectively.

Example 2.12. Let N be the stationary Poisson point process with parameter c > 0. Then by the independence (see Definition 2.7), (2.31)

F(k, S) = qk(S) = P(N(0, S] = k) = E(k, S) = e-cs (cS)

(2.32)

d P(k,S) = -dS

k

.

(cS S) k

F(i S) = ce_CS i=0

and hence (2.33)

p2(u) = c;

Y2(u) = 0;

E2(S) = CS-

Example 2.13. For each w in the probability space (1, 8[0, 1), ]E), where 1 = [0, 1), B[0, 1) is the Borel o-field on [0, 1), and IF is the Lebesgue measure, define (2.34)

Nu,(dx) :=

0"

6(n+w)1a(dx)7

N. MINAMI

364

where A > 0. The resulting point process is obviously simple and stationary. It is just the random translation of the lattice A-1Z. For any bounded measurable function f (x) with compact support, one has 00

1

f (x)Nu, (dx)J =

E LJ I

fo

,n=_o

00

f (A-1(n + w)) dw = A

f (x) dx.

In particular letting f (x) = 1(o,1] (x), IE[N(0,1]] = A.

Now fix k > 0 and x > 0. Then for sufficiently small h > 0, Nu, (-h, 0] > 0 implies N, (0, x] < k when x < (k + 1)/A, and N, (0, x] > k when x > (k + 1)/A. On the other hand, P(N(-h, 0] > 0) = Ah for small h > 0. Hence Qk(x) = lim 1 P(N(0,x] < k,N(-h,0] > 0) = 1[0,(k+l)/A)(x).

(2.35)

h\,o Ah

Thus we get (2.36)

F(k, S) = qk(S) = 1[k/a,(k+l)/A)(S);

/

P(k, S) = 6I S_ k +

1).

\

By direct computation, one can verify E2(S) = {AS}(1 - {AS}),

(2.37)

where {AS} is the fractional part of AS.

2.4. Superposition of independent point processes. In several situations, we can consider the spectrum of a quantum system as a superposition of spectra of independent or very weakly interacting subsystems (see, e.g., [4,14,15], and also [23, 28]). In our language, this amounts to consider a point process of the form (2.38)

N(dx) = Nl (dx) + N2 (dx) +

+ NP(dx),

. , Ni,-. (dx) are stationary point processes which are statistically independent from each other. Let mi, m, A , A be respectively the mean density and the intensity Ni and N respectively. Also let EZ (S), E2 (S), y(i) (u)

where N1(dx), N2 (dx), .

.

Y2 (u) be the corresponding number variances and two level cluster functions, and let Ei(S), E(S), Fi(S), F(S) be defined similarly. From (2.38), we obviously have (2.39)

m=m1+m2+"'+MP

and

(2.40)

E2 (S) = Ei (S) + E2 (S) + ... + E2 (S)

from the independence. For small h > 0, we have from (2.38), the independence, and Proposition 2.3, P7

P(N(0, h] > 0) = 1 - P(N(0, h] = 0)

=1-11 P(Ni (0, h] = 0)

i=1

=1-H(1-Aih+o(h)) i=1

A)h+o(h),

=1 z=1

/


365

whence (2.41)

A = L Ai. i=1

Now suppose N1,.. , Np are all simple, so that IP(Ni (0, h] > 2) = o(h) as h \ 0 for .

i = 1,...,p. Since {N(0, h] > 2} C U {Ni(0, h] > 2} U U {Ni(0, h] > 1, Nj (0, h] > 1}, 1 1)F(Nj (0, h] > 1) i<j

= o(h) + 6(h2) = o(h),

so that by Proposition 2.6, the superposition N is also simple. In this case, if we compare (2.40) with (2.28) and (2.29), we obtain (as in [15]) AY2 (S) _

(2.42)

P

AiY('') (u)

i=1

If we apply (2.19) to p

(2.43)

p

E(S) = IP(N(0, S] = 0) _ [J P(NN (0, S] = 0) = fJ Ei (S), i=1

i=1

we get P

(2.44)

F(S) = E A.z Fi (S) [j Ei (S). i=1

j54i

(Compare [4,14].)

Remark 2.14. It is natural to consider the case where the system is decomposed into infinitely many non-interacting small subsystems, so that the spectrum of the system is a superposition of a large number of subspectra. To model this situation, suppose for each n = 1 , 2, ... , we are given mutually independent point processes Ni l (dx), ... , Npni (dx) such that pn ----> oo and that (2.45)

lim max P(Ni(n)(I) > 1) = 0

'n.-roc 1 0, where

Ann)

{1 - Qpn'j) (u)} = 0

j=1

is the intensity of

Qon,j) (u) := lim

h\O

(dx) and

(0, u] = 0 1

(-h, 0] > 0).

The condition (2.47) is exactly the same as "Case 1" which is formulated in [14].

3. Ergodic point processes; statistics along the sequence 3.1. Ergodic point processes on R. In this section, we strengthen the assumption on the point process Na,, and suppose that our probability space (S2, F, IP) is equipped with a measure preserving ergodic flow {Bt; t c RI such that NBt (A) = Nu, (t + A)

(3.1)

holds for all t c I[8 and any Borel set A E B(IR). Here t + A = It + x; x E Al is the translation of the set A. A point process satisfying the condition (3.1) is called a {Bt}-ergodic point process on the probability space (Il, F, IE; {Bt}). As before, we shall assume m := IE [N(0,1]] < oc. Example 3.1. Suppose N is a stationary Poisson point process with parameter c > 0, and let PN be its law induced on the measurable space (MP(IR), Mp (R)). Let us then forget the original probability space on which N = Nu, was given, and take (MP(ll ), MP(IR), PN) as our new probability space, the mapping MP(I8) D w H Nu, being the identity, and the transformation Otw being the spatial translation of w E MP(11 ). Then the flow {Bt} is ergodic, and the new point process satisfies (3.1) and the conditions of Definition 2.1 with the same parameter c > 0. Thus a stationary Poisson point process can always be supposed to be ergodic.

Example 3.2. Define the transformations {Bt; t E RI on the unit circle SZ = [0, 1) by Otw = It + w}. It is well known that {Bt; t E RI is an ergodic flow on the probability space (52,B[0,1), IE) of Example 2.13, and it is easy to see that the point process defined by (2.34) is a {Bt}-ergodic point process.

Let Nu, be an ergodic point process, which we consider as representing an unfolded quantal spectrum. The individual ergodic theorem then tells us that various statistics taken along the spectral sequence has non-random limits with probability one. In the following proposition, we formulate some of such limiting relations, for later comparison with level statistics in §4:

Proposition 3.3. (3.2)

(i) With probability one, we have

lim

1 Nu, (0, L] = m.

L-oo L (ii) For any n > 1, any bounded intervals I1, ... , In, and any integers k1,. .. , kn > 0, define

Fu,:={t>0;N,(t+Ij)=kj,j=1,...,n}.


Then we have with probability one lim

(3.3)

L 1F (t) dt = F(N(Ij) = L Jo

j = 1, ... , n).

1

In particular, for any k = 0, 1, ... , and c > 0, the limit L

L

L

1{ivy (t,t+cI=k} dt

exists almost surely, and is equal to P(N(0, c] = k). (iii) For any k > 1 and any bounded intervals I1, ... , I,,,, ( 3.5)

lim

f

L

N[k]

(t + Ik)) dt

((t + I1)

1

L

L

Nek (I1 x ... x Ik) dt = m[k] (I1 x ... X Ik). 1 = lim L=oo L o In particular, for any k = 0, 1, ... , and c > 0, the limit

/tk(c) _

(3.6)

1

L-oc L

f

LliM (NW(t,t+c])[k{dt

exists almost surely, and is equal to E[(N'(0,c])[k]], the kth factorial moment of the random variable Nu,(0,c].

3.2. The Palm measure. To each ergodic point process N, defined on a probability space IE; {Bt}), there corresponds a measure P on (1k, .F) called the Palm measure of the point process Na,. The Palm measure clarifies the basic structure of a given ergodic point process. Although P and P are mutually singular, it will turn out that the level spacing should be observed under P, but not under P. The basic materials of this section are taken from [31], to which the reader is referred for detailed proofs.

Theorem 3.4 (Existence and definition of the Palm measure). If Nu, P; {Bt}), then is a {Bt}-ergodic point process defined on the probability space there is a unique measure P on (Q, .F) with the following property: for any jointly measurable function f (w, s) > 0 on 52 x ) 0 jointly measurable, the defining the Palm measure is equivalent to

condition (3.7) (3.8)

Jin

P(dw)

sz

Ja

N,,(ds)g(w, s) =

J

F(dw)

f

Ja

ds g(O_sw, s).

In particular, taking g(w, s) = 1A(s), with A E 8(R), we get E[N(A)] = JAIP(1l).

Letting A = (0, 1], we obtain

Proposition 3.5. P(Q) = lE [N(0, 1]] = m.

N. MINAMI

368

Hence IP is a finite measure if m < oc, but is a probability measure only if m = 1. Proposition 3.6. The Palm measure P is concentrated on the set

Q{wEQIN,({0})>0}. To see this, take a strictly positive function u with fR u(s) ds = 1. Then for

k=0,1 ...

P(N({0}) = k) =

J

9(dw)

J

ds 1{N ({o})=k}u(s)

= if P(dw) J N,,,(ds)1{N,,, ,({o})=k}u(s)

= fI P(dw) J

N_(ds)1{N,({s})=k}u(s).

This shows ?(N({0}) = 0) = 0, and when N is simple, also IP(N({0}) > 2) = 0. For non-negative continuous functions cp(x) on R, we set LN(G) := ]E[exp(-N(co))]

(3.9)

where

f(x)Nw(dx) ,

N,,,(cp) :=

and call it the Laplace functional of the (not necessarily stationary) point process N. The probability law of a point process is uniquely determined by its Laplace functional. By definition, a sequence {N (n) } of point processes converges weakly to a point process N if their law induced on the metric space MM(R) converge weakly to that of N. It is well known that this is equivalent to the convergence of LN(G) (cp) to LN(cp) for all cp in the class Co (R), the totality of non-negative continuous functions with

a compact support. The following proposition then tells us that the conditional law IE(N E I N(-h, 0] > 0) of a simple ergodic point process N converges weakly to the probability law IP(N E )/TP(1) as h \ 0. Proposition 3.7. Let a point process N, be {Bt}-ergodic and simple, so that lP(1l) = m =A < oo. Then for any cp E Co (ll ), one has (3.10)

P(S)

jeP(Nw())F(dw) = lim h-,0

J

exp(-N,(cp))P(dw I N(-h,0] > 0).

PROOF. If we set H(w) := exp[-Nu,(co)], then H(Otw) is continuous in t. Hence letting g(w, s) = H(w)1(_h,o](s) in (3.8), we get Ah

st P(dw)H(w)Nu, (-h, 0]

Ah

fz P(dw) J h

ds H(B_Sw).

Since N, is simple, we have m = A, TP(N(-h, 0] > 0) - Ah = IE [N(-h, 0]] as h \ 0. Hence N, (-h,0] in the above equality can be replaced by 1{N,(-h,o{>o} in the limit h \ 0, and the conclusion follows.


369

Remark 3.8. If we let f (w, s) = 1(_,,,0](s)1{N,(o,x-8]=k} in (3.7), we obtain the equality

P(N(0, x] < k) =

(3.11)

J

P(N(0, s] _ k) ds.

Since the integrand on the right-hand side of (3.11) is easily seen to be right continuous, we can conclude from Palm-Khinchin formula (Theorem 2.10) that (3.12)

TP(N(0, x] = k) =.qk(x) := A

lim

P(N(0, x] = k I N(-h, 0] > 0)

holds for k = 0, 1, 2, ... , and x > 0. (Note that qk (x) is also right continuous by Proposition 2.9.) This is not a direct consequence of the weak convergence of P(N E I N(-h, 0] > 0) to IP(N E )/\. In fact, weak convergence of a sequence of point process is in general equivalent to the convergence IP(N(-)(Ij) = k;,7 = 1,2,...,p) ---> lP(N(Ij) = k;,j = 1,2,...,p) for all p > 1, all bounded intervals II, . . . , I p such that P(N(8Ij) > 0) = 0, and for all integers k 1 , . . . , kp. But P always has a mass at the end point 0 of (0, x].

Example 3.9 (Palm measure of the stationary Poisson point process). Let N, be a stationary Poisson point process with parameter c > 0 which is {Bt}ergodic. (See Example 3.1.) Then Proposition 3.7 can be applied to obtain its Palm measure in the following way. If we define for h > 0,

Nw'hO :=N,,,(-n(-h,0]); N 2,h (.) := N,,,(- \ (-h, 0]), then N1 h and N2" are independent. Hence for each cp E Co (I[8), in exp[-N,,, (co)]P(dw I N(-h, 0] > 0)

I N(-h, 0] > 0)

= Jn =

exp[-NI°h(o)]]/TP(N(-h, 0] > 0).

It is easy to see that this converges to 1 e-w(0)R c

[eXp[-N(AP)]]

as h \ 0. Hence, under the probability measure (1/c)§, N has the distribution of the stationary Poisson point process with an extra non-random point placed at the origin.

Example 3.10. For the point process treated in Examples 2.13 and 3.2, it is easy to see that TP(dw) = .X50(dw). Under (1/A)P, N is the non-random lattice AZ.

In the remaining part of this section, we assume that our ergodic point process Nu, is simple and has the representation (2.9), with x,,(w)'s satisfying (2.1). Its Palm measure is concentrated on the set

Q:={wesZIxo(w)=0}. In terms of x0(w) and xI(w), the defining condition (3.7) can be rephrased in the following way:

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370

-

Proposition 3.11. For any jointly measurable function g(w, s) > 0 on 11 x

[0

00)

P(dw)g(Bxo(,, )w, -xo(w)) =

(3.13)

ff

fxl(w)

dsg(w, s).

P(dw)

In particular, if we let g(w, s) = f (w), we get (3.14)

fn

F(dw) f (Bxo(W)w)

= f l(D(dw)xl (w) f (w), Z

and letting g(w, s) = f (O w) and noting g(Bxo(u,)w, -xo(w)) = f (w), we obtain (3.15)

in

x

IP(dw) f (w) = f P(dw) sz

ds f (O w)

fo

PROOF. The function ao(w, s) =

satisfies the relation

ao(O w, -s) = ao(w, s). If we take f (w, s) = ao(w, -s)9(w, -s)

in (3.7), we get

J

P(dw)

J

Nu, (ds) ao(Bsw, -s)g(Bsw, -s) =

J

9(dw)

J

ds ao(w, -s)g(w, -s).

But Nu,(ds) ao(9Sw, -s)9(O.sw, -s)

= f N, (ds) ao(w, s)9(Bsw, -s) = 9(O 0(L,)w, -xo(w))

and since xo(w) = 0 for w E S2 on which IP is concentrated, we have mo(w)

ds ao(w, -s)g(w, -s) = f JR

ds g(w, s).

0

Hence we obtain (3.13). As corollaries, we obtain the following formulas for the probability distributions of xo and xI ([31]):

Proposition 3.12. If we define the measure F(dv) on (0, oo) by F(dv) = IP(xl E dv), then 00

(0)

fv(dv)=1;

(i)

]P(-xo E ds, xl - xo c dv) = 1[o,v) (ds)F(dv);

(ii)

(iii)

P(xi - xo c dv) = vF(dv); IP(-xo c ds) = TP(xl E ds) = p(s) ds,

where p(s) = F((s, oo)).

In particular, the distributions of x0 and xI are absolutely continuous.


371

PROOF. Applying (3.13) to the function of the form g(w, s) = h(x1(w) xo(w), s), with h(v, s) an arbitrary non-negative Borel function on 1[8+, we obtain, after noting (u,)w, s) = g(w, s) and that xo(w) = 0 for w E 1 , f

(3.16)

E[h(xi(w) - xo(w), -xo(w))] =

j 00

ds

Jn

s),

which proves the assertion (i). When h - 1, (3.16) reduces to the assertion (0), and when h(v, s) _ 0(v), with 0 > 0 Borel, then we have from (3.16)

E[ (xi(w) - xo(w))] =

](D(dw) xl(w)o(xi(w)),

proving (ii). On the other hand if h is of the form h(v, s) = cp(s), we get

E[p(-xo(w))] = 100" dsp(s) n(x1 > s), or

P(-xo E ds) = F((s, oo)) ds. Finally, let h(v, s) = 1(u, .... ) (v - s), where u > 0 is arbitrarily fixed. Then we have g

/(0."(W)w,

-x0(w)) = 1(u,oo) (x1(w))

for w E S2, and

g(w, s) = 1(u+S,") (xl(w)) for w E S2 and s > 0. Thus (3.16) reduces to

P(x1 > u) = which completes the proof of (iii).

f

P(x1 > s) ds,

0

Now we state a theorem which describes the fundamental structure of ergodic simple point processes. For its proof, we refer the reader to [31] or [12].

Theorem 3.13. Let Nu, = >nE7Z 8 be a simple, {Bt}-ergodic point process. Then for each n c Z, the mapping O , defined by 8.,,(u,)w, maps Il onto f2. If we denote by Bn the restriction of 0-In to S2, then {8,,, 1 n E Z} is an ergodic group of measure preserving transformation on (S2, (1/.)IP). xn(w) - xn_1(w) (n E Z). Then obviously rn = T o Bi -1 holds on Let -r,, (W) S2, for all n E Z. By the theorem above, {r,,}, which is the sequence of spacings between points {xn(w)}, is stationary and ergodic under the probability measure (1/A)F.

Example 3.14. When the random variables {rn} are independent, the point process N, is called a renewal process. The stationary Poisson point process (Example 3.1) and the randomly shifted periodic lattice (Examples 2.13 and 3.2) are both renewal processes. For the case of the Poisson point process, the distribution of 'rn under (1/.X)IP is the exponential distribution )e-at dt. Remark 3.15. If NN,, is a renewal process, the distributions of Tn for n 4 1 under IF and (1/.X)IF coincide. But according to (ii) of Proposition 3.12(ii), r1 =

x1 - xo and rn (n # 1) have different distribution in general. When {xn} is Poissonian, this phenomena is called the "waiting time paradox" ([7,8]). In fact, the

N. MINAMI

372

only renewal process such that 7-1 and Tn (n 1) are identically distributed under F is the randomly shifted lattice .X-1Z discussed in Examples 2.13, 3.2 and 3.10. This is seen by solving the equation

AF(dv) = vF(dv).

3.3. The level spacing distribution. Let Nw = >

nEZ

8x,,(w) be a simple,

ergodic point process. In the previous section, we saw that the spacings Tr,,(w) := x,,,(w)-x,,,_1(w) of points are identically distributed under the probability measure (1/) )F. But under the original probability measure F, the distribution of rn can be different for different n. To see this, first note that T,,,(w) = Tn(O .w). By (3.14) of Proposition 3.11, for any t > 0, (3.17)

> t) =

P(rn(w) > t) = P(Tn o O

f(dw)xl(w)1{()>t}.

If N is not a renewal process, x1 and rn (n :,4 1) are correlated under P. Hence the right hand side of (3.17) can be different for different n. This poses a question: Suppose we are given a sample < x_1(w) < xo(w) < 0 < xl(w) < x2(w) < ...

of an ergodic point process, which we regard as an unfolded quantal spectrum. If you wish to determine the level spacing distribution experimentally, then what you naturally do would be to collect a large set of data Tk(w) = xk(W) - xk_1(w), 1 < k < n, of level spacings, and then draw a histogram of their frequency, expecting that it will approximately represent some smooth curve when the data size n is large enough and the histogram is fine enough. Mathematically speaking, this amounts to considering the empirical distribution FW (dt) :=

(3.18)

n

E 8Tk (W) (dt), k=1

and expecting this to converge, as n -> oc, to some fixed distribution F(dt) almost surely. But if the random variables T,,, are observed under the probability measure F, they are not identically distributed. Hence mixing these to form empirical distributions seems meaningless. On the other hand, if the data {Tn} are observed

under (1/\)P, they form a stationary ergodic sequence. In this case, theergodic theorem works well, and FW (dt) converges to (1/.X)P(T1 E dt) as n -> oo, F-almost surely. But P and P are mutually singular. Which of them should be considered as the real law governing {rn}? Fortunately, this puzzle is easily solved:

Proposition 3.16. The empirical distribution Fw (dt) defined by (3.18) converges weakly, as n - oo, to (1/))P(T1 E dt) for F-almost all w E Q. PROOF. For each t > 0, define Jim

At = { w E Sl fill

I

n' k=1

t)}.


373

As was already pointed out, Tk = Tl oOi-1 holds on (Il, (1/.\)1[D), Bl being an ergodic automorphism. Hence by the individual ergodic theorem, lim

1

n-oo n

1{-k(W)>t} = lim

1 E 1{Tl>t} 0 9i 1(w) = 11[D> t) (Ti

n-oo n

k=1

A

k=1

holds for (1/)t)IID-almost all w E S2, namely

AP(S2nAt)=1. It remains to show P(At) = 1. But since w c A if and only if B,,ow E A, we have from Proposition 3.11, f f IA(A) =

J

F(dw)1A(w) =

J

= J P(dw)xl(W)1A(w)

lP(dw)1A(B,ow)

=

J

P(dw)xl(w)

1,

the last equality following from Proposition 3.12(0).

Remark 3.17. By the Palm-Khinchin formula, we have

P(N(0, s] = 0) ds =J "0 P(xl > t) dt.

F(N(0, x] = 0) = FX

Thus if we put 7ro(x) := IP(N(0, x] = 0), and p(t) := IED(x1 > t), we get oo

iro(x) = f p(t) dt = 1 -

(3.19)

x

J

p(t) dt,

or

P(x) _ -x7ro(x)

(3.20)

The consideration in the proof of Proposition 3.16 can be easily generalized to yield the following assertion:

Proposition 3.18. For any non-negative measurable function one has (3.21)

lim 1 f L-oo L

f

on (9, Y, I(D),

N (dt) f (Otw) = f (dw) f (w) o, L]

S2

for F-almost all w c Q. PROOF. If we let 'n.

lim

n-oo n > f (Bxkw) =: 1 k=1

IED(dw)f (w) },

then by the ergodic theorem applied to (S2, IP, B1), we have lP(SZ n A) = A. But since w E A is equivalent to B,,ow E A, the argument in Proposition 3.16 gives IA(A) = 1.

On the other hand, we have lim

1

L-oo L

Nu, (0, L] _ A

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374

F-almost surely. Hence for P-almost all w E SZ, we have N, (0, L] -> oc and

i

L

1

f N (dx)f (Otw) =L-oo i LE1(o,L](xj(w))f(O jw) o,L]

j

lim

NW (0, L]

L-.oo A

.

L

f (0" w)

lim

n-oo 1 n

j=1

1 f P(dw)f (w) = fn P(dw)f

(w).

El

Let us apply this proposition to the limiting k-level correlation, for later comparison with level statistics in §§4 and 6.

Proposition 3.19. For realizations {xn(w)} of a simple ergodic point process, define the k-level correlation (3.22)

RL (I2, ... , Ik)

=

1 #{ (xil (w), ... , xik (w))

I

xi1 (w) < L, il, ... , ik are distinct and xij (w) - xij (w) E Ij, 7 = 2, ... , k}

where L > 0 and 12i ... , Ik are any bounded intervals. Then

i m RL(I2, ... , Ik) = j(dw)N_h1(Ix ... x Ik) _: Rk(I2, ... , Ik)

(3.23)

z

holds for F-almost all w E Il, where we have set Ij* := Ij \ {0}. PROOF. Noting (2.10), we can write LRL (I2, ... , Ik)

_

=

1(o,L] (xi (w))

f

µµ

(xz2 (w), ... , xik (w))

I

i2, ... , ik are distinct,

xij (w) :A xi, (w), xij (w) E I j + xij (w), j = 2, ... , k}

((I2 + s) \ {s}) x ... X ((Ik + s) \ {s})

= JNw(ds)1(o,LI(s)Nh1(I2 x ...

X I%),

from whi ch the conclusion follows by Proposition 3.18.

Remark 3.20. In case I2 = . = Ik = (0, c], let us write Rk (12, ... , Ik) _ Rk(c). By the Palm-Khinchin formula, we can compute E[(N(0, L] )[k]] =

1 (k

n>k

I P(N(0, c] > n)

1)

n-1

_ n>k

(k - 1

(k - 1)! Jo

10c P(N(0, s] = n - 1) ds

if

2

P(dw)(N.(0, s])!k-11,


375

namely (3.24)

Pk(c)=k f Rk(s)ds. 0

See [6] for related formulas.

Example 3.21. If Nu, is the stationary Poisson point process with parameter A > 0, then by Example 3.9 and Proposition 2.11, (3.25)

Rk (I2.... , Ik) _ A

f

k P(dw)NIk-ll (I2

x ... x Ik) = Ak [J IIj I.

si

In particular, Rk(c) _

j=2

Akck-1

Remark 3.22. If {xj (w)} is a renewal process, then R2 (c) = A2c implies that N is Poissonian. Indeed, if we set

F(c) _

AP-(N(0,

c] > 0)

P _ (xn+1 - x,, < c),

which does not depend on n, then 00 1

AR2(c) =

1

A

f ]P(dw)N,,,(0, c] =

00 _ n

A

E

1

A

n=1 00

1

xj) < cJ =

n =1

j =0

E(Fn*)(C)

n =1

where Fn* is the n-fold convolution of F. Hence if Laplace transform of dF(c), then from R2(c) = A2c, we get

f 0°

is the

00

E n=1

1 1

G(L

)

f e-C°dR2(c)

namely L A/(A + 6). Hence dF(c) = Ae-A°dc. But the Poisson point process is the only renewal process for which xn+1 - xn obeys the exponential distribution.

4. Statistics for asymptotically uniformly distributed sequences Suppose we are given a finite or infinite sequence

0 < xl(L) < x2(L) < . . . < xn(L) < depending on a parameter L, which we shall let tend to infinity. We think of this sequence {xj(L)}j as the unfolded spectrum of a quantum Hamiltonian depending on a parameter, or more directly a realization of some fictitious simple ergodic point process with mean density (or intensity) A, if it is asymptotically uniformly distributed in the following sense:

Condition 1. There is a positive constant A < oo and A < oo such that for any 0 < a < A, (4.1)

{ j > 11 xj (L) < aL} - AaL (L -> oo).

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376

In particular, when the sequence {xj(L)}j does not depend on L and satisfies

#{j > 11 xj < L} - AL (L - oo),

(4.2)

then it satisfies Condition 1.

For each c > 0 and for each interval I = (a, b], we consider the following quantity (statistics): (4.3)

#{j > 11 xj(L) E I};

NL(I)

L

(4.4)

'7rk(L; I) := L

f

(k > 0);

1{NL(t+r)-k} dt

0

(4.5)

µk(L;I) = L

L

(k> 1);

NL(t+I)[k]dt=Ej[k]7rj(L;I)

j>k

(4.6) (4.7)

L #{ j > 11 xj (L) < L, xj+1(L) - xj (L) > c}

p(L; c)

Rk(L; c) := L#{ (xi, (L), ... , xik (L)) xi1 (L)

I

(c > 0);

il, ... , ik are distinct,

< L, xij (L) - xi1 (L) E (0, c] }

(c > 0, k > 2).

As an immediate consequence of Condition 1, we obtain the following

Proposition 4.1. For any c > 0, 1

MI(c) := limo L L-o

(4.8)

L jNL(t,t

+ c]dt = lim pi(L; (0,c]) _ L-oo

PROOF. By the definition,

f

(4.9) M, (L; (0, c]) = 1 L

0

L

L

1 1(t,t+cl (xj(L)) dt =

f 01[Xj(L)-c,xj(L))(t) dt.

L

But when L > c > 0, one has L 1

J0

[X' (

L

)- `'x' ( L ))

(t) dt =

C'

11c<xj(L) L+c,

while pL

J

1[xj(L)_c,x;(L))(t)dt < c,

if xj(L) < c or L < xj(L) < L+c.

0

Hence for each fixed c > 0, it is obvious from Condition 1 that pi (L; (0, c]) - cL#{j > 11 xj(L) < L}

as L --+ oo,

so that pi (c) = lim L-oopi (L; (0, c]) = Ac, which was to be proved. Except for this trivial case, Condition 1 does not imply that any of the statistics (4.4) - (4.7) converges to some limit as L -> oc. But when some of these convergence- occur, we can prove equalities between them, which may be called deterministic analogues of the Palm-Khinchin formula, as the following two propositions show. (Compare Remarks 3.17 and 3.20.)


377

Proposition 4.2. Let k > 2 and assume that

> (c - xjk (L)) = o(L) (L - oo)

(4.10)

(L) 0.

(i) If the limit (4.11)

µk (c) := L-oo liM /Lk(L; (0, c])

exists for each c > 0 and is differentiable with respect to c, then lim Rk(L; c)

Rk(c)

(4.12)

L-.oo

also exists and one has (4.13)

Rk(c)

dcµk(c)

k

(ii) If on the other hand, Rk (c) exists for each c > 0, then Ak (c) also exists and µk(c) = k

(4.14)

J0

c

Rk(c')dc'.

Remark 4.3. The condition (4.10) is obviously satisfied when the sequence {xj (L) }j does not depend on L. PROOF OF PROPSITION 4.2. Since (NL(t, t + c])[k]

_

1(t,t+c] (xj,(L)) ... l(t,t+c] (xik (L)) yjl (L) t}. Then we have v(v-1(t)) = t and the equivalence (5.7)

v(E) < t If we let ej (h)

E < v-1(t).

v (Ej (h)), then for each t > 0,

{j > 1;ej(h) < t} = {j > 1; Ej(h) < v-1 (t)}

= Nh (v-1(t)) ' v (v-1(t)) h-a =

th-'

as h \ 0 by Condition 2. Hence if we let L = h-', and xj (L) = Lei (L-1/0), then we have for any a > 0,

#{ j > 1; xj (L) < aL} = 01i > 1; ej (L-1"a) < a} " = aL as L -> oo, so that the parametrized sequence {xj(L)}j satisfies Condition 1 with a(L-1"a)-'

=1. 5.3. The unfolding of the second kind. The unfolding formulated above is what is usually practiced in quantum chaos (see, e.g., [14]), and looks like a quite natural idea in that it leads one to directly observe energy levels, although under a non-linear transformation ej = v(Ej). But with regard to this, we notice a seemingly strange remark made by Berry and Tabor in their celebrated paper [3]. They argue that the energy E itself "is an unsuitable parameter to describe the levels, for two reasons. The first is that the mean level density n(E) depends on energy.... The second reason is that the statistics of the levels might themselves


385

depend on the region of energy being studied." As a solution to this problem, Berry and Tabor propose to observe the quantum system at a fixed energy value E, and to "quantize the Planck's constant h." They apply this procedure only to regular spectra discussed in Example 5.2, but in a general setting, it would be formulated in the following way: In addition to Condition 2, let us assume that Ej (h) is a non-decreasing function of h > 0 for each j. Let us fix E such that E0 < E < El, where

Eo := inf{E; v(E) > 0} > -oo

(5.8)

and

El := sup{E; v(E) < v(oo)} < oo.

(5.9)

Now define

hj (E) := inf {h > 0; Ej (h) > E}

(5.10)

for each j > 1, where we let hj (E) = 0 if Ej (h) > E for all h > 0, and hj (E) = +00 if Ej (h) < E for all h > 0. Actually, the last two possibilities are ruled out by the following argument: Since v(E) > 0, Condition 2 implies in particular that

lim { j > 1; Ej (h) < E} = oo.

(5.11)

h\O

If hj (E) = 0 for some i, then Ej (h) > E for all j > i and h > 0. Hence

o{j>1;Ej(h)<E} 0 must hold for all j > 1. On the other hand,

since E < EI, there is an E' > E such that v(E) < v(E') < v(oo). Again by Condition 2, we have (5.12)

lim { j > 1; E < Ej (h) < E'} = oo.

This shows that for only finitely many j's, one can have sup Ej (h) < E. h>0

Thus there is a jo > 1 such that hj (E) < oo for all j > jo. We now define (5.13)

xj = xj(E) := v(E)hj(E)-`Y

for j > jo. In the sequel we assume jo = 1. By the definition, we have the following equivalence and implication for each L > 0:

xj < L

hj (E) > (v(E)/L)I"c = Ej ((v(E)/L)'10') < E hj(E) > (v(E)/L)'"" #=:> xj < L.

This shows

{j > 1 ; xj < L} < { j > 1; Ej ((v(E)/L)I1`) < E} < {j > 1; xj < L}. But by Condition 2, we have

i

1/a -a 1

((v(E)1L)I1a)<EJ=L ooLv(E)v(L))

so that lim sup 1 { j > 1; xj < L} < 1 < lim inf {j > 1; xj < L}, L-oo L-.oc L

-

}

=1'

N. MINAMI

386

namely

b{j>1;xj 0. Then we put (5.16)

hj (E) := inf{h; h2ej = E} = (E/ej )1/2

and cdIDled/2 xj (E) v(E)hj (E) d = cdIDIEd/2 (E/e;)-d/2 = Thus both kinds of unfoldings yield the same sequence. 5.4.2. A one-dimensional Schrodinger operator with S-potentials in the semiclassical limit. Consider the one-dimensional Schrodinger operator 2 +V n (5.17)

(5.18)

E 6(x - xs), 0 < x < 1

Hv (h) = -h2 d 22

s=1

under Dirichlet boundary conditions at x = 0, 1. Here v > 0, and

0=x0 <x1 0 (5.19)

Nh(E) :_ #{j > 1; Ej (h) < E} - 1 -\I-Eh-1, Ir

h \, 0.

Thus Condition 2 is satisfied with a = 1 and v(E) = v/E-, which is continuous. Moreover, we can verify that each energy level Ej(h) is non-decreasing in h > 0 by the min-max principle. Hence both kinds of unfoldings are applicable to our spectrum {Ej(h)}: For each L > 0, the unfolded levels of the first kind are given by

(5.20)

xj(L) := Lv(Ej(L-1)) _

L v-

Ej(L-1),

while for each fixed E > 0, the unfolded levels of the second kind are given by (5.21)

xj (E) := v(E)hj (E) 1 = VIE hj (E)

where hj(E) is defined by the equation Ej(h) = E.


387

It was shown in [22] that E is an energy level of H (h), if and only if it is a zero of the function

/ - n-p

n

(5.22)

F' (E)

P++(1

11 sin (xjs - xjs-1) Vh 1

1 /E)

1<ji 0 is an unfolded level of the FLwhile first kind if and only if it is a zero of the function on the other hand ( > 0 is an unfolded level of the second kind if and only if it is a zero of the function 'YE(() := rr 1 (E). If we define n+1

f (z) := 11 sin{7r(xs - xs-1)z},

(5.23)

s=1

then we can write and -YE(S)=f(S)+SE(C),

(5.24)

is small for large L and for = CA(L), while 6E(O is small for large values of . Hence by the argument in [22] using Rouche's theorem, the statistics for the zeros of the functions GL () and yE (() are both reduced to the statistics for the zeros of the function f (z), which are where

; m > 1, s = 0,1,-, n m l xs - xs_1

(5.25)

{

. JJJ

Thus two kinds of unfoldings applied to the spectrum {Ej (h)} lead to the same level statistics. To proceed, consider as in §4 the counting function n

o0

N(t, t + c]

(5.26)

1(t,t+c] s=0 m=1

m

xs - xs-1 )

and the probability 1

7rk(L; (0, c]) := L

(5.27)

fL

J0

1{N(t,t+c]=k} dt

for k = 0, 1, ... , c > 0, and L > 0. Then we have

MM 0 is small, then Mc = 0 and n+1

7ro(c) = 11 (1 - c(xs - xs-1)).

(5.30)

S=1

We can now apply Proposition 4.4, to obtain (5.31)

n+1

d

P(c)

dc7ro (c)

_

xs - XS-1

(> 1 - c(xs - X'-1)) 11(1 - c(xs - xs-1)) s=1

S=1

For large n and for typical configuration {xs} of the 6-potentials, Irk (c), k = 0, 1, ... , and p(c) are different from, but very close to being Poissonian. In fact, if X1, X2....

are independent random variables which are uniformly distributed on (0, 1), and 1 is a rearrangement of X1, ... , Xn according to their if 0 < xln) < < magnitude, then one can show ([22]) that with probability one, k

(5.32)

li

7rk(c;xln),...,x(n)) =e

k=0,1,...,

kk,

and (5.33)

lim p(c; x1n), ... ,

e

5.4.3. Regular spectra. Under the assumptions on the classical Hamiltonian of Example 5.2, v(E) is a continuous function of E, and En(h) = H(hn) is increasing in It for each n. Hence both kinds of unfolding are applicable to our regular spectrum

{En(h); n E 7G+}. To apply the unfolding of the second kind, we first note that

for each x E R+ \ {0} and E > 0, the equation H(hx) = E can be solved for If we denote this solution by hE (x), then the unfolded energy level is given by xn(E) = UE(n), where we have defined UE(x) = v(E)hE(x)-d, x E R+. (5.34) Since /3hE (,3x) = hE (x) for any,3 > 0, we see that UE(x) is a homogeneous function of degree d, namely It.

(5.35)

UE(,3X) = /3dUE(x),

for any /3 > 0. (See [3].) To perform the level statistics of §4 for this sequence {UE(n)}n, which is asymptotically uniformly distributed, we need to consider for I = (a, b] (5.36)

NL(t+I) = E 1(t+a,t+b) (UE(n)) nEZ+

If we define (5.37)

B(x)= IXI,

xEll

\{0}

and

(5.38) III(t) := {x E R+ I (t+a)11 dUE(O(x))-11d < Ixl < (t+b)1/dUE(O(x))-1/d}, then (5.39)

NL(t+I) _ E 1rl,(t)(n), nEZ+


389

which is the number of lattice points contained in the d-dimensional domain III (t).

It is then easy to see III (t) I= b- a= III, so that as t gets large, the domain III (t) expands in the space ][8+, but at the same time gets thinner and thinner to keep its volume constant. Hence provided the boundary of III (t) is not too regular, e.g., flat or spherical, then each point in 7G+ would randomly belong to III (t) if t were chosen at random from a long interval [0, L], so that the total number of lattice points in III (t), namely NL (t + I), would obey Poisson distribution with mean III. Namely we expect that (5.40)

Irk(L;I)=e-IIII

7rk(I) := lim

II-,

k=0,1,2,...

holds. This is conceivable if we recall how Poisson's law of small numbers is proved

in elementary probability theory. Actually, it is a difficult problem to rigorously justify this intuition. In fact, no example of classically integrable Hamiltonian is known for which (5.40) is verified. Let us now apply the unfolding of the first kind to our regular spectrum. This gives H)(L-1/dn), L> 0, n E 7L+, xn(L) := L L. v(En(L-lid)) = L . (v o as unfolded levels. To proceed further, we assume that our Hamiltonian H(x) satisfies the following

Condition 3. For any x, z c R+ and /3 > 0, H(/3z) < H(,3x) implies H(z) < H(x). Under this condition, the function G := v o H is homogeneous with degree d. In fact for any 0 > 0, G(,3x)

=

... f

= /3d

J

...

1{H(I) 0, we have v(E) = CHEd/7 with CH =

J... I Rd1{H(x) 0, or at least that {xy (L); j > 1} converges in law to the Poisson point process on (0, oo) with mean density 1. But for the moment, this is only a conjecture.

Remark 5.4. McKean actually proved that as L -> oo, the distribution of xi (L) = LN(El (L)) converges weakly to the exponential distribution with mean 1. See [11] for related topics.

Let 0E' (t) be the solution of the equation Eli satisfying the initial condition 0(0) = 0, '(0) = 1. Then by the Sturm's oscillation theorem, E _ Ej' (L) if and only if t = L is the jth zero of the function zL (t). Moreover Ej- (L) is increasing in h := L-1, and hence we can apply the unfolding of the second kind. For this purpose, fix an energy value E E 1[8 and let L'? (E) be defined by the

equation Ej(L) = E. Then LY(E) has the following meaning. If the stochastic processes {rE(t)} and {OE(t)} are introduced through the Prufer's transformation (5.46) OE (t) = rE(t) sinOE(t); WE (t) = rE(t) COSOE(t), then {OE(t)}t is a diffusion process on the circle 1[8/7rZ ([9, 19]) with OE(0) = 0 (mod 7r), and Lj (E) (j > 1) are the jth hitting times to the origin 0 of this process. Namely letting Lo(E) := 0, we have LS(E) := inf{t > Lj_1(E);OE(t) = 0 (mod 7r)} (5.47) for j > 1. By the strong Markov property of {OE(t)}t, TA (E) := LS(E) - Lj_1(E), j = 1, 2, ... , are independent and identically distributed random variables, and L,Y (E) = =1Tk(E). Moreover one has IE[Ti(E)] = N(E)-1 ([9]). Hence the unfolded levels of the second kind, given by xj = xj (E) := N(E)Lj" (E), j = 1, 2, .. . (5.48) constitute a renewal process on (0, oo) with mean density 1, so that the corresponding level statistics is rather trivial. For example, one has (5.49)

i

p(L; c)

i

L

b{j > 1; xj < L, xj+1 - xj > c} = P

\E[Ti (E)] > c with probability one. In this case, the result of unfolding of the second kind does depend on the fixed energy value E, the distribution of Ti(E)/]E[Ti(E)] being de-

pendent on E. As we let E -> -co, this distribution tends to the exponential distribution 1(o,00) (x)e-"dx, as was proved by McKean ([19]). Accordingly the renewal process {xj (E) }j converges weakly to the Poisson point process on (0, oo) with mean density 1.

6. Clustering and repulsion of levels As we noted in §5.4.3, no concrete example of regular spectrum is known for which Berry-Tabor conjecture is literally true in the sense that (6.1)

p(c) := L-oo lim p(L; c) = e

c>0

N. MINAMI

392

holds for the unfolded spectrum. The assertion (6.1) on the level spacing distribution function p(c) is what Berry and Tabor called the "level clustering." We have also seen in §5.4.2 that there is even an example of quantum Hamiltonian for which level statistics is carried out in full details, yet yielding a level spacing distribution function p(c) which is close to, but different from e-c. This situation suggests us that we should give some wider, or looser definition to the term "level clustering" than the strict assertion like (6.1). To formulate this problem, suppose we are given a sequence {xj(L)}j depending on a parameter L > 0 which satisfies Condition 1 with A = 1, and which we regard as an unfolded quantal spectrum. For this sequence {x j (L) } j, let us consider statistics defined by (4.4)-(4.7) with I = (0, c]. Except that we always have pi (c) := limL . pi (L; (0, c]) = c, none of these is guaranteed to converge to some limit as L ---> oo. Let us then define ?rk(c) := lim of -7rk(L; (0, c]),

(6.2)

L-oo pk(c) := 1im of pk(L; (0, c]), L-oo p(c) := lim of p(L; c), L-oo

(6.3) (6.4)

lrk(c) := lim sup7rk(L; (0, c]); L-.oo µk (c) := lim sup Mk (L; (0, c]); L-.oo

p(c) := lim sup p(L; c) L-.oo

and

(6.5)

Rk (c)

lim of Rk (L; c), L-oo

Rk (c) := lim sup Rk (L; c) L-.oo

When the corresponding upper and lower limits coincide, we shall denote their common value by Irk (c), pk (c), p(c), or Rk(c) as before. Now one of the characteristic of the exponential distribution is that its density is strictly positive at the origin c = 0. Hence if (6.1) were true, we would have relatively many closely adjacent pairs of levels. This situation would be well described in the following way.

Definition 6.1. We shall say that the unfolded spectrum {xj(L)}j exhibits level clustering [resp. repulsion] in the rwide sense if the condition

li . of 1 - p(c) > 0

(6.6)

I resp. lim sup 1 - P(c) = 0 L

1

holds.

We can give sufficient conditions for repulsion or clustering in the wide sense in terms of pk (L; (0, c]), k = 2, 3, as follows.

Proposition 6.2.

(i) If µ2(c) = o(c2) as c \ 0, then the spectrum ex-

hibits level repulsion in the wide sense. (ii) If for some a > 0, p2(c) > ac2 for all c > 0 and if µ3(c) = o(c2) as c \, 0, then the spectrum exhibits level clustering in the wide sense.

PROOF. Applying [6, Lemma 2.2] to the probability distribution lrk(L; (0, c]) (k > 0) on non-negative integers, we obtain the following equality which is valid for n > 1 and k > 0: (6.7)

P,, (L; c) := 1: rj (L; (0, c]) = Sn,k(c) + (-1)k+1T, j>n


393

where n+k

Sn,k(c) = 1(-1

(6.8)

(n - 1) j! µj (L; (O, c]),

j=n

and T = Tn,k > 0 is given by (6.9)

Tn,k =

j-1)(j-n-1)Pi(L;c), n-1 k j>n+k+1 (

though this explicit expression for T will not be used in the following argument. When n = 1, formula (6.7) reduces to k+1

1 - ro(L; (0, c]) =

(6.10)

1:(-1)j-1

iµj (L; (0, c]) + (-1)k+1T,

j=1 and hence

(6.11) -7ro (L; (0, c]) 0, define N by XN(L) < L < xN+1(L), namely by

1;x(L) < L},

N := n(L) and put xo(L) = 0. Then we have (6.13)

(6.14)

Liro (L; (0, c] )

N-1

E (xj+1(L)-xj(L) - c)+ + {(XN+1(L)-xN(L)-c)+A(L - XN(L)) }

j=0 N-1

N-1

_ E (xj+l (L) - xj (L) - c) + E (c - (x j+1(L) - xj (L))) + j=0

j=o

+ {(xN+1(L) - XN(L) - c)+ A (L - XN(L))}

xN(L) - cN + c #{j > 0; xj+1(L) < L, xj+1(L) - xj(L) c} + (L - xN(L)) = L - c #{j ? 0; xj+1(L) < L, xj+1(L) - xj (L) > c}. Suppose that the condition of (ii) is satisfied. Then dividing both sides of (6.14) by L and letting L -* oo, we obtain from (6.12) (6.15)

1 - cp(c) > z:0 (c) > 1 - µ1(c) +12µ2(c)

1

- 3µ3(c) > 1 - c+ 2c2 +o(c2),

or ace c(1 - p(c)) > 2 + o(c2).

Hence we get (6.16)

lim inf c\,o

1 - p(c) > a > 0 c

2

N. MINAMI

394

Next suppose that the condition of (i) is satisfied. For each 0 < S < 1, we have (6.17)

Liro(L; (0, c])

> xN (L) - cN + (L - XN(L) - c) N-1 + E 1(0,5C] (xj+1 (L) - xj (L)) {c - (xj+1 (L) - xj (L)) } j=0

>(L-c)-cN + (1 - S)c[N - 0{j > 0; xj+1(L) < L, xj+1(L) - xj(L) > Sc}]. Again dividing by L and noting N/L - 1 as L -> oo, we obtain from (6.11) (6.18)

1 - c + (1 - 6)c[1 - p(Sc)] < ro(c) < 1 - pi (c) + 2µ2(c) = 1 - c+ o(c2),

and hence lira sup

(6.19)

1 - p(c) c

= lim sup c\o

1 - p(Sc) Sc

= 0,

as desired.

Corollary 6.3.

(i) If R2(c) = o(c) as c \ 0, then the spectrum exhibits

level repulsion in the wide sense.

(ii) If for some a > 0, R2(c) > ac for all c > 0 and if R3(c) = o(c) as c \ 0, then the spectrum exhibits level clustering in the wide sense.

PROOF. By the inequality (4.16) in the proof of Proposition 4.2, we see that (6.20)

kc

Rk

-) I < µk(c) < µk (c) < kc1

f=0

holds for all integer e > 1. Letting £ -4 oo, we get (6.21)

k

Rk (c(1

-)

I

t=O

f

J0

r

Rk(c') do < µk(c) < µk (c) < k f Rk(c') dc'. 0

It is now obvious that the conditions of the corollary imply those of Proposition 6.2.

If we impose conditions on Rk(c), we get more precise estimates on p(c) and p(c).

Proposition 6.4. If the limits Rj (c) exist for j = 2, ... , k, k > 2, then we have k

(6.22)

P(c) < 1 + E(-1)i 1 Rj+l (c) j j=1

when k is even, and k

(6.23)

p(c) > 1+E(-1)j j=1

when k is odd.


1(o,L](xj(L)) and define the probability distribution

PROOF. Let n(L) {Pk(L;c)}k>o by (6.24)

Pk(L;c)

395

n(L) E 1(0,L](xj(L))1{NL(x1(L),xj(L)+c]=k}

Then (6.25)

po (L; c) = n(L)P(L; c).

Note that n(L) - L, and hence the upper and lower limits of po(L; c) as L -> 00 coincide with p(c) and p(c) respectively. If we let (6.26)

n(L) Rk (L; c)

rk (L; c)

for k > 2, then it is not difficult to see (6.27)

rk(L; c) _

1

n(L ) E 1(o,L] (xj (L)) (NL (xj (L), xj (L) + c]) [k-1] j

E

j[k 1]pi(L; c).

i>k-1

Namely rk(L; c) is the (k - 1)th factorial moment of the probability distribution {pi (L; c) }i>0. Thus again by [6, Lemma 2.2], we can write (6.28)

Q. (L; c)

E pi (L; c) = Qn,k(c) + (-1)k+1 T, j>n

where T > 0 and n+k (6.29)

.)j1)j[rj+1 (L; c).

Qn,k(c) = 1(-1) j=n

When n = 1, this reduces to k

(6.30)

1 - po(L; c) = k(-1) j=1

1jrj+1(L; c) + (-1)k r, 7

which is valid for k > 1. The assertion of the proposition immediately follows from this equality. Example 6.5. In [41], VanderKam considered the level statistics for the Laplacian on a flat d-dimensional torus, which is the quantization of the uniform motion on that torus. The corresponding Hamiltonian, expressed in the action variables, is a positive definite quadratic form (6.31)

P(x1, ... , xd) = txAx,

where A is a d x d real symmetric, positive definite matrix, and the eigenvalues of the Laplacian are values taken by P on the lattice hZd. To eliminate the obvious degeneracy, x and -x are identified. Let us denote this equivalence relation by-.

Then the spectrum {P(hn); n E Z'/ -} is unfoldable with a = d and v(E) = cpEd12, where cp is the volume of the domain {x E Rd; P(x) < 1}/'. Since P(x)

N. MINAMI

396

is a homogeneous function of degree 2, both kinds of unfoldings discussed in §5 give the same unfolded sequence {l;n; n E 7Zd}, where (6.32)

t cPP(11)d/2 S.:=

Then VanderKam proves that for almost all realization of the coefficient matrix A

of P(x), one has Rk(c) = ck-1 for k = 2, ... , [d/2], for the correlation functions of the unfolded spectrum (6.32). Hence if d > 6, Corollary 6.3 tells us that {I;n} exhibits level clustering in the wide sense. Moreover, by Proposition 6.2, we can assert that when d is large and c > 0 is small, p(c) and p(c) are very close to each other, and are approximated by e-c. Acknowledgments. The author is grateful to Professor V. Jaksic for giving him the opportunity to make this contribution, and to Dr. E. Giere for careful reading of the manuscript and useful comments.

References 1.

V. I. Arnol'd, Mathematical methods of classical mechanics, Grad. Texts in Math., vol. 60, Springer, New York, 1978.

R. Aurich, A. Backer, and F. Steiner, Mode fluctuations as fingerprints of chaotic and nonchaotic systems, Internat. J. Modern Phys. B 11 (1997), no. 7, 805-849. 3. M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. Roy. Soc. London Ser. A 356 (1977), no. 1686, 375-394. 4. M. V. Berry and M. Robnik, Semiclassical level spacings when regular and chaotic orbits coexist, J. Phys. A 17 (1984), no. 12, 2413-2421. 5. 0. Bohigas and M.-J. Giannoni, Chaotic motion and random matrix theories, Mathematical and Computational Methods in Nuclear Physics (Granada, 1983) (J. S. Dehesa, J. M. G. Gomez, and A. Polls, eds.), Lecture Notes in Phys., vol. 209, Springer, Berlin, 1984, pp. 12.

99. 6.

H. Cramer, M. R. Leadbetter, and R. J. Serfiing, On distribution function - moment relationships in a stationary point process, Z. Wahrscheinlichkeitstheorie and Verw. Gebiete 18 (1971), 1-8.

D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Ser. Statist., Springer, New York, 1988. , An introduction to the theory of point processes. I: Elementary theory and methods, 8. 2nd ed., Probab. Appl. (N. Y.), Springer, New York, 2003. 9. M. Fukushima and S. Nakao, On spectra of the Schrodinger operator with a white Gaussian noise potential, Z. Wahrscheinlichkeitstheorie and Verw. Gebiete 37 (1977), no. 3, 267-274. 10. L. P. Gor'kov and G. M. Eliashberg, Minute metallic particles in an electromagnetic field, Soviet Physics JETP 21 (1965), no. 5, 940-947. 11. L. N. Grenkova, S. A. Molchanov, and Yu. N. Sudarev, On the basic states of one-dimensional disordered structures, Comm. Math. Phys. 90 (1983), no. 1, 101-123. 12. A. Hanen, Processus ponctuels stationaires et hots speciaux, Ann. Inst. H. Poincare Sect. B (N.S.) 7 (1971), 23-30. 13. H. Hasegawa, Level statistics for quantum systems: Introduction to quantum chaos, Butsurigaku Saizensen, vol. 28, Kyoritsu Shuppan, Tokyo, 1991 (Japanese). 14. H. Makino and S. Tasaki, Level spacing statistics of classically integrable system: Investigation along the lines of Berry-Robnik approach, Phys. Rev. E (3) 67 (2003), 066205. 15. , Long-range spectral statistics of classically integrable systems: Investigation along the line of Berry-Robnik approach, Progr. Theoret. Phys. 114 (2005), no. 5, 929-941. 7.

16. P. Major, Poisson law for the number of lattice points in a random strip with finite area, Probab. Theory Related Fields 92 (1992), no. 4, 423-464. 17. J. Marklof, The Berry - Tabor conjecture, European Congress of Mathematics (Barcelona, 2000), Progr. Math., vol. 202, Birkhauser, Basel, pp. 421-427. 18. K. Matthes, J. Kerstan, and J. Mecke, Infinitely divisible point processes, John Wiley & Sons, New York, 1978.


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19. H. P. McKean, A limit law for ground state of Hill's equation, J. Statist. Phys. 74 (1994), no. 5-6, 1227-1232. 20. M. L. Mehta, Random matrices, 3rd ed., Pure Appl. Math., vol. 142, Elsevier, Amsterdam, 2004.

21. N. Minami, Schrodinger operators with potential which is the derivative of a temporally homogeneous Levy process, Probability Theory and Mathematical Statistics (Kyoto, 1986) (S. Watanabe and Yu. V. Prokhorov, eds.), Lecture Notes in Math., vol. 1299, Springer, Berlin, 1988, pp. 298 - 304.

, Level clustering in a finite system (1994), no. 116, 359-368. , Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177 (1996), no. 3, 709-725. , On the Poisson limit theorems of Sinai and Major, Comm. Math. Phys. 213 (2000), 24. no. 1, 203-247. 25. , Level statistics for quantum Hamiltonians: Some preliminary ideas toward mathematical justification of the theory of Berry and Tabor, Proceedings of the Second ISAAC Congress (Fukuoka, 1999), Int. Soc. Anal. Appl. Comput., vol. 7, Kluwer, Dordrecht, 2000, pp. 7 -14. 26. , On level clustering in regular spectra, Surikaisekikenkyusho Kokyuroku (2000), no. 1156, 113-130. , Mathematical foundation of energy level statistics in terms of point process theory, 27. Bussei Kenkyuu 73 (2000), no. 6, 957-1011 (Japanese). 28. S. A. Molchanov, The local structure of the spectrum of the one-dimensional Schrodinger operator, Comm. Math. Phys. 78 (1980/81), no. 3, 429-446. 29. S. A. Molchanov and A. Ya. Reznikova, Limit theorems for random partitionings, Teor. Veroy22. 23.

atnost. i Primenen. 27 (1982), no. 2, 296-307 (Russian); English transl., Theory Probab. Appl. 27 (1982), no. 2, 310-323. 30. K. Nagai, The integrated density of states of one-dimensional random Schrodinger operator with white noise potential and background, Tsukuba J. Math. 30 (2006), no. 2, 383-400. 31. J. Neveu, Processus ponctuels, Ecole d'Ete de Probabilites de Saint-Flour (1976), Lecture Notes in Math., vol. 598, Springer, Berlin, 1976, pp. 249-445. 32. I. C. Percival, Regular and irregular spectra, J. Phys. B 6 (1973), no. 9, L229-L232. 33. C. E. Porter (ed.), Statistical theory of spectra: Fluctuations, Academic Press, New York, 1965.

34. A. Ya. Reznikova, The central limit theorem for the spectrum of random Jacobi matrices, Theory Probab. Appl. 25 (1980), no. 3, 504-513. , The central limit theorem for the spectrum of the random one-dimensional Schro35. dinger operator, J. Statist. Phys. 25 (1981), no. 2, 291-308. 36. A. M. Savchuk and A. A. Shkalikov, Sturm - Liouville operators with singular potentials, Mat. Zametki 66 (1999), no. 6, 897-912 (Russian); English transl., Math. Notes 66 (1999), no. 6, 741- 753.

37. P. Sarnak, Values at integers of binary quadratic forms, Harmonic Analysis and Number Theory (Montreal, 1996) (S. W. Drury and M. R. Murty, eds.), CMS Conf. Proc., vol. 21, Amer. Math. Soc., Providence, RI, 1997, pp. 181-203. 38. Ya. G. Sinai, Mathematical problems in the theory of quantum chaos, Chaos (Woods Hole, MA, 1989) (D. K. Campbell, ed.), American Inst. Phys., New York, 1990, pp. 395-414. 39. , Mathematical problems in the theory of quantum chaos, (Unknown Month 1989), Geometric Aspects of Functional Analysis (J. Lindenstrauss and Milman V. D., eds.), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 41-59. 40. , Poisson distribution in a geometric problem, Dynamical Systems and Statistical Mechanics (Moscow, 1991) (A. B. Sossinsky, ed.), Adv. Soviet Math., vol. 3, Amer. Math. Soc., Providence, RI, 1991, pp. 199-214. 41. J. M. VanderKam, Correlation of eigenvalues on multi-dimensional flat tori, Comm. Math. Phys. 210 (2000), no. 1, 203-223. 42. I. Veselic, Integrated density of states and Wegner estimates for random Schrodinger oper-

ators, Spectral Theory of Schrodinger Operators (Mexico City, 2001) (R. del Rio and C. Villegas-Blas, eds.), Contemp. Math., vol. 340, Amer. Math. Soc., Providence, RI, 2004,

N. MINAMI

398

pp. 97-183. 43. J. Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Math., vol. 1258, Springer, Berlin, 1987. INSTITUTE OF MATHEMATICS, UNIVERSITY OF TSUKUBA, TSUKUBA-SHI, IBARAKI 305-8571, JAPAN

Current address: School of Medecine, Keio University, Hiyoshi, Kouhoku-ku, Yokohama, 223-8521, Japan E-mail address: minamimsakura.cc.tsukuba.ac.jp


On the Law of Addition of Random Matrices: Covariance and the Central Limit Theorem for Traces of Resolvent L. Pastur and V. Vasilchuk ABSTRACT. We consider the ensemble of n x n random matrices Hn = An + where An and Bn are Hermitian (real symmetric), having the limiting Normalized Counting Measure of eigenvalues (NCM), and Un is unitary (orthogonal), uniformly distributed over U(n) (O(n)). We give first a more short and transparent proof of our result of [9] on the existence of the NCM

of Hn. This is based on a new inequality, analogous to the Poincare-Nash inequality for Gaussian random variables. We then find the leading term of the covariance and establish the Central Limit Theorem for traces of resolvent

ofHnasn ->oo.

1. Introduction The paper deals with Hermitian (real symmetric) n x n random matrices

Hn=An +UtB,Un,

(1.1)

where An and Bn are Hermitian (real symmetric) matrices such that if {AA"}

1

and {AB" } 1= 1 are eigenvalues of An and Bn and NA,,, and NB" are their Normalized 1

Counting Measures (NCM), defined as (1.2)

NA" (0) =

#{AA" E A,l = 1,...,n}n-1, EA,l=1,...,n}n-1 NB"(A)=#{AB"

for any interval A C III, then we have the weak convergence (1.3)

NA" -> NA,

NB" -p NB

of NA" and NB" to probability measures NA and NB. We assume further that Un in (1.1) is the random unitary matrix, whose probability law is given by the normalized to unity Haar measure on the unitary group U(n) in the case of Hermitian An and

Bn and Un is the random orthogonal matrix whose probability law is given by corresponding Haar measure on 0(n) in the case of real symmetric An and Bn. 2000 Mathematics Subject Classification. Primary 15A52; Secondary 60F05. This is the final form of the paper. ©2007 American Mathematical Society 399

400

L. PASTUR AND V. VASILCHUK

We mention also that following the general ideas of random matrix theory one can define analogous matrices using quaternion real An and Bn and symplectic Un (see, e.g., [7] for corresponding definitions). We will confine ourselves to the technically simplest case of Hermitian An and Bn and unitary Un in (1.1). Our results for symmetric and quaternion real matrices

and the groups O(n) and S(n) have a similar form, although their proof is more involved technically. Matrices of the above form are of interest by themselves (see, e.g., [9] for various motivations) and arise also in the free probability studies and related combinatorial problems (see review works [11,12] and references therein).

Our goal in this paper is the analysis of the eigenvalue distribution of Hn of (1.1), given that of An and Bn. The simplest but important for practically any random matrix study is the problem of weak convergence of the Normalized Counting Measures of eigenvalues {A[ "} 1 of H,, (1.4)

Nn(A) = #{Axn E 0, 1 = 1,. .. ,

n}n-I

to a non-random measure as n -> oo. Following general ideas of spectral theory, we study Nn via the resolvent (1.5)

Gn(z) _ (Hn - z)-1,

sz 54 0,

of Hn and its normalized trace (1.6)

gn(z) = n-1 Tr Gn(z),

related to the Normalized Counting Measures of eigenvalues of Hn by spectral theorem (1.7)

Nn(dA)

gn(z)= f A-z ,

sz54 0.

Here and below integrals without limits denote integrals over R. To study the asymptotic behavior of gn we use an approach, based on certain differentiation formulas (matrix analogs of the integration by parts), leading to certain identities for the moments of gn and to bounds for the variance of gn, allowing one to convert the identities into functional equations, determining uniquely 9n, hence the limiting measure (see [4,8] for various realizations of the approach). The approach proved to be rather efficient in studies of Normalized Counting Measures of eigenvalues

of a wide variety of random matrix ensembles since the paper [6], where it was introduced. For its modern forms see [4, 8]. A version of the approach was also used in [9] to treat random matrices (1.1). Here we partly use the ideas of [9], namely we relay again on a differentiation formula (see Proposition 2.1) to derive identities for the moments of gn. However, to obtain the bounds for the variance of gn we use not the same differentiation formula, applied to more complex expressions

containing the resolvent, but on a "unitary" analog of the well-known PoincareNash inequality for Gaussian random variables [2, 5]. This makes the proofs of bounds more short and transparent and enlarges the domain of validity of the bounds. Moreover, this allows us to find the leading term of the covariance of gn(zl) and gn(z2) for `£z1,2 $ 0, and to prove the Central Limit Theorem for any finite collection {9n(z1)}p 1.

ON THE LAW OF ADDITION OF RANDOM MATRICES

401

An important property of a wide variety of random matrix ensembles, the ensemble (1.1) in particular, is that the variance of 1

gn(z) = n c-1

1 A,H

-z

sz

1

:A

0

is of the order O(n-2), rather than O(n-1), as in the case, where AH' are replaced by the i.i.d. random variables. This implies that the Central Limit Theorem is valid not for V "gn but for ngn. The paper is organized as follows. In Section 2 we present our basic tools and rederive a system of functional equations, obtained in [9] and determining uniquely the limiting Normalized Counting Measures of eigenvalues of Hn. In Section 3 we outline the derivation of a compact form of the leading term of the covariance of gn(z1,2) and in Section 4 we do the same for the Central Limit Theorem. We mention in conclusion that our presentation is a kind of announcement. The detailed proofs and their discussion will be publishes elsewhere [10].

2. Convergence of Normalized Counting Measures of eigenvalues We denote E{... } the expectation with respect to the normalized to unity Haar measure of U(n). Our proofs are based on the following two facts on this expectation. The first one is the differentiation formula, given by Proposition 2.1. Denote 7-ln the linear space of n x n Hermitian matrices. Let 7-ln --- C be a continuously differentiable function. Then the following relation holds for any element X E 7-ln: :

E{V(UtMU) [X, UtMU]} = 0, where

[M1, M2] = M1 M2 - M1 M2 is the commutator of M1,2 E 7-ln.

The proof of the proposition is given in [9]. The second important technical mean is a "unitary" analog of the Poincare - Nash inequality. Proposition 2.2. Let (D: U(n) -* C be a continuously differentiable function. Then E{

Var{(D (Un)}

n j,k=1

E {I' Un) . E(j,k)Unl2},

where {E(j,k)}j,nk-_1 is canonical basis in the space Mn of all nxn matrices: E(j,k) {Epq'k)}p,4=1, ''pqk) _ (Sjpvkq.

The proof of proposition will be given in [10]. We will use the resolvent identity for resolvents Gl and G2 of two Hermitian matrices M1 and M2: (2.1)

G2 (Z)

- Gi(z) = Gi(z)(Mi - M2)G2(z) = G2(z)(MI - M2)Gl(z),

and the formula for the derivative of the resolvent of a Hermitian matrix M: (2.2)

valid for any X E fn.

G' X = -GXG,


402

We will also need the notion of the Nevanlinna functions (see, e.g., [1]). Namely,

an analytic in C \ IR function f is a Nevanlinna function if

f (z) = f (z),

(2.3)

f (z) z > 0,

z # 0.

Any Nevanlinna function admits the representation (2.4) f (z) = az + b + J + µz m(dµ), 'U

-z

where a > 0, b E IR, m is a finite non-negative measure and we write here and below integrals without limits for the integrals over R. The representation takes the form (2.5)

m(dµ) J µ-z

f (z) =

with a finite non-negative m if and only if sup,>l Iy f (iy) I < oo, and in this case

lim Iyf(Iy)I = m(R) < oo.

y-oo We have

Theorem 2.3. Consider the random matrices of the form (1.1) and assume (1.3). Then there exists a non-random probability measure N and for any continuous and bounded function co: 1[8 -> C we have with probability 1: (2.6)

li m if 2(A)NHn (d,\) = if

Moreover, the Stieltjes transform f (z) =

f a (d

z)'

z# 0

of N is a unique solution of the system

1f(z) = fA(hB(z)), f(z) = fB(hA(z)), (f (z))-I = z - hA(Z) - hB(z),

(2.7)

where

fA,B (z) = if

NA,B (d.)

A-z ' f (z) is a Nevanlinna function and hA,B (z) are analytic in C \ 1[8 functions verifying the conditions (2.8)

f (z) =

-z-1

+ o(z-I),

hA,B(z) = z + o(z),

Z--+00-

Remark 2.4. An analogous theorem was proved in [9] under the additional condition (2.9)

sup

f

IAINA,. (dA) < oo.

for the convergence in probability. It was also proved ([9, Theorem 3.1]) that if (2.10)

sup if I AI4NA-B, (dA) < 00

then NH, converges to N weakly with probability 1.


403

We will prove the theorem in two steps. On the first step we will prove the convergence with probability 1 of N,, under the condition (2.10). This is the analog of [9, Theorem 3.1]. On the second step we will remove condition (2.10) and prove the convergence in probability, using some truncation procedure and a compactness argument. On the both steps we need the following

Lemma 2.5. The system (2.7) for the triple (f, hA, hB) is uniquely soluble in the class of triples (f, hA, hB), where f is a Nevanlinna function, satisfying (2.8), and hA,B are analytic in C \ R and satisfy (2.8). PROOF. Indeed, introduce 1

(2.11)

r(z)

f(z),

where we have in view of (2.8)

r(z) = z + o(z),

z -* oo,

we can rewrite (2.7) as

r(z) - rA(hB(z)) = 0, r(z) - rB(hA(z)) = 0,

(2.12)

r(z) - hA(z) - hB(z) = -z, where (2.13)

rA,B (z) = -

B1 ,

fA , (z)

rA,B (z) = z + o(z),

Z`00-

This implies the following linear system for the derivatives r'(z), hA B(z):

Ir'(z) - r'a(hB(z))h'B(z) = 0, (2.14)

r'(z) - ra(hA(z))h'A(z) = r'(z) - hA(z) - h'B(z)

0,

In view of (2.8) and (2.13) the derivatives rA B(z) have the asymptotics

rA,B(z) = 1+o(1), Z-f00. This and condition (2.8) imply that the determinant J(z) of (2.14) has the same asymptotics:

(2.15) J(z) = rA(hB(z))+r'B(hA(z))-r'4(hB(z))r (hA(z)) = 1+o(1),

z -> oo.

Now suppose that the system (2.7) has two different solutions, hence there are two solutions (r('), 0)B) and (r(2)) hA2B) of (2.12). Using the integral representation of the Nevanlinna functions rA,B(z) (see (2.4)) we can obtain the linear system (2.14) with zero r.h.s. for the differences r(l) - r(2), hAI,B - hA2B. Its easy that in view of condition (2.8) the asymptotic of the determinant of this system is the same as that of J. Thus, this system has only trivial solution for z in some neighborhood of infinity and in view of analyticity of solutions in C \ R (r(1), hAIB) and (r(2), h(2,B) coincide.


404

Corollary 2.6. The linear system (2.14) has the unique solution

r'(z) = (2.16)

1

(rA(hB(z))) 1 + (rB(hA(z))) hA B (z)

_

1

-1

(r' z rB,A (hA,B (z))

In what follows we will omit the subscript n in An, Bn and Gn(z) and the argument z of the resolvent in the cases where there will be no confusion. PROOF OF THEOREM 2.3. Denote {Gjk}j'k=l the matrix of G. Taking in Proposition 2.1 4) = Gac and using (2.2), we obtain

E{(G[X, UtBU]G)ac} = 0,

a, c = 1, ... , n.

Then take X = E(a,b), rewrite this relation in the form E{Gaa(UtBUG)bc} = E{(GUtBU)aaCbc},

and apply to the result the operation n-1 Ea=1 This yields the matrix equality E{SBG} = E{gUtBUG}, where

SB = n-1 Tr GUtBU.

(2.17)

Introduce the centralized quantities (2.18)

bB = bB - E{bB},

g,' = gn - E{gn}, and rewrite UtBUG using (2.1):

E{SB} E{G} + E{b G} = E{gn}(I + z E{G} - E{AG}) + E{g°UtBUG}, or, after regrouping terms (2.19)

fn(z)(A - hB,,I) E{G} = fn(z)I + (E{gnUtBUG} - E{SBG}),

where (2.20)

fn(z) = E{gn(z)},

hBn(z) = z -

E{8B(z)} fn (z)

In view of relations;

fn(Z)=-Z

I zI

Ign(z)I

1

(2.21)

I6B(z)I C

I

zl

mB)

= supJ

jAjkNB

.

n

valid for any z in the domain (2.22)

r,,,,a = {z c C :

QzI

aI zI, IFjzI > a},

Q > (a + 1)mBl)

and k=0,---,4, we have hB,. (z)I > /3 - (a + 1)mBli and, hence, the matrix A - hB (z)I in (2.19) has the inverse (2.23)

GA (hBn (z)) = (A - hB, (z)I)

1


405

uniformly in n bounded for z E r,,,p: (2.24)

II GA(hB.(z)) II 0 nonvanishing principal curvatures. Then, we do a similar study for Cauchy principal values of such Fourier transforms. Finally, we deduce the decay of Green's functions of generalized Laplacians for energies inside their associated spectra. We adopt the following conventions: most of our theorems establish the existence of a neighborhood on which a certain phenomenon occurs. For sake of

GREEN'S FUNCTIONS OF GENERALIZED LAPLACIANS

419

simplicity (and without loss of generality), we consider only nonempty, bounded, open, cubic neighborhoods and we call them cubes.2 Given a real-valued phase, co(x, t), we define I f (r, t) =

(2.1)

J.d

e'r` (x,t) f (x, t) dx,

where r > 0, t E Rm, and f (x, t) is a complex-valued function-provided that this integral makes sense.

Smooth is used in this text for infinitely differentiable. The vector space of all (complex-valued) smooth functions on Rd is denoted by C°° (Rd). It contains two important subspaces: C"(Rd), consisting of all analytic functions on Rd, and C°° (Rd), consisting of all compactly supported smooth functions on Rd. The transpose of a linear transformation on C°° (Rd) is defined by duality with respect to the following bracket:

(1(x) I g(x)) = f d f (x)g(x) dx. For instance, denoting by 8,,(j) the differentiation with respect to x(j), integration by parts gives at (j) = -8x(j). Moreover, denoting by F(x) the multiplication by a smooth function of the same name, F(x)t = F(x). Our notations regarding asymptotic behavior are standard: for instance,

f (r) = O(r-a) when r --4 00 means the existence of a positive constant, C, such that If (r) I S Cr-" when r is sufficiently large. If such a constant (depending on a) exists for any a > 0, we write

f (r) = O(r-°°)

when r ---> oo.

Finally,

f(r) - 00E

ajr-i

when r --> oo

j=0

means that for any N > 0 N

1(r) -

Eajr-j = 0(r-N-1).

j=0

If the function f also depends on a parameter, t, we say that the previous estimates/asymptotics are uniform in t if the constants C may be chosen independently of t.

3. Oscillatory integral without stationary phase point The following theorems are stated in the way they are used when studying Fourier transforms over surfaces. They establish the existence of neighborhoods on which a certain phenomenon occurs, given a fixed phase. The given phase, cp(x, t), is supposed to be smooth in (x, t) E Rd x JRtm. 2Recali that a cubic neighborhood in Rd is a subset of the form Il x is an interval.

X Id, where each Id

P. POULIN

420

Lemma. Let d = 1. Suppose axco 0 at a given (xo, to) E R x Rm. Then, there exists an arbitrarily small cube, U x B, containing (xo, to) such that the following holds: if the amplitude, f (x, t), is smooth in the neighborhood of R x B and vanishes on UC x B, then IIf (r, t) I = 0(r-°°) uniformly in t E B.3

0 on a certain cube U' x B' containing PROOF. By continuity, axcp(x, t) (xo, to). The operator D = 1/ (axcp(x, t)) oax is thus well defined on this cube, where ire'rw(x,t) and Dt = -ax o 1/ (axcp(x, t)). Let U x B D (xo, to) it satisfies = De'rw(x,t)

be a cube whose closure is in U' x B'. If f (x, t) satisfies the asserted properties,

then for anyN>, 0andtEB IIf (r, t) I = rN rN

Ju

J

(DNe'rw(x't) )f (x, t) dx eirp(x,t)

U

(Dt)Nf (x t) dx 1

CN rN '

where the constant CN does not depend on t E B. The multidimensional analogue follows:

Theorem 3.1. Suppose Vxcp(xo, to) 0 for a given (xo, to) E Rd x Rm. Then, there exists an arbitrarily small cube, U x B, containing (x0, to) such that the following holds: if the amplitude, f (x, t), is smooth in the neighborhood of Rd x B and vanishes on U° x B, then IIf (r, t) I = 0(r-°°) uniformly in t E B. PROOF. By assumption, ax(k) cp(xo, to)

0 for a certain 1 < k 0 such that the following holds: if the amplitude, f (x, h, t), is smooth in the neighborhood of Rd x [-6,6] x B and vanishes on U° x [-6,6] x then p. V. fb If (r h, t) dhl = 0(r-') uniformly in t E B.

PROOF. By the lemma it suffices to estimate 8hI f (r, h, t). In fact, the dominated convergence theorem implies 8hIf(r, h, t) =

eirW(x,h,t) (ah f(x, h, t) + it f (x, h, t)8hcp(x, h, t)) dx.

Hence, by Theorem 3.1 there exist an arbitrarily small cube, U x B, containing (xo, to) and an arbitrarily small S > 0 such that for f (x, h, t) as stipulated

ahIf(r,h,t)

CN

uniformly in (h, t) E [-S, S] x B. The result follows. Applying the above to the phase cp(x, h; e, t) = cp(x, h + e, t) and the amplitude f (x, h; e, t) = f (x, h + e, t), one obtains:

P. POULIN

422

Corollary. Suppose V., p(xo, eo, to) : 0 for a given (xo, eo, to) E Rd x JR x R'. Then, there exist an arbitrarily small cube, U x B, containing (xo, to) and an arbitrarily small S > 0 such that the following holds: if the amplitude, f (x, e, t), is smooth in the neighborhood of Rd x [eo - 6, eo + S] x B and vanishes on U° x [eo 5, eo + 5] x B, then p. V. f'17-ej (x(j))2 - E (x(k))2, k=s+1

j=1

where 01

ifr 0 independent of rl satisfying irX2 xl f

(x t) dx , C1 r -

1+1

2

00

uniformly in t E B.

PROOF. Let x(x) be a compactly supported smooth function on lib such that 0 < x(x) 0, J e1T52x1 f (x, t) dx =l\ I + II, where

I=J xlE

Concerning I, there exist constants (generically denoted by Const) independent of t E B, but depending on 1, satisfying 2E

II < Const

f

Ixll dx =

Consul+l.

2E

Concerning II, let D = (1/x) o 8x, which is well defined on the support of the integrand. Since 1/(2ir)D fixes eirx2, /N

II =

\ 2ir1

/ \ eirx2 (Dt)Nxl f (x, t) 1 - x I I

dx.

J,- I>E

For F(x, y, t) = f (x, t) (1 - x(y)), notice that (Dt)Nx1F(x, x/e, t) is of the form x1-2NFo(x, x/E, t) + x1-2N+1E-1F1(x x/E t) + ....+ xI-NE-NFN(x, x/E, t), where Fo, ... , FN are bounded on lib x lib x B. Hence, for a fixed N > l + 1 and for IxI > E,

I(Dt)Nx1F(x, x/E, t) I
E Ixll-NE-N dx = Constlrl-Ne1 2N+1

In total,

II + III < Const (E1+1 + Irl-NE1-2N+1) for such an N. Choosing E = Irl-1/2 then completes the proof.

Lemma. Consider a function, f (x, t), smooth in the neighborhood of lib x B (where B C 1R" is a cube) and vanishing when IxI < e (where e > 0). Suppose that

for all N > 0 there exist a DN > 0 and an aN > 0 such that I9 f(x, t) I < DN(1 + IxI'N)

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424

uniformly in t E B. Then,

f00e(_1)x2f(x,t)dx = O(r-°°) uniformly in t E B.

PROOF. The derivation D = (1/x) o 8x is well defined on the support of the integrand. Notice that 1/(2ir)D fixes e'rx2 and Dt = -8x o (1/x). Then, e('r-1)xZf (x t) dx

e;rx2 fix I

(Dt)N(e-x2f (x, t)) dx

->E

Our assumption on the derivatives of f (x, t) makes the integral on the right side of the previous equation uniformly bounded in t E B, where N is arbitrarily fixed. The result follows.

In dimension one we are interested in the phase x2, so let 00

if (r, t) =

-cc

elrx2 f(x, t) dx.

According to our convention r > 0. However, since we are also interested in the phase -x2, we allow r to be negative. We are ready to compute the asymptotic expansion of If (r, t) (resp. If(-r, t)) when Irl --> oo: Theorem 4.1. Consider an amplitude, f (x, t), smooth in the neighborhood of x B and vanishing on Uc x B, where U C R and B C R' are cubes. There exist constants, aj (t), depending smoothly on t E B such that

l1

00

eirx2

1700

f

(x, t) dx

' I rl -1/2 E aj (t) I rl -j/2 j=0

uniformly in t E B when r --> oo. The same result holds when r ---> -oo, with different constants aj(t). PROOF. Let x(x) be a smooth, bounded, compactly supported function such that x(x) = 1 on an interval containing U U {0}. Then, for f (x, t) as stated If (r, t) = f e(ir-1)x2 (ex2.f (x, t))X(x) dx. R

By Taylor's theorem (in dimension 1), for any fixed N there exists a polynomial, N

Pt (x) _

bi(t)x1, c=0

and a smooth remainder, Rt (x), both depending smoothly on (x, t) E R x B, such

that ex2 f(x, t) = Pt(x) + xN+1Rt(x).


425

Thus, If (r, t) decomposes into I + II + III, where N

I = E bl(t)

J

l=0

II = III =

e(ir-1)x2xl

dx,

R

e(ir-1)x2xN+1Rt(x)X(x) dx,

J

f

e(ir-1)x2Pt(x)(X(x) - 1) dx.

R

Concerning I the first lemma gives the existence of c(jl) E C such that for any IrI > 1 N 00 Irl-(l+1)/2 E c7(l) Irl-j I = E bt (t) j=0

l=0 N

=

lrI-1/2Eb1(t)IrI-l/2

cjl)IrI

-i

j(N--1)/2

1=0

Notice that N

E cjl)Irl-j

bl(t)lrl-(l+1)/2

=O(lrl-N12-1)

j>(N-1)/2

1=0

uniformly in t E B when r -+ oo (resp. r --> -oo). Consequently, we have found coefficients, ak(t), smooth in t E B, satisfying N

I = Irl-1/2 E ak(t)I rl -k/2 +

0(Irl-N12-1)

k=0

uniformly in t E B. Moreover, the last two lemmas give respectively

II =

0(Irl-N12-1)

and

III = O(lrl-°°)

uniformly in t E B. In total, N

I.f (r, t) = Irl-1/2 E ak(t)I rI

-k/2 +

0(lrl-N12-1)

k=0

uniformly in t E B when r --> oo (resp. r -+ -oo), as desired.

0

Our treatment of a parameter t E ][gym permits to deduce from the previous theorem its multidimensional analogue by induction. Indeed, let us consider the phase d

s

Q(x) = E(x(j))2 - E (x(k))2, k=s+1

j=1

where x E Rd and 0 < s 0, we let

if (r, t) _ Then,

ei

IRa

r) f(x, t) dx.

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426

Corollary. Consider an amplitude, f (x, t), smooth in the neighborhood of Rd X

B and vanishing on U' x B, where U C Rd and B C R' are cubes. There exist constants aj (t) depending smoothly on t E B such that 00

f

(t)r-j/2

eirQ(x) f (x t) dx ti r-d/2 E aj d

j=0

uniformly in t E B when r -> oc. PROOF. Suppose the result holds for a certain d-1. Let x1 = and Q1(x1) be defined by

(X(1),...,

x(d-1))

(d))2.

Q(x) = Qi(xi) ± (x Recall that U may be written Ul x . x Ud. Then, by the inductive hypothesis, for an arbitrary N >, 0 e±ir(.(d))2

if (rt) = J d

r

e +ir(x(d))2

f

eirQ,(x1) f(xl, x(d), t) dxl dx(d) d -1

N+1

(r- (d-1)/2jJ[ bk (x(d), t)r -k/2 + O(r- (d+N+l)/2)

dx (d)

k=o

Ud

uniformly in (x(d), t) E Ud x B. The following estimates then hold when r --> oc, uniformly in t E B: N+1

if (r t) _

r-(d+k-1)/2 f E k=0

e±ir(.(d))2 bk (x(d), t) dx(d) +

O(r-(d+N+1)/2)

Ud

N+1

E

N

r-(d+k-1)/2 (r-1/2 [ ek l

(t)r-l/2)

+ O (r-(d+N+1)12 )

k=0

=

r-d/2

N

E aj

(t)r-j/2

+ 0(r- (d+N+1)12 )

j=0

Since N is arbitrary, this completes the proof.

Morse lemma. The investigation of oscillatory integrals with nondegenerate stationary phase points reduces to the above case by means of Morse's lemma, which we now prove.

Let the anticipated phase, O(h, t), be a smooth function in (h, t) E Rd x R"` satisfying

0(0, t) = 0, V hc5(0, t) = 0, and det t) # 0 for all t E B, where B C Rm is a given cube containing a given to E Rm. Such a function can be expressed like a quadratic form, but with coefficients varying smoothly in (h, t): Lemma. Under the above circumstances there exist functions, qjk (h, t), smooth in (h, t) E Rd x Rm and satisfying d

0(h, t)

d

= ET.Ojk(h,t)h(j)h(k)

j=1 k=1

where lpjk (h, t) = Oki (h, t).


427

PROOF. Using our hypotheses on cb(h, t), the fundamental theorem of calculus and integration by parts give for any t E B 1

h t)

1

83 ,,//, sh, t))

ds =

(1

s) 8

sh, t)) ds.

Expanding 0,2 (O(sh, t)) in the above gives the result.

The next step consists of applying Lagrange's algorithm, which is better understood using matrices. Using the standard basis on Rd, h is represented by a column (also denoted by h), while the "quadratic form" given by the previous lemma is represented by a d x d matrix, denoted by 4) (h, t). The (j, k)th element of (D (h, t) is then given by the function Oik(h,t), so the previous lemma gives 0(h, t) = ht-D (h, t) h.

For any given t E B the rank of D(0, t) is read through the Hessian of 0(h, t) using the following, straightforward relation: Lemma. Under the above circumstances, for any t E B, D20(0, t) = 2(D (0, t).

In order to perform Lagrange's algorithm one uses the following elementary line/column operations: Given a scalar c # 0, to multiply the jth row and then the jth column by c, which is denoted by CLj(c); Given a scalar c E C, to add c times the kth row to the jth row and then c times the kth column to the jth column, which is denoted by CLjk(c);

To interchange the jth row with the kth row and then the jth column with the kth column, which is denoted by CLjk. By the previous lemma, since det D10(0, t) 0 and since 4) (0, t) consists of smooth elements, there exists a cube Vo x Bo C Rd x B containing (0, to) such that -1) (h, t) is invertible for all (h, t) E Vo x Bo. Without loss of generality 011 (h, t) $ 0 on a certain cube V1 x B1 C_ Vo x Bo containing (0, to). Otherwise, 011(h, t) vanishes at (0, to). However, considering

the Laplace expansion of the above determinant along the first row, there exists an index 1 < k < d such that 01k(0, to) 0. If Okk(0, to) = 0, one replaces ' (h, t) t) with CLIk)(h, t). In t); if Okk(0, to) 0, one replaces with CLIk both cases the resulting upper-left element does not vanish at (0, to), and hence on a certain cube V1 x Bi C Vo x Bo containing (0, to), -which then takes the place of V1 x B1.

Then, one may reduce the upper left element to ±1 (depending on the sign of

011(h, t), which does not change on V1 x B1) by applying CL1(1011(h, t) 1-1/2) to (the

possibly refreshed) 4)(h, t). Finally, this resulting constant on the upper left corner permits to cancel the rest of the first line and column, by applying CLk1(f (h, t) ) f o r k = 2, ... , d successively - where f (h, t) is equal to the element to cancel (up to the sign). All these operations are represented by matrices having smooth elements in (h,t) E V1 x B1. They transform (P (h,t) in a block diagonal matrix, having ±1 as its first block and a square (d - 1) x (d - 1) matrix as its second block. Repeating this procedure for the second block, (D(h, t) is transformed in a block diagonal matrix having ±1 as its first two blocks and a square (d-2) x (d-2) matrix as its third block. All the operations used for this second step are represented by

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428

matrices having smooth elements in (h, t) E V2 x B2, where

(O,to)EV2xB29V1xB1CV0xBo. So on and so forth one transforms D(h, t) into a diagonal matrix having elements

±1 only. All the required operations are smooth (in the previous sense) for (h, t) varying in a cube Vd x Bd containing (0, to). One then applies permutations CLik, so the resulting matrix becomes diag(1,... ) 1 , -1, ... , -1), where the element 1 is repeated, say, s times. Since s does not depend on (h, t) E Vd x Bd, the previous lemma and the Sylvester Inertia Theorem permit to recover s from the signature of D2 0(0, to), which is then (s, d - s, 0). In summary, we have proved:

Lemma. Under the above circumstances there exist a cube, Vd x Bd, containing (0, to) and a nonsingular linear map, Q(h, t), whose matrix elements are smooth in (h, t) E Vd x Bd such that

Q(h, t)t4(h, t)Q(h, t) = diag(1,...-, 1-1, ... , -1). In the above the element 1 is repeated s times, where (s, d - s, 0) is the signature of D2 q5(0, to).

It is thus tempting to consider the nonlinear mapping h H Q(h, t)-1h defined on Vd as a potential change of variables given a fixed t E Bd. Considering t E Bd as a parameter, {Q(h, t)-1h}tEBd is indeed a family of smooth mappings depending

smoothly on t in the strong sense that (h, t) H Q(h, t)-1h is jointly smooth on Vd x Bd. Moreover, all these mappings map 0 to 0. They really consist of invertible changes of variables when restricting suitably the ranges of h and t, as shown below:

Lemma. Under the above circumstances there exists a cube, W x B', such that

(0,t0)EV'xB'CVdXBd, on which

(h, t) H (Q(h, t)-1h, t) is a smooth diffeomorphism. In addition, this diffeomorphism maps V' x B' onto an open bounded set, hence contained in a certain cube V' x B'. PROOF. Let F(h, t) = (Q(h, t)-1h, t) for (h, t) E Vd x Bd. A direct computation shows that det D(h,t)F(0, to) = det Q(0, to)-1,

which is not zero. Thus, F(h, t) is a local diffeomorphism in a neighborhood of (0, to). Choosing a cube, V' x B' -D (0, to), whose closure is included in this last neighborhood (and in Vd x Bd) then yields the result.

The previous diffeomorphism maps V' x B' onto a bounded open set V1

C

xB'. We want to fix tEB',so let Vt={hER d;(h,t)ED}. Then, DtC V' is

an open set in Rd. Let us consider the restriction V' -+ Dt, h H Q(h, t) h, which is also a smooth diffeomorphism. It may be used as a smooth invertible change of variables by setting

h = Q(h, t)-'h


429

for h c V. Notice that 0 is then mapped to 0. Let Pt be the inverse change of variables, so

h=Pt(h) for h E V,. Then, h = Pt 1(h) = Q(h, t)-1h, which implies h = Q(h, t) h and hence O(Pt (h), t) = 0(h, t) = O(Q(h, t)h, t)

s

t)Q(h, t)h = E(h(j))2

= htQ(h,

j=1

d

-

E (h(k))2. k=s+1

We have thus proved the Morse lemma with special care of the parameter t:

Theorem 4.2. Given a cube B C Rm containing a fixed to, suppose 0(h, t) is smooth in (h, t) E Rd x B and satisfies ¢(0, t) = 0,

and

VhO(0, t) = 0,

det D2 0(0, t) 54 0

for all t E B. Then, there exists a cube V x B' C Rd x B containing (0, to) such that the following holds: for all t E B' there exists an invertible change of variables

h = Pt 1(h) on V', smooth and with smooth inverse, mapping 0 to 0, which satisfies

- r d

2

0(Pt(h),t) _ j=1

(h(k)) 2.

k==S+1

The resulting family of changes of variables, {Pt(h)}tEB,, depends diffeomorphically on t E B' in the following sense: setting Dt = Pt 1(V'),

D = U Dt x {t} tEB'

is an open set in Rd x RI (contained in a cube V' x B') on which

b , V' x B',

(h, t)

-

(Pt (h), t)

is a diffeomorphism.

Continuation. The corollary of Theorem 4.1 joined with the Morse lemma finally yield:

Theorem 4.3. Suppose cp(x, t) is smooth in (x, t) E Rd x R"n and satisfies V,,cp(xo, to) = 0

and

det D2cp(xo, to)

0.

Then, there exists an arbitrarily small cube, U x B, containing (xo, to) such that the following holds: if f (x, t) is smooth in the neighborhood of Rd x B and vanishes

on U° x B, then

if(r t) - eire(t)r-d/2 E00 aj (t)r-j12, j=0

where 0(t) is real-valued, the aj(t)'s are complex-valued, and all these functions are smooth in t E B. Moreover, these estimates are uniform in t E B.

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430

PROOF. By the Implicit Function Theorem there exists a smooth function Bo C R' --+ Rd,

t F_+ X(t)

defined on a cube B0 E) to such that (V.,p) (x(t), t) = 0 for all t E Bo and x(to) = xo. Without loss of generality we also suppose det D.W(x(t), t)

0

for all t E Bo. Let 0(h, t) = cp(x(t) + h, t) - cp(x(t), t),

which is smooth in (h, t) E Rd x Bo. Then, 0(h, t) satisfies the hypotheses of Morse's lemma. Hence, there exists a cube, V' x B', containing (0, to) and whose closure is in Rd x Bo, and a family of diffeomorphisms

Pi-1: V'->VtCV' (wheretEB') such that, letting h = Pt 1(h), one obtains s

d

j=1

k=Gss+1

(Pt(h),t) - E(h(7))2 -

(h(k)) 2.

Notice that 0 = xo - x(to) E V. Consequently, there exists an arbitrarily small cube, U x B E) (xo, to), such that

(x - x(t), t) E V x B' for all (x, t) E U x B. In other words, U x B is mapped onto a region whose closure

is in V x B' via the change of variables (h, t) = (x - x(t), t). Let us consider an amplitude, f (x, t), satisfying the asserted properties. Since for a fixed t E B the integrand in if (r

t) _

f

ei(xt) f(x, t) dx d

is supported in U, it follows that the right-hand side in

if

(r, t) =

f

eir`F(h+x(t),t) f (h + x(t), t) dh = eirp(X(t),t) d

f

eirO(h,t) f

(h + x(t), t) dh

d

is supported in V'. Then, by Morse's lemma the change of variables h = Pt 1(h) is available, yielding

I f (r t) = eirw(X(t),t) f e() f (Pt (h) + x(t), t) Jt (h) dh, t

where Jt (h) is the Jacobian and

Q(h) _ (h(7))2 j=1

d

_E

(h(7))2.

9=s+1

Since the amplitude in the above extends smoothly on Rd x B and vanishes for h V', the corollary of Theorem 4.1 then completes the proof. Finally, the following result is an interesting application of our treatment of a parameter:


431

Theorem 4.4. Suppose cp(x, t) is smooth in (x, t) E ]!8d x Rm and satisfies Vxcp(xo, to) = 0

and rank D2cp(xo, to) = n,

where , > 1. Then, there exists an arbitrarily small cube, U x B, containing (xo, to) such that the following holds: if f (x, t) is smooth in a neighborhood of Rd x B and vanishes on U° x B, then

if (r, t) = O(r-"/2)

uniformly in t E B. PROOF. Since the rank of D2:,;cp(xo, to) is equal to the order of its largest nonzero

principal minor, there exist indices, j1, ... , j., such that, letting

_ (x(il) ...,x(M), det D£cp(xo, to)

0. After permuting the variables we write and xo =

x = (l;, X),

Xo),

with the obvious definitions of x, o, and Xo. Interpreting (X, t) as a parameter the previous theorem gives the result. 0

Remark. Using the previous decomposition one cannot derive the complete asymptotics of the considered oscillatory integral, since the resulting coefficients would be oscillatory integrals themselves! Their decay is not known a priori. Cauchy principal value. We now derive similar results for Cauchy principal values of oscillatory integrals with nondegenerate stationary phase points. To this end let us consider first p. V.

JI

eirh f (h) dh = lim h

Ej0

ei,h f (h) dh, IhI>E

h

where the amplitude, f (h), is smooth in h E R and compactly supported. Notice that the lemma of Theorem 3.1 generalizes to a complex-valued phase provided that the path of integration remains in C+, explicitly: Given a smooth regular4 path, y(t), lying in the closure of C+, and an amplitude, f (z) = f (x, y), smooth in z and compactly supported along this path,

f

eirz f(z) dz =

O(r-o°)

7

whenr -->oo.

Lemma. Suppose f (h) is smooth in h E R, compactly supported, and analytic at 0. Then, p. v.

J 00

eirh f() dh = iri f (0) + 0(r-') h

whenr ->oc. PROOF. For e > 0, let CE be a smooth regular path starting at -2E, going through [-2e, -e], then avoiding the origin, but staying in

{zeC;IRe(z)l <E and0 0 and a cube B C Rm, consider a function, f (h, t), smooth in a neighborhood of R x B and vanishing on x B. Then, p. V. f 00 eirh f (h, t)

J

dh = 7ri f (0, t) + 0(r-')

uniformly in t E B, when r - oo. PROOF. Let x(x) be a smooth function in x E R, compactly supported, such

that 0 0. The previous theorem thus applies, which completes the proof.

Let us now consider a phase, co(x, h, t), smooth in (x, h, t) E Rd x R x R1, and

if (r, h, t) _ f d eit) Theorem 4.6. Suppose

f(x, h, t) dx.

0, to) = 0, det D'cp(xo, 0, to) 54 0, and ahcp(xo, 0, to) :,4 0.

Then, there exist a fi > 0 and an arbitrarily small cube, U x B, containing (xo, to) such that the following holds: if f (x, h, t) is smooth in the neighborhood of Rd x R x B and vanishes on (U x ]]-8, fi[)c x B, then p. V.

J

aj (t)r-j/2

If (r, h, t) dh - e're(t)r-d/2 j=0

uniformly in t E B, where 0(t) and aj (t) are smooth in the neighborhood of B.

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434

PROOF. Interpreting (x, t) E Rd x !IR' as a parameter, the previous lemma 0. Moreover, interpreting (h, t) E R x RY'' as a parameter, Theorem 4.3 also holds, since Oxcp(xo, 0, to) = 0 and det D2cp(xo, 0, to) # 0. Consequently, there exist a b > 0 and an arbitrarily small cube, U x B, containing (xo, to) such that for f (x, h, t) of the stipulated form holds, since 8hcp(xo, 0, to)

(4.4)

p. v.

eirW(x,h,t) f (x, h,

J

t)elrW(x,o,t) + 0(r-oo)

dh = airi f (x, 0,

uniformly in (x, t) E U x B and (4.5)

J

e(o,) f(x, 0, t) dx

ee(t)r-2a(t)r/2 00

j=0

uniformly and smoothly in t c B, both when r -+ oo. By Fubini's and the dominated convergence theorems if (r, h, t) dh = fad p. V. J

p. V. J

e'rp(x,h,t) f

(x, h, t)

dh dx,

where the dominator is given by the lemma of Theorem 3.2. By the equation (4.4), the above is equal to 'kd

Q,7rif(x, 0,

t)e'rW(x,o,t)

dx +O(r-°)

uniformly in t E B. The equation (4.5) then yields the result. In the same way as we derived from Theorem 3.2 its corollary, Corollary. Suppose Oxcp(xo, eo, to) = 0, det D2cp(xo, eo, to) # 0, and 8ecc(xo, eo, to)

0.

Then, there exist a 6 > 0 and an arbitrarily small cube U x B containing (xo, to) such that the following holds: if f (x, e, t) is smooth in the neighborhood of Rd x R x B

and vanishes on (U x ]eo - 6, eo + 8[)c x B, then p. V. J

1 If (r, rl, t) dij xt7-e

eire(e,t)r-d/2 E aj (e t)r-1l2 j=o

uniformly in (e, t) E [eo - 6, eo + 6] x B, where O(e, t) and aj (e, t) are smooth in the neighborhood of [eo - 6, eo + 6] x B.

More generally, by the argument used in Theorem 4.4 (which we repeat!), Theorem 4.7. Suppose Oxcp(xo, eo, to) = 0, rank Dxcp(xo, eo, to) = ,c, and 8ep(xo,eo,to) 54 0.

Then, there exist a b > 0 and an arbitrarily small cube, U x B, containing (xo, to) such that the following holds: if f (x, e, t) is smooth in the neighborhood of Rd x R x B

and vanishes on (U x ]eo - 6, eo + 6[)c x B, then p. V.

xe

If(r) 77, t) d77 =

uniformly in (e, t) E [eo - 6, eo + 6] x

O(r-k/2)


435

PROOF. There exist indices, jl,... , j,,, such that, for

= (x(jl)...,x(M), detD2(xo,to)

0. After permuting the variables we write

x=

X)

and

xo =

Xo)

with the obvious definitions of X, o, and Xo. Interpreting (X, t) as a parameter, the above corollary gives the existence of a 8 > 0 and an arbitrarily small cube, U' x (U" x B) E) Xo, to), such that, letting U = U' x U" E) xo, for f (x, t) of the stipulated form p. V.

X, 71, t d d

1

f77 - e fR.

r0(r-

uniformly in (X, e, t) E U" x [eo - 8, eo + 8] x B. Therefore, faa_ w

pv

f

1

71 - e

= - O(r2)

fw

uniformly in (e, t) E [eo - b, eo + 8] x B. By Fubini's and the dominated convergence theorems this last integral is equal to

J

eirW(-;Tl,t) f (x. ,q t) dii dx

p. V. J

and hence (for the same reasons) to

p.v.ff71 d rl-e

t)dxd7]

where both times the dominator is given by the lemma of Theorem 3.2. The proof is thus complete.

5. Fourier transforms over level surfaces of analytic functions We now consider a real-valued function, J (x), analytic in x E Rd and its level surface I'(e) = {x E Rd; 4D(x) = e}.

Given a function, f (x), summable on I'(e), its Fourier transform over r(e) is defined as

F(r(e), f)(n) =

f

eon

f(x) dS(x)

(e)

for n E Zd, where dS(x) denotes the element of surface on I'(e). In this section we derive the decay of .F(r(e), f)(n) when Inj -> oo and show its uniformity when e varies on an appropriate interval. We then derive an analogous result for the Cauchy principal value of such a Fourier transform. Let a' < b' and S' = UeE]a,,b,[ r(e) be given. We assume:

Assumption A.

D(x) is real-valued and analytic in Rd; 0 for all x E S'; IF (e) is compact for all e c ]a', b'[.

V (x)

The second statement in Assumption A and the Implicit Function Theorem ensure that F(e) is a regular smooth surface for any e E ]a', b'[. In particular, r(e) may be covered by real-analytic local parametrizations, U 14 I'(e), where each

P. POULIN

436

U C Rd-1 is open, each a(u) is a smooth homeomorphism between U and o-(U) in the topology of F(e), and

U a(U) = F(e). (U,a)

Moreover, if two of the previous local parametrizations, (U, a) and (V, r), have a nonempty overlap, D = a(U) fl T(V) C F(e), then the change of parametrizations 7-1 o a(u) is a real-analytic diffeomorphism from o,-1(D) to T-1(D). Given a local parametrization, (U, o), the restriction of the Fourier transform to Q(U) gives

F(o, (U), f)(n) =

f

ei"'f(x) dS(x) = u e;n-a,(u)f(o,(u))J(u) du (U)

for any f (x) summable on o,(U), where J(u) du is the element of surface. Of course the decay of such an integral when Inj -> oo is studied by means of the stationary phase method. Let n = rw be the polar form of n E Zd, so r = InI and w E Sd-1. The phase in the previous integral then becomes c'(u, w) = w o,(u). Let us consider a point, xo = Q(uo), in o,(U). Remarkably, the fact that uo is or is not a stationary phase point depends on intrinsic properties of F(e) only; if uo is stationary, the rank of Ducp(uo, w) is also intrinsic. Indeed,

Theorem 5.1. In the above circumstances uo is stationary if and only if w is perpendicular to F(e) at xo = o,(uo). Then, the rank of the Hessian of co(u, w) at uo is equal to the number of nonvanishing principal curvatures of F(e) at xo. PROOF. Notice that uo is stationary if Vuw o, (uo) = (w du(1) o, (uo), ... , w - du(d-1)a(uo)) = 0. Since the tangent plane of F(e) at xo is generated by -

{du(j) a(uo); j = 1, ... , d - 1},

the first statement follows. Suppose uo is a stationary phase point and consider any other local parametrizaLet D = o, (U) fl T(V) and F(v) = tion, (V, -r), of a neighborhood of x0 = or Q-1 o T(v). Then, F(v) is a smooth diffeomorphism from T-1(D) to a-1(D) satisfying

w T(v) = w o(F(v)).

(5.1)

Let vo = T(xo), so (Vu(w a)) (F(vo)) = Vu(w o,) (uo) = 0. Then, the chain rule applied to (5.1) gives Dv(w . T) (vo) = Du(w . o,)(uo)(DvF(vo))2.

Since F(v) is a diffeomorphism, DvF(vo) is invertible and hence

rankD'(w T)(vo) = rankD2(w Q)(uo), which shows that this last rank is intrinsic. Finally, since uo is a stationary phase point, w is perpendicular to F(e) at xo, so

±VX-D(xo) 11VA'(xo)II


437

0. Suppose without loss Moreover, there exists a j E {1, ... , d} such that wU) Ix (d-1)) of generality w(d) 54 0, and hence 88(d) (xo) 54 0. Let w = (x(') and w0 = (xol) ... , xod-1)). By the Implicit Function Theorem there exists a function, h(w), smooth in the neighborhood of w0i such that D(w, h(w)) = e and h(wo) = Hence, -y(w) = (w, h(w)) gives a smooth local parametrization of a neighborhood of x0 as a graph of a smooth function. Differential geometry then shows that rank Dwh(w) is equal to the number of nonvanishing principal curvatures at xo E F(e). Since D2,(w . y)(w) = w(d)D2 h(w),

the proof is complete.

Joint system of parametrizations. Given a fixed e0 E ]a', b'[, we now construct a system of parametrizations for F(eo) compatible with all F(e)'s for e varying in a small neighborhood of e0 (so the derived estimate for Fourier transforms will be uniform in e). We make the following hypothesis:

Assumption B. For every e E ]a', b'[, F(e) admits at least i nonvanishing principal curvatures at any point, where i > 1 is a fixed integer. The plan is the following: starting from an arbitrary system of real-analytic parametrizations for F(eo), we will parametrize F(e) using the local coordinates of F(eo), by lifting them orthogonally to F(eo) (for e very close to eo). Let {(Up, -yp)}p 1 be a system of real-analytic parametrizations covering F(eo). Since F(eo) is compact, we assume M is finite. Without loss of generality, we also assume ryp(u) is analytic in a neighborhood of 90-1 so the expression -y,3(u) for u E aup makes sense. Let x0 E F(eo), say, x0 = -y,3(u0) for a given uo E U13 and a given 1 0 such that E(x, e) is injective for every e E ]eo - 6, eo + 6[. Then, there exist a sequence, {en}, converging to eo, and points, xn, on F(en), such that xn = E(Ynr en) = E(zn) en)

for distinct points Yn and zn on I'(eo). Since the xn's accumulate towards a certain x* E P(eo), yn and zn are eventually in the same coordinates neighborhood, contradicting the fact that each o-a is a diffeomorphism. We limit our considerations to the system {(U0 x ]eo -b, eo+6[, o-"')} 8 is specified by the previous theorem. We have proved:

1, where

Theorem 5.4. Let wo E Sd-1, eo E ]a', b'[, and e > 0 be arbitrarily fixed. Under Assumptions A and B, there exists a finite family of cubes of diameters less than e,

{Ua x]eo-6,eo+8[x BIN ct=1 r where [eo -6,e0+6] C ]a', b'[ and wo E B C Rd-1, and functions,

oa: Ua x]eo-6,eo+b[-- U Ir(e) Ie-eo1 (cr0(u, e))D(u e)o (u, e) = [0

...

0

1]

,

which we abbreviate DI>DQ0 = ed. Since o0(u, e) is a diffeomorphism, Do's is invertible. Thus, 04) = ed(Do-,,,)-1 =

det Do-,

et (adj Do-.),

where adj stands for the classical adjoint. The result follows from

ed(adj DQ0) _ (-1)d-1J0(u, e).

Corollary.

II Ja (u, e) II

is smooth in (u, e) E U0 x ]eo - S, eo + S[.

For fixed wo E Sd-1 and eo E ]a', b'[, let {(U0 x ]eo - 35, eo +36[,o,4, )} 1, B wo, and {Xo}a summing at 1 on Uie-eoj 2, in which case cosh = -e/(2d - 4) and cosu(') = e/(2d - 4) for j = 1, ... , d - 1. Then, cos2 h = cost U(i) for any j, so sin uW sin u(k) = ± sine h. Consequently, the (j, k)-th element of the considered matrix, (7.3), is 0 when j = k, but ±e/(2d-4) otherwise. This last quantity differs from zero (since e V E), which provides a contradiction. In conclusion, at any point in a neighborhood of x0, F(e) admits at least one nonvanishing principal curvature. Since x0 is arbitrary, this completes the proof.

0

Theorem 6.2 thus gives a polynomial decay for the Green's function, G(n, e), associated with the standard Laplacian:

Corollary. Let E _ {-2d, -2d + 4, ... , 2d - 4, 2d} U {0} and suppose e E [-2d, 2d] \ E. Then, G(n, e) = lim G(n, z) = O(InJ -I/2) zEC+

when InI --+ oo, uniformly in e on each compact and uniformly in w E Sd-1, where n = InIw is the polar form of n zA 0.

Molchanov - Vainberg Laplacian. In order to avoid convexity problems, Molchanov and Vainberg have suggested to change the discretization of the Laplacian. They have based their construction on the 2d full-diagonal neighbors of elements in Zd, instead of their 2d immediate neighbors. The constant-energy surfaces of the resulting operator are strictly convex in any dimension, as shown below. Explicitly, the Molchanov - Vainberg Laplacian (or diagonal Laplacian) is the adjacency operator of the translational invariant graph specified by the following set of points adjacent to the origin: V

By an elementary combinatorial argument, n E Zd is in the component of the origin if and only if the n(3)'s are all even or all odd. Indeed, the considered graph consists of 2d-1 connected components with set of representatives {(0, n(2), ..., n(d)); n() E {0, 1} for j = 2, ... , d}.

The graph is also specified by the following metric: d(m, n)

- tI Im - nl,,, 00

if the components of m - n have the same parity, otherwise,

where InI( = maxd=l InU) 1. Remarkably, the symbol of the Molchanov - Vainberg Laplacian factorizes:

Theorem 7.3. 2i is the operator of multiplication by (P(x) = 2d COS x(1) ... COS x(d),

where x = (X (1), ... , x(d)) 6 Td.

P. POULIN

450

PROOF. Let us denote by {e1, ... , ed} the standard basis of Zd. By Theorem 7.1 the symbol of A is the multiplication by d

v(i)ej

-1,(x) _ vEV

.El

vEVj=1

vEV

_

d

d

2d 11 COS x(j) j=1

j=1

as claimed.

Consequently, the spectrum of A is purely absolutely continuous and equal to [-2d, 2d]. Moreover,

Lemma. Suppose 0 < lel < 2d. Then, for all x E F(e), Vx4b(x) particular, r(e) defines a regular surface for such an e.

0. In

PROOF. If x E F(e), then cos x(j) 54 0 for j = 1,... , d, so ax(;) fi(x) _ Vx4D(x) II2 =4 d e2

-2de tan x(j). Thus, JI

Ed=1 tang x(j), which differs from 0, since

e#±2d. Let us investigate the constant-energy surfaces associated with 4)(x). Firstly,

let us consider the covering r(e) = {x E Rd; 4D(x) = e}. If e = 0, f(e) consists of the hyperplanes of equation x(j) = (2k + 1) for k E Z z we call and j = 1, . . . , d. These hyperplanes divide ll into open hypercubes, which cells. The cells admit a good bicoloration in the following sense: starting from a set of two colors, say, red and blue, it is possible to paint each cell in such a way that the 2d neighbors of any red cell are blue and vice versa. Let us accomplish this, the cell containing the origin being painted in red. If e = 2d, then f(e) is a discrete set consisting of the centers of the red cells. On the other hand, if e = -2d, then I'(e) consists of the centers of the blue cells. When x varies continuously, D(x) changes sign each time one of the previous hyperplanes is crossed. It follows that the connected components of f(e) are enclosed in the red cells when e > 0, each red cell containing one component. Moreover,

these components are all congruent. The situation is the same when e < 0, but replacing the red cells with the blue ones. Finally, F(e) is obtained from the previous surface by restricting f(e) to the torus, where e c ] -2d, 2d[\{0} is fixed. It follows that r(e) consists of 2d-1 identical connected components. As Molchanov and Vainberg conjectured,

Theorem 7.4. For 0 < lei < 2d any component of F(e) is strictly convex. PROOF. Suppose 0 < e < 2d, the other case being similar. Then, without loss of generality the considered component is d

( 7r 7r d SxE1-2 , 2 [ ;2dftcosx(j) =e).

j=1

ll

JJJ

Let m = d-1 and h = x(d). The equation defining the previous component becomes 2d cos x(1)

cos x('') cos h = e.


451

Since each factor in the above is positive, (7.4)

+ In cos x(m) + In cos h - In e = 0.

d In 2 + In cos x(l) +

Since the considered component is symmetric with respect to the hyperplanes x(j) = 0 and x(j) = x(l), where j, l E {1, ... , d} are distinct, it suffices to show the result on the fundamental domain h < x(l) < x('n) 4. Its Green's function is denoted by G(n - m, z) = (6,, I (A

- z)-'6n),

where m, n E Z' , z c (C+, and 6 is the Kronecker delta. By Kato's formula (A.1)

G(n, z) = i

f0

J

00

e-'tz (a0 I e'tA6n) dt.

P. POULIN

452

Recall that the symbol of A is the multiplication by OP(x) = 2 Ed x = (x(1), ... , x(d)) E Td. Hence, d

(5o

eitA6n) I

_ (27) -d

-

fi

1

Cosx(j), where

d

ein(')ke 2it cos k dk =

j=1

i(r,(3))J,n(3)

11

(2t)

j=1

at any n = ((1),. .. , n(d)) and z c C+, where Jm (t) = (1/2ir) f

,,eimke-it sink dk

is the Bessel function. It is well known that there exists a universal constant, C, such that Jm(t)I

C/3

and

I Jm(t)I

C3

for any m E Z and t E R (see [4]). Consequently, since d >, 4, for an arbitrarily fixed E > 0 I(6o

I eit05n)I cdltl-1/3ItI-1/3ItI-1/3(Itl-e/3 In(j4) I-(I-E)/3)In(j5) I-1/3 ... In(jd) I-1/3,

where (01),...,n (id) ) is a permutation of (n(1), ... , n(d)). This last estimate and the equation (A.1) then give I G(n, e + i0) I 0 in dimension 4. References

L. Erdds and M. Salmhofer, Decay of the Fourier transform of surfaces with vanishing curvature, preprint. 2. L. Erdds, M. Salmhofer, and H.-T. Yau, Quantum diffusion of the random Schrodinger evolution in the scaling limit, preprint. 3. V. Jak"sic and P. Poulin, Scattering from sparse potential on graphs, preprint. 4. L. Landau, Monotonicity and bounds on Bessel functions, Electron. J. Differ. Equ. Conf. 4 (2000), 147-154. 5. W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766-770. 6. S. Molchanov and B. Vainberg, Scattering on the system of the sparse bumps: multidimensional case, Appl. Anal. 71 (1999), no. 1-4, 167-185. 7. P. Poulin, The Molchanov - Vainberg Laplacian, Proc. Amer. Math. Soc. 135 (2007), no. 1, 77-85. 8. W. Shaban and B. Vainberg, Radiation conditions for the difference Schrodinger operators, Appl. Anal. 80 (2001), no. 3-4, 525-556. 9. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., vol. 43, Princeton Univ. Press, Princeton, NJ, 1993. 10. B. R. Vainberg, Asymptotic methods in equations of mathematical physics, Gordon and Breach 1.

Science Publisher, New York, 1989. DEPARTMENT OF MATHEMATICS AND STATISTICS, MCGILL UNIVERSITY, 805, RUE SHERBROOKE

OUEST, MONTREAL, QC H3A2K6, CANADA

Current address: Department of Mathematical Sciences, NTNU-Norwegian University of Science and Technology, 7491 Trondheim, Norway E-mail address: poulin.philippe®hotmail.com


Orthogonal Polynomials with Exponentially Decaying Recursion Coefficients Barry Simon ABSTRACT. We review recent results on necessary and sufficient conditions for measures on JR and an to yield exponential decay of the recursion coefficients of

the corresponding orthogonal polynomials. We include results on the relation of detailed asymptotics of the recursion coefficients to detailed analyticity of the measures. We present an analog of Carmona's formula for OPRL. A major role is played by the Szeg6 and Jost functions.

1. Introduction: Szego and Jost functions In broad strokes, spectral theory concerns the connection between the coefficients in differential or difference equations and the spectral measures associated to those equations. The process of going from coefficients to the measures is the direct problem, and the other direction is the inverse spectral problem. The gems of spectral theory are ones that set up one-one correspondences between classes of measures and coefficients with some properties. Examples are Verblunsky's form of Szego's theorem [25] and the Killip-Simon theorem [12]. In this paper, our goal is to describe (mainly) recent results involving such gems for orthogonal polynomials whose recursion coefficients decay exponentially. These are technically simpler systems than the L2 results just quoted but have more involved details. The two classes we discuss are orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC). For the OPRL case, we have a probability measure, dp, on R of bounded but infinite support whose orthonormal polynomials, pn(x), obey (1.1)

xpn(x) = an+lpn+1(x) + bn+lpn(x) + anpn-1(x)

with bn E R and an E (0, oo) and called Jacobi parameters. {an, bn}° 1 is a description of dp in that there is a one-one correspondence between bounded sets of such Jacobi parameters and such dps. For background discussion of OPRL, see [4,9,18,24]. 2000 Mathematics Subject Classification. Primary 42C05, 42C25, 34L05; Secondary 33C45, 33C47, 39A10.

This work is supported in part by NSF grant DMS-0140592. This is the final form of the paper. ©2007 Barry Simon 453

B. SIMON

454

For the OPUC case, IID = {z E C I Izl < 1}, and dµ is a probability measure on c7D whose support is not a finite set. The orthonormal polynomials, (Pm(z), obey the Szego recursion relation:

Z(0- (Z) = P. on+I(z) + ancn(z)

(1.2) (1.3)

c* (z) = zncpnWz) Pn = (1 - Jan 12)1/2

(1.4)

with an E lID and called Verblunsky coefficients. {an}°°_o is a description of dp in that there is a one-one correspondence between sequences of an obeying and < 1 and such d/.as. For background discussion of OPUC, see [10,17-19, 24]. The measure theoretic side of the equivalences will be in terms of a derived

object, rather than the measures themselves. For OPUC, the object is D(z), the Szego function [18, Section 2.4]. One says the Szego condition holds if and only if

dp(9) = w(O) 2B + dp

(1.5)

where dµ,, is singular and

flog(w(O)) 2B > -00

(1.6)

(which is known to be equivalent to by (1.7)

D(z) = exp

E°°_0Ianl2

(f_ ele

< oo). In that case, D(z) is defined

+ z log(w(0))

491 J

which obeys

v*(z) - D(z)-1

(1.8)

iflzl n). One can ask about the analogous approximation for OPRL. We will get the function Sn used by Dombrowski - Nevai [8]:

Let Jt be the Jacobi matrix with parameters an(Jt) =

(2.17)

(2.18)

{an(J)

bn(Jt) _

n=1 ... t-1

1

n>2

bn(J)

n = 1,...,t

0

n>.t

According to Theorem 13.6.1 (with at replaced by 1), its Jost function is (x z + 1/z) (2.19)

9e(z) = ze pe l z +

z - zpQ_l z +Z I)J

Define Se(x) by

Sel z+ z =9e(z)gezJ

(2.20)

Then, by (2.19), (2.21)

St(x) = pe(x)2 + pe-1(x)2 - xpe(x)pe-1(x)

Taking into account the different normalization (for us, "free" is ak = 1; for them, ak = 2), this is the function St(x) of Dombrowski-Nevai [8]. The approximating measure has a.c. part related to dx/Ige(z)12 on [-2,2] which is dx/Se(x). The eigenvalues of Jt are zeros of Se(x) but not all zeros since Se also vanishes if 9e (1/z) = 0, that is, at antibound state and resonance energies. For most purposes, (2.7) is a more useful representation than the one associated

to S. 3. Necessary and sufficient conditions on exponential decay The starting point of the recent results on exponential decay is the following result of Nevai-Totik for OPUC:

Theorem 3.1 ([14]). Let dp be a nontrivial probability measure on 8TD and R > 1. Then the following are equivalent: (a) (3.1)

limsupI can(dp)1l"n < R-1 n-.oo

(b) dµ,, = 0, the Szeg6 condition (1.6) holds, and D(z)-1 has an analytic continuation to {z I jzj < R}.

Remark. Since R-1 < 1, (3.1) is an expression of exponential decay.

OPS WITH EXPONENTIAL DECAY

459

The proof is easy. If (3.1) holds, Szego recursion first implies inductively that

for Iz = 1, (3.2)

-Pn+l(ei0)I

(1 + Ianl)I-IDn(eie)I

so (3.3)

sup I4)*(z)I = supI

n,Izl oo, to the Poisson point process with

POISSON STATISTICS FOR EIGENVALUES

467

intensity measure n(Eo) dx, where dx denotes the Lebesgue measure and n(Eo) is the density of states at E0. Hence, Molchanov's result [16] shows that there is no local correlation between the energy levels of the random one-dimensional operator (2.1).

The next challenge for mathematical physicists was to see whether a similar result holds for multidimensional Schrodinger operators. A positive answer to this question was given by Minami in [14], who showed that near energies where Anderson localization is expected (more precisely, in regimes where the Anderson localization holds via [3]), there is no correlation between the eigenvalues of the Anderson tight-binding model. In order to describe Minami's result, let H = -A + Vu, be the standard Anderson tight-binding model acting in 12(7Gd) and let HA = XAHXA be the truncated Schrodinger operator corresponding to the hypercube A C V. Also let GA(z) = < En(A) z)-1 be the resolvent associated to HA. If E1(A) < E2(A) < (HA are the eigenvalues of HA, then we can define the integrated density of states:

-

N(E)=n II #{j,ES(A)<E}

(2.4)

and the density of states:

n(E) = dN(E)/dE

(2.5)

Note that if n(E) > 0, then the average spacing of the eigenvalues near the energy E is of order JAI-1. Consider the rescaled spectrum (near E): (2.6)

3 (A, E) = IAI (Ej (A) - E) and the corresponding point process

(A, E) _

(2.7)

(A,E)

Minami proved in [14] that, under suitable conditions: (2.8)

(A, E) Arm (Poisson point process of intensity n(E))

This means exactly that there is no correlation between the eigenvalues of HA if A is large.

We should mention here that one of the main ingredients of the proof is the following technical lemma: (2.9)

det

]E I

Im GA (z; x, x)

(Im

GA (Z; y, X)

Im GA (z; x, y)

ImGA(z;y,y))I

< C,

-

where the constant C depends only on the distribution of the potential V = Vu,. This result is referred to as the "Minami trick." It is worth to mention that (2.9) encodes a spectacular and somewhat mysterious cancellation that has found applications to other questions in mathematical physics; see for example [13].

3. Random CMV Matrices Recent work of Cantero et al. [5] emphasized the importance of a new class of unitary random matrices called now CMV matrices. These matrices are intimately connected with the orthogonal polynomials on the unit circle (see the monographs [17,18] on the theory of orthogonal polynomials on the unit circle).

M. STOICIU

468

The CMV matrix is a five-diagonal matrix realization for the unitary operator z -p z f (z) on L2 (T; µ), where u is a nontrivial probability measure on the unit circle T (we call a measure nontrivial if it is not supported on finitely many points). For any such measure p we can apply the Gram-Schmidt procedure to the set of polynomials {1, z, z2, ... } E L2 (T, p) and get the set of monic orthogonal polynomials {(Do (z, dµ), 4), (z, dµ), 4)2(z, dµ), ... } C L2 (T; µ).

These polynomials obey the recurrence relation (3.1)

(Dk+1(z, dµ) = z(Dk (z, df2) - ak"bk (z, dµ)

k>0

where, for ck (z) _ bj zi, the reversed polynomial D* (z) is defined by 4)* (z) _ k E =o bk- z3. The recurrence coefficients {an},>o are called Verblunsky coefficients; they are complex numbers of absolute value < 1. If we apply the Gram- Schmidt algorithm to the sequence { 1, z, z-1, z2, z 2, ... } we get the set {Xo(z), X1 (Z), X2 (Z), ... }, which is a basis of L2 (T; µ). The CMV

matrix associated to the measure p is the matrix representation of the operator f (z) -* z f (z) on L2 (7;,u). It has the form: do Po 0

a1Po

PiPo

-alao -P1ao

0

0

0

0

a2Pi

-alai

c3P2

P3P2

0

P2Pi

-P2a1

-a3a2

-P3a2

0

0

0

a4P3

-c4a3

where Pk = V1 - I ak I2

Note that the Jacobi matrices obtained in a similar way for orthogonal polynomials on the real line are tridiagonal matrices. As in the case of orthogonal polynomials on the real line, an important connection between CMV matrices and monic orthogonal polynomials is (3.3)

cn(z) = det(zI -

where C(n) is the upper left n x n corner of C.

If Ian._i 1 = 1, then the CMV matrix decouples between (n - 1) and n. The upper left corner is an (n x n) unitary matrix (3.4)

C(n)

= C(n) {ao,a I'...,an-1}

We will consider random CMV matrices and study the statistical distribution of their eigenvalues. We will randomize the matrix C(n) {ao,al,...,«.n-1} by taking independent identically distributed random variables ao, a1, ... , an-2. The last variable, an-1 will be chosen to be uniformly distributed on the unit circle. As in the case of random Schrodinger operators, for various classes of n x n random CMV matrices, the local statistical distribution of the eigenvalues of these matrices will converge (as n oo) to the Poisson distribution. This fact indicates that, as n gets large, there is no local correlation between these eigenvalues. The following theorem can be found in [23] (and will be extended to singular distributions in Theorem 3.2):

Theorem 3.1. Consider the random CMV matrices C(n) = C{no,a1,...,«n_1} where ao, a1, ... , an-2 are i. i. d. random variables distributed uniformly in a disk of radius r < 1, and an-1 is another random variable independent of the previous ones and uniformly distributed on the unit circle.

POISSON STATISTICS FOR EIGENVALUES

469

Consider the space

Q. = {a = (ao, al, ... , an_2i a,_1) E D(0, r) x D(0, r) x ... x D(0, r) x T} with the probability measure IPn obtained by taking the product of the uniform (Lebesgue) measures on each D(0, r) and on T. Fix a point e'00 E T and let ((n) be the point process defined by ((n) = X:k=l Szk where {zl, z2i ... , zn} are the eigenvalues of the matrix C(n) (each eigenvalue zi depends on ao, al, ... , an-1)Then, on a fine scale (of order 1/n) near e'00, the point process ((n) converges to the Poisson point process with intensity measure ndO/(27r) (where dO/(27r) is the normalized Lebesgue measure). This means that for any fixed al < bl < a2 < b2 oo. PROOF. The proof is similar to the proof of the Theorem 3.1, with some modifications required by the fact that we use a different probability distribution for the random Verblunsky coefficients. With the exception of (3.10), all the steps outlined before hold when we replace

the uniform distribution on the disk or radius r with the uniform distribution on the circle of radius r. Lemma 3.3 (see below) shows that the conditional moments of order s with s E (0, 2) are uniformly bounded. A standard application of Holder's theorem implies that the result in Lemma 3.3 holds for all s E (0, 1). We should also mention that Aizenman's theorem for orthogonal polynomials on the unit circle (see [19]) holds for Verblunsky coefficients uniformly distributed

on the circle of radius r. As in the proof of Theorem 3.1, we can now derive the exponential decay of the fractional moments of the matrix elements of the resolvent of C(n')

Following the same route as in the proof of Theorem 3.1 (the eigenfunctions are exponentially localized, the matrix C(n) can be decoupled, and the decoupled

matrices contribute at most one eigenvalue in each interval of size 0(1/n)) we can conclude that (3.15) holds, that is, the local statistical distribution of the eigenvalues of

is Poisson.

We will now give the analog of (3.10) for the case of Verblunsky coefficients uniformly distributed on C(0, r).

M. STOICIU

472

Lemma 3.3. For any s E (0, a ), any k, 1 < k < n, and any choice of a = a0, al, ... , a,,_1 as in Theorem 3.2, (3.16)

E(lFkk(z,C(n))13

{ai}jlk)

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