Lecture Notes in Applied Differential Equations of Mathematical Physics

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Author: Luiz C. L. Botelho

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Lecture Notes in

Applied Differential Equations of Mathematical Physics

This page intentionally left blank

Lecture Notes in

Applied Differential Equations of Mathematical Physics Luiz C L Botelho

Federal University, Brazil

World Scientific NEW JERSEY

•

LONDON

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI


Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

LECTURE NOTES IN APPLIED DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-281-457-9 ISBN-10 981-281-457-4

Printed in Singapore.

ZhangJi - Lec Notes in Applied.pmd

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Foreword

“. . . Both Mathematics and Physics are in a state of crisis in present days, from intellectual and social sides simultaneously. Both have much to contribute to each other’s disciplines, and I believe that a renewal of their traditional liaison would go far to cure at least the intellectual side of the malaise felt by workers in both disciplines . . . ” — Robert Hermann in Lectures in Mathematical Physics — Vol. I & Vol. II. My aim in this set of informal lecture notes on applied mathematical analysis is to present mathematical author’s original research material (Chaps. 3–9 and appendixes) revised and amplified of our two previous work on path-integrals in statistical and quantum physics and strongly focused at a pure and applied mathematical audience of readers. We have a strong believe that mathematics students can find a source of inspiration to stimulate mutual interaction of their discipline and physics and thus, undertaking the reading of physics books, talking with physicists and try to read our previous two monographs in classical and quantum physics. The special emphasis in this supplementary volume 3 is on advanced mathematical analysis methods with its applications to solve partial differential equations in finite and infinite dimensions arising in applied settings. Another objective behind writing this set of author’s mostly original results in the form of lectures is the hope it could serve to show that the “aridity” and “sterility” that one finds in much modern “estimate mathematical analysis” can be conterweighted side by side by mathematical analysis formulae exactly used in the modern applications. A word on the “methodology” in our set of lecture notes as a set of complementary notes we feel that the students should read the chapter with a pencil and paper at hand, after which they should make additional studies in the specific topic in the literature and thus present their studies in seminars to others students with all lectures mathematical details worked out. Luiz C.L. Botelho Professor vii

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Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1.

Elementary Aspects of Potential Theory in Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . .

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Laplace Differential Operator and the Poisson–Dirichlet Potential Problem . . . . . . . 1.3. The Dirichlet Problem in Connected Planar Regions: A Conformal Transformation Method for Green Functions in String Theory . . . . . . . . . . . . . . . . . . . . . . . 1.4. Hilbert Spaces Methods in the Poisson Problem . . . . . . 1.5. The Abstract Formulation of the Poisson Problem . . . . 1.6. Potential Theory for the Wave Equation in R3 — Kirchhoff Potentials (Spherical Means) . . . . . . . . . . . 1.7. The Dirichlet Problem for the Diffusion Equation — Seminar Exercises . . . . . . . . . . . . . . . . . . . . . . 1.8. The Potential Theory in Distributional Spaces — The Gelfand–Chilov Method . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Light Deflection on de-Sitter Space . . . . . . . . A.1. The Light Deflection . . . . . . . . . . . . . . . . . . . . . A.2. The Trajectory Motion Equations . . . . . . . . . . . . . A.3. On the Topology of the Euclidean Space–Time . . . . . . Chapter 2.

vii

1

.

1

.

1

. . .

13 18 21

.

23

.

27

. . . . . .

28 30 31 31 34 36

Scattering Theory in Non-Relativistic One-Body Short-Range Quantum Mechanics: M¨ oller Wave Operators and Asymptotic Completeness . . . . . . . . . . . . . . . . . 39 ix

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Lecture Notes in Applied Differential Equations of Mathematical Physics

2.1. The Wave Operators in One-Body Quantum Mechanics . . 2.2. Asymptotic Properties of States in the Continuous Spectra of the Enss Hamiltonian . . . . . . . . . . . . . . . . . . . . 2.3. The Enss Proof of the Non-Relativistic One-Body Quantum Mechanical Scattering . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3.

4.1. 4.2. 4.3. 4.4.

44 48 49 50 54 56

On the Hilbert Space Integration Method for the Wave Equation and Some Applications to Wave Physics . . . . . . . 58

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The Abstract Spectral Method — The Nondissipative Case . . . . . . . . . . . . . . . . . . . . 3.3. The Abstract Spectral Method — The Dissipative Case 3.4. The Wave Equation “Path-Integral” Propagator . . . . 3.5. On The Existence of Wave-Scattering Operators . . . . 3.6. Exponential Stability in Two-Dimensional Magneto-Elasticity: A Proof on a Dissipative Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. An Abstract Semilinear Klein Gordon Wave Equation — Existence and Uniqueness . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Exponential Stability in Two-Dimensional Magneto-Elastic: Another Proof . . . . . . . . Appendix B. Probability Theory in Terms of Functional Integrals and the Minlos Theorem . . . . . . . Chapter 4.

39

. .

58

. . . .

. . . .

59 63 65 68

. .

70

. . . .

73 75

. .

76

. .

80

Nonlinear Diffusion and Wave-Damped Propagation: Weak Solutions and Statistical Turbulence Behavior . . . . . . . . . . . . 84

Introduction . . . . . . . . . . . . . . . . . . . The Theorem for Parabolic Nonlinear Diffusion The Hyperbolic Nonlinear Damping . . . . . . A Path-Integral Solution for the Parabolic Nonlinear Diffusion . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

84 85 91

. . . . . . .

94

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Contents

4.5. Random Anomalous Diffusion, A Semigroup Approach References . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D. Probability Theory in Terms of Functional Integrals and the Minlos Theorem — An Overview . . . . . . . . . . . . . . . . . . Chapter 5.

. . . . .

. . . . .

. . . . .

. . . 107

Domains of Bosonic Functional Integrals and Some Applications to the Mathematical Physics of Path-Integrals and String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The Euclidean Schwinger Generating Functional as a Functional Fourier Transform . . . . . . . . . . . . . . . . 5.3. The Support of Functional Measures — The Minlos Theorem . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Some Rigorous Quantum Field Path-Integral in the Analytical Regularization Scheme . . . . . . . . . . . 5.5. Remarks on the Theory of Integration of Functionals on Distributional Spaces and Hilbert–Banach Spaces . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. On the Support Evaluations of Gaussian Measures Appendix C. Some Calculations of the Q.C.D. Fermion Functional Determinant in Two-Dimensions and (Q.E.D.)2 Solubility . . . . . . . . . . . . . . . Appendix D. Functional Determinants Evaluations on the Seeley Approach . . . . . . . . . . . . . . . Chapter 6.

96 102 103 105 106

110 111 114 124 131 143 144 148

150 160

Basic Integral Representations in Mathematical Analysis of Euclidean Functional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.1. On the Riesz–Markov Theorem . . . . . . . . . . . . . . . . 169 6.2. The L. Schwartz Representation Theorem on C ∞ (Ω) (Distribution Theory) . . . . . . . . . . . . . . . . . . . . . 173

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6.3. Equivalence of Gaussian Measures in Hilbert Spaces and Functional Jacobians . . . . . . . . . . . . . . . . . 6.4. On the Weak Poisson Problem in Infinite Dimension . . 6.5. The Path-Integral Triviality Argument . . . . . . . . . . 6.6. The Loop Space Argument for the Thirring Model Triviality . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Path-Integral Solution for a Two-Dimensional Model with Axial-Vector-Current-Pseudoscalar Derivative Interaction . . . . . . . . . . . . . . Appendix B. Path Integral Bosonization for an Abelian Nonrenormalizable Axial Four-Dimensional Fermion Model . . . . . . . . . . . . . . . . . . Chapter 7.

. . 193 . . 196

. . 197

. . 202

Nonlinear Diffusion in RD and Hilbert Spaces: A Path-Integral Study . . . . . . . . . . . . . . . . . 209

7.1. Introduction . . . . . . . . . . . . . . . 7.2. The Nonlinear Diffusion . . . . . . . . . 7.3. The Linear Diffusion in the Space L2 (Ω) References . . . . . . . . . . . . . . . . . . . Appendix A. The Aubin–Lion Theorem . . . Appendix B. The Linear Diffusion Equation Chapter 8.

. . 178 . . 183 . . 186

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

209 209 216 222 223 225

On the Ergodic Theorem . . . . . . . . . . . . . . . . . . . . . . 227

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. On the Detailed Mathematical Proof of the RAGE Theorem 8.3. On the Boltzmann Ergodic Theorem in Classical Mechanics as a Result of the RAGE Theorem . . . . . . . . 8.4. On the Invariant Ergodic Functional Measure for Some Nonlinear Wave Equations . . . . . . . . . . . . . . . . . . 8.5. An Ergodic Theorem in Banach Spaces and Applications to Stochastic–Langevin Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . 8.6. The Existence and Uniqueness Results for Some Nonlinear Wave Motions in 2D . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. On Sequences of Random Measureson Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . .

227 228 231 234

241 245 249 249

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xiii

Appendix B. On the Existence of Periodic Orbits in a Class of Mechanical Hamiltonian Systems —An Elementary Mathematical Analysis . . . . . . . . . . . . . . . . 251 B.1. Elementary May be Deep — T. Kato . . . . . . . . . . . . 251 Chapter 9.

Some Comments on Sampling of Ergodic Process: An Ergodic Theorem and Turbulent Pressure Fluctuations . . . . . . . . . . . 254

9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. A Rigorous Mathematical Proof of the Ergodic Theorem for Wide-Sense Stationary Stochastic Process . . 9.3. A Sampling Theorem for Ergodic Process . . . . . . . . . . 9.4. A Model for the Turbulent Pressure Fluctuations (Random Vibrations Transmission) . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Chapters 1 and 9 — On the Uniform Convergence of Orthogonal Series — Some Comments . . . . . A.1. Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Regular Sturm–Liouville Problem . . . . . . . . . . . . . . . Appendix B. On the M¨ untz–Szasz Theorem on Commutative Banach Algebras . . . . . . . . . . . . . . . . . . . B.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. Feynman Path-Integral Representations for the Classical Harmonic Oscillator with Stochastic Frequency . . . . . . . . . . . . . . C.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . C.2. The Green Function for External Forcing . . . . . . . . . . C.3. The Homogeneous Problem . . . . . . . . . . . . . . . . . . Appendix D. An Elementary Comment on the Zeros of the Zeta Function (on the Riemann’s Conjecture) . . . . . . D.1. Introduction — “Elementary May be Deep” . . . . . . . . . D.2. On the Equivalent Conjecture D.1 . . . . . . . . . . . . . .

254 255 257 262 265 266 266 270 272 272

274 274 274 278 282 282 283

Chapter 10. Some Studies on Functional Integrals Representations for Fluid Motion with Random Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 287

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10.2. The Functional Integral for Initial Fluid Velocity Random Conditions . . . . . . . . . . . . . . . . . . . . 10.3. An Exactly Soluble Path-Integral Model for Stochastic Beltrami Fluxes and its String Properties . . . . . . . . 10.4. A Complex Trajectory Path-Integral Representation for the Burger–Beltrami Fluid Flux . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. A Perturbative Solution of the Burgers Equation Through the Banach Fixed Point Theorem . . . Appendix B. Some Comments on the Support of Functional Measures in Hilbert Space . . . . . . . . . . . . .

. 287 . 292 . 298 . 304 . 304 . 309

Chapter 11. The Atiyah–Singer Index Theorem: A Heat Kernel (PDE’s) Proof . . . . . . . . . . . . . . . . . 312 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Appendix A. Normalized Ricci Fluxes in Closed Riemann Surfaces and the Dirac Operator in the Presence of an Abelian Gauge Connection . . . . . . . . . . 320 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

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Chapter 1 Elementary Aspects of Potential Theory in Mathematical Physics 1.1. Introduction Many problems of elementary classical mathematical-physical analysis are connected with the solution through integral representations of the Dirichlet, Newmann, and Poisson problems for the Laplacian operator, including its generalization to the Cauchy problem for wave and diffusion equations. In this chapter, we present the basic and introductory mathematical methods analysis used in obtaining such above-mentioned integral representations.1

1.2. The Laplace Differential Operator and the Poisson–Dirichlet Potential Problem Let Ω be an open-bounded set in RN . We define C 2 (Ω) to be the vectorial space composed by those functions f (x) ≡ f (x1 , . . . , xn ) ≡ f (r ). (Here we i ei denoting the canonical adopt the physical notation r = N i=1 x ei with N vectorial basis of R ), continuously differentiable until the second order in Ω, namely, ∂ 2 f (x) ∈ C(Ω), ∂xk ∂xm

for any (k, m) ∈ {1, . . . , n}.

Let us consider the following linear transformation with domain C 2 (Ω) and range C(Ω) ∆(N ) : C 2 (Ω → C(Ω)

f (x) → (∆f )(x) =

n ∂2 ∂x2i i=1

f (x1 , . . . , xn ).

(1.1)

We now have the following simple result about the kernel of the above linear transformation in R3 (holding true for any RN , with N ≥ 2). 1

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Lemma 1.1. The kernel of the Laplacean operator as defined by Eq. (1.1) has infinite dimension in R3 . 2 ¯ f Proof. Let f¯ ∈ C 2 (C × R) such that ∂∂2 w (w, t) exists and the function 1 2 3 defined below (x = x, x = y, x = z — the usual classical notation in potential theory in the physical space R3 )

Ut (x, y, z) = f¯(ix cos t + iy sin t + z, t).

(1.2)

We wish to show that ∆(3) Ut (x, y, z) = 0.

(1.3)

This result is a simple consequence of the following elementary identities: ∂2 ¯ ∂2 f (w, t)|w=z+ix cos t+iy sin t , U (x, y, z) = t ∂2z2 ∂2w

(1.4a)

∂2 ∂2 Ut (x, y, z) = 2 f¯(w, t)|w=z+ix cos t+iy sin t ((i cos t)2 ), (1.4b) 2 2 ∂ x ∂ w ∂2 ∂2 Ut (x, y, z) = 2 f¯(w, t)|w=z+ix cos t+iy sin t (i sin t)2 . 2 2 ∂ y ∂ w

(1.4c)

As a consequence ∂2 ∆(3) Ut (x, y, z) = 2 f (w, t)w=z+ix cos t+iy sin t (1 − cos2 t − sin2 t) ≡ 0. ∂ w (1.5) ˜ Note that a whole class of harmonic functions H(Ω) in the class of C 2 (Ω)-functions in the kernel of the Laplacean operator Eq. (1.3) may be obtained from Eq. (1.3):

b

dt g(t)(f¯(w, t)|w=ix cos t+i sin ty+z ),

U (x, y, z) = a

and for the special function below: f¯(w, t) = wn eiαt ,

(α ∈ R, n ∈ Z),

(1.6)

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3

Elementary Aspects of Potential Theory in Mathematical Physics

we have the harmonic legendre polynomials in the three-dimensional spherical ball π U (x, y, z) = eiαt (z + ix cos t + iy sin t)n dt −π

π

dt eiαt (r cos θ + ir sin θ cos ϕ cos t + ir sin θ sin ϕ sin t)n

= −π

= 2rn eiαϕ

π

dt cos(αt)(cos θ + (sin θ sin t)n 0

= rn eiαϕ Pn,α (cos θ).

(1.7)

After these preliminary elementary remarks we pass on the discussion of the famous classical Poisson–Newton problem: Let Ω ⊂ R3 be a bounded open set of R3 with a boundary ∂Ω being an orientable simply connected C 2 -surface. We search for the solution of the nonhomogeneous linear problem in Ω through an integral representation (the Laplacean Green function). −∆3 U (x, y, z) = f (x, y, z).

(1.8)

¯ (Ω ¯ = Ω ∪ ∂Ω). Here f (x, y, z) ∈ C 1 (Ω) We have the basic result in answering the Poisson–Newton problem. Theorem 1.1. The Laplacean operator ˜ ¯ → C 1 (Ω) −∆3 : C 2 (Ω) H(Ω)

(1.9)

is inversible, and its inverse has the following integral representation: f (x , y , z ) 1 dx dy dz U (x, y, z) = 4π (x − x )2 + (y − y )2 + (z − z )2 Ω def

≡

Ω

d3r f (r ) · G(0) (|r − r |).

(1.10)

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Proof. Let us re-introduce the “classic mathematical-physicist” vectorial notation in our formulas r = (x, y, z);

r = (x , y , z ).

(1.11)

Let us consider the δ-approximant solution for the problem 1 (δ) d3r f (r ) G(0) (|r − r |). U (δ) (r ) = 4π Ω Here the “regularized” Green function is given by  1  for |r − r | > δ,   |r − r | , (δ) G(0) |r − r | =

 1 |r − r |2   , for |r − r | ≤ δ. 3− 2δ δ2

(1.12)

(1.13)

Note that the above “regularized” Green function is a continuous function in both variables, i.e. G(0) (|r − r |) ∈ C(Ω × Ω)(since G(0) (δ + ) = G(0) (δ − )). (δ)

(δ)

We now have the estimate for δ → 0: 1 (δ) 2 sup |U − U |(r ) ≤ f L2(Ω) ×

r∈Ω

√ ≤ f L2(Ω) ( 2πδ),

(δ)

2 d rG(δ) (|r − r |) 3

12

Bδ (0)

1 2

(1.14)

which shows the uniform convergence of the sequence of approximate solutions to the (well-defined) function U (r ) by the Weierstrass uniform convergence criterium. That the integral representation Eq. (1.10) defines a well-defined function comes straightforwardly from the estimate around our infinitesimal ball Bε (r ) = {r | ||r − r| ≤ ε}. ε 1 1 r ) 3 f ( 2 2 d r × 4π dv · v · ≤ f 2 L (Ω) |r − r | v2 0 Bε ( r) 1

1

≤ (4πε) 2 f L2 2(Ω) < ∞.

(1.15)

As a second step to the proof of Theorem 1.1, we point out that the sequence of functions 1 ∇ · U (δ) = d3r f (r )∇ · G(δ) (|r − r |) (1.16) 4π Ω

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5


¯ converges uniformly to the (well-defined) function in C(Ω) I=+

3

d r f (r )∇r Ω

1 4π|r − r |

.

(1.17)

The proof of the above made claim is a consequence of the following estimate:

2 |r − r |2 (x − x ) d3r ∇rx − + = 0. δ2 |r − r |3 Bδ ( r)

lim

δ→0

(1.18)

Let us now pass on the problem of evaluating second derivatives. In this case we have the Gauss theorem (integration by parts in R3 !)

1 |r − r | Ω

1 3 d r f (r ) ∇r =− |r − r | Ω

1 ∠r ) dS(r ) cos(N f (r ) = −O |r − r | ∂Ω 1 . d3r ∇r · f (r ) + | r − r | Ω

d3r f (r ) ∇r

(1.19)

∠r ) is the cosine between the normal field of the globally oriHere cos(N ented surface ∂Ω and the point r, argument of the function U (r ). At this ¯ However, Eq. (1.19) has a rigpoint we are using the fact that f ∈ C 1 (Ω). 2 ¯ orous meaning for f ∈ L (Ω) (left as an exercise to our diligent reader!). It is important to remark that at this point of our exposition all these topological–geometrical constraints imposed on the surface ∂Ω by means of Gauss theorem must be imposed as complementary hypothesis on the geometrical nature of the Poisson problem. By deriving a second time the left-hand side of Eq. (1.19) and taking into account that in the surface integral we always have r = r (we do not evaluate the solution U (r ) at the boundary points) and obviously ∇r f (r ) ≡ 0, we arrive at the following result (without bothering ourselves

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with the 4π-overall factor in Eq. (1.10)):

1 ∠r )dS(r ) ∆r U (r ) = − O f (r ) ∇r cos(N |r − r | ∂Ω

1 d3r ∇r f (r ) − f (r ) ∇r + |r − r | Ω

1 = − O f (r ) ∇r cos(N ∠r )dS(r ) |r − r | ∂Ω

1 ∠r )dS(r ) f (r ) − f (r ) ∇r + O cos( N |r − r | ∂Ω

1 3 − (f (r ) − f (r )) ∆r (1.20) d r . |r − r | Ω Now we have that f (r ) O

∇r

∂Ω

1 |r − r |

= f (r ) O ∂Ω

cos(N ∠r )dS(r )

· r N ds(r ) = 4πf (r ) |r |3

(1.21)

and (with xa = (x, y, z) and a = 1, 2, 3):

3 (xa − xa )2 3 (f (r ) − f (r ) − + d r |r − r |3 |r − r |5 Bε ( r) ≤

sup r ∈Bε ( r)

|Df |(r ) × 3

+ Bε ( r)

d3r

Bε ( r)

|xa − xa |2 |r − r | 3 d r |r − r |5

|r − r | |r − r |3

ε→0 . →0

(1.22)

Precisely, to obtain this estimate, we need the condition that the data ¯ (including the continuity of the f (r ) must belong to the space C 1 (Ω) derivative at the boundary ∂Ω!). As a general exercise in this section we left to our reader to prove Theorem 1.1 in the general case of the domain Ω ⊂ RN (N ≥ 3) with the Green function in RN . Γ N2 √ N × U (r ) = dN r f (r )|r − r |2−N . (1.23) (N − 2)2( π) Ω

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Note that the regularized Green function G(δ) is now given by G(δ) |r, r )  1    |r − r |N −2 =

N  N −2 δ 2−N |r − r | 2   − 2− (N − 2)wN N 2 δ

if |r − r | > δ, if |r − r | ≤ δ. (1.24)

In R2 , we can show that (left to our readers)

1 1 d2r g ) . U (r ) = − f ( r 2π |r − r | Ω

(1.25)

This case can be seen as coming from the general case Eq. (1.23) after considering the (distributional) limit 2 Γ 2 1 1 (2−N )g| r− r | √ e g|r − r |. lim (1.26) =− N →2 2( π)2 N − 2 2π We consider now the well-known Dirichlet problem: Determine a U (r ) ∈ ¯ such that C 2 (Ω) ∩ C 1 (Ω)

∆ U (r ) = 0, U (r ) = g(r ) ∂Ω

for r ∈ Ω, ∂Ω

(1.27)

,

a very useful problem in string path integrals for Ω ⊂ R2 . By a simple application of the first Green identity we obtain the uniqueness property for the Dirichlet problem Ω

2 3 U · ∆U +|∇U | (r )d r = O (U · ∇N U )(r )dΓ(r ) = 0. 0

(1.28)

∂Ω

As a consequence, |∇U |2 (r )d3r = 0 ⇔ U (r ) ≡ 0

in Ω.

(1.29)

Ω

An explicit solution for Eq. (1.27) is easily obtained by considering the Poincaré method of considering a Green function of the structural form below.

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Let G(r, r ) = H(r, r ) +

1 , 4π|r − r |

(1.30)

where H(r, r ) is a harmonic function in Ω in relation to the variable r , satisfying the following condition: 1 . (1.31) H(r, r )|r ∈∂Ω = − 4π|r − r | r ∈∂Ω Then we have the following explicit representation:

Theorem 1.2. We have the integral representation for the Dirichlet problem 1 − g( r )∇ G( r , r ) dS( r ) U (r ) = O 4π N ∂Ω (−1/2π if r ∈ ∂Ω − exercise)

1 1 =− ) . (1.32) g(r )∇N H(r, r ) + O dS( r 4π 4π|r − r | ∂Ω Proof. Let us consider the regularized form for the Green function as proposed by the Poincaré ansatz

Here

with

G(δ) (r, r ) = H (δ) (r, r ) + U (δ) (r − r ).

(1.33)

 ∆r H (δ) (r, r ) = 0, H (δ) (r, r ) = −U (δ) (r − r ), r ∈∂Ω

(1.34)

   

1 , 4π| r − r | (δ) U (r − r ) =

 1 1 |r − r |2   , 3− 4π 2δ δ2

if |r − r | > δ, (1.35) if |r − r | ≤ δ.

By applying the second Green identity to the pair of functions U (r ) and G(δ) (r, r ), we obtain the relation U (r ) ∆r U (δ) (r, r ) d3r Ω

= O ∂Ω

U (r ) ∇N G(δ) (r, r )dS(r ).

(1.36)

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By noting that we have explicitly the Laplacean action on the regularized function U (δ) :  0, if |r − r | > δ,

(δ) ∆r U (r, r ) = (1.37) 3 − , if |r − r | ≤ δ, 3 4πδ and by taking the limit of δ → 0, it yields the integral representation Eq. (1.30). Our problem now is to obtain an explicit procedure to determine the Harmonic piece of the Green function, namely, H(r, r ). A solution was first given by H. Poincaré through the use of the so-called double-layer potential Φ(r, r ), satisfying the Poincaré integral equation in L2 (∂Ω, dS):

1 1 1 Φ(r, r ) − O ∇ − = Φ(r, w) × dS(w). N 4π|r − r | 2 4π |r − r | ∂Ω (1.38) Note the somewhat geometrical complexity of the geometrical measure dS on the surface ∂Ω, when the integral equation above is solved by the standard Fixed Point Theorems.1,2 After solving Eq. (1.38), Poincaré showed that one easily has a formula for the function H(r, r ):

1 H(r, r ) = O Φ(r, r )∇N − (1.39a) dS(r ). | 4π| r − r ∂Ω Another point worth to call the reader’s attention is that it appears more straightforward to consider the following integral equation for the surface derivative of the harmonic function U (r ). Since

1 1 1 O ∇ U (r ) − U (r )∇N dS(r ), U (r ) = 4π ∂Ω |r − r | N |r − r | (1.39b) we have (r ) · N (r ) = ∇ U (r ) = ρ(r ) r U ∇ N

r )

ρ( (r − r ) 1 = N (r ) · ∇N U (r ) O − 4π ∂Ω |r − r |3

1 −g(r )∇N ∇N dS(r ). |r − r |

(1.39c)

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One can in principle solve the above-written integral equation in L (∂Ω, dS(r )) by the Banach fixed point theorem at least for small volumes and for the normal derivative of the harmonic function U (r ) and then using again the Liouville integral representation Eq. (1.39b) to determine U (r ) in Ω. A more useful approach for the explicit determination of the harmonic term H(r, r ) in the Poincaré–Green function is by means of the eigenvalue problem for our Laplacean operator with a Dirichlet condition 2

−∆ ϕn (r ) = λn ϕn (r ) ϕn (r ) = 0 ∂Ω

for r ∈ Ω,

(1.40)

for r ∈ ∂Ω,

or in the equivalent integral form (see Theorem 1.1) ϕn (r ) d3r µn ϕn (r ) = 4π|r − r | Ω with 1 µn = = λn

(1.41)

|∇ϕn | (r )d r

ϕ2n (r )d3r Ω

2

3

> 0.

Ω

Now it is straightforward to see that

(∇N ϕn (r ))dS(r ) ϕn (r ) 1 d3r G(r, r )ϕn (r ) = − O λn λn 4π|r − r | Ω ∂Ω

r )) 3 r ϕn ( (−∆ + H(r, r ) d r . λn Ω (1.42) At the point we left as an exercise to our reader to show that the surface integral and the volume integral cancel out. By the usual Mercer theorem (if supr∈Ω¯ |ϕn (r )| ≤ M , for ∀ n ∈ Z+ ), we have the uniform convergence of the spectral λ series defining the Dirichlet n < ∞, Green Function as a result that limn→∞ |n| 2 |ϕn (r )ϕn (r )| 1 2 ≤M < ∞. (1.43) |G(r, r )| ≤ λn |n|2 3 3 n∈Z

n∈Z

The same result holds true for the case of the Newmann problem written below:  ∆ U (r ) = 0, for r ∈ Ω,

(1.44a) ∇N U (r ) = g(r ) O g(r )dS(r ) ≡ 0 , ∂Ω

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with

U (r ) = + O

g(r )G(r, r )dS(r ),

ch01

11

(1.44b)

∂Ω

where G(r, r ) =

βn (r )βn (r ) n

λn

,

and with the eigenfunction/eigenvalue problem −∆ βn (r ) = λn βn (r ), for r ∈ Ω, ∇N · βn (r ) = 0,

for r ∈ ∂Ω.

(1.44c)

(1.45)

Finally, it is left to our readers as a highly nontrivial exercise to generalize all the above results (their proofs!) for the case of Ω possessing a Riemannian structure: The famous Laplace–Beltrami operator acting on scalar functions into the domain (Ω, gab (x)), Ω ⊂ RN , ∂a = ∂x∂ a (a = 1, . . . , n) 1 √ −∆g U = − √ ∂a ( g g ab ∂b ) U (xa ) (1.46) g mathematical object instrumental in quantum geometry of strings and quantum gravity.4 For instance, for two-dimensional Riemann manifolds, one can always locally parametrize the Riemann metric into the conformal form in the trivial topological Riemann surface sector gab (x, y) = ρ(x, y)δab . An important application in potential theory is the use of the famous multipole expansion inside the Laplacean Green function (r = r|, r = |r |):

3 r · r 1 r · r 1 (r )2 = + 3 + − + ··· . (1.47a) |r − r | r r 2π r2 2r3 A useful explicit expression for the Laplace Green function can be obtained for the Ball Ba (0) = {r ∈ R3 | |r| = r ≤ a} in spherical coordinates 2π

π f (θ , ϕ ) a(r2 − a2 ) U (r, θ, ϕ) = dϕ sin θ dθ , 4π (a2 + r2 − 2ar cos Ω) 0 0 (1.47b) where the composed angle in the above-written formula is defined as (1.47c) Ω (θ, θ ); (ϕ, ϕ ) = cos θ cos θ + sin θ sin θ cos(ϕ − ϕ ).

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Another point worth remarking is that Eqs. (1.38), (1.39a) and (1.39c) have potentialities for problems involving weak perturbations of the domain boundary ∂Ω(0) → ∂Ω(0) + ε ∂ Ω(1) , since we have for instance the explicit perturbation result for the integrand of Eq. (1.38) for a given parametrization of the perturbed surface  (0) = W1 (u, v)ˆ (1) (u, v) W e1 + W2 (u, v)ˆ e2 + W3 (u, v)ˆ e3 = W + εW        (1) (0) (1) (0)   (u, v) = (W + εW ) × (W + εW ) = N (0) + εN (1) + · · ·  N   (0) + εW (1) ) × (W (0) + εW (1) )| 3  |(W R

 ·(0) εW (1) )  1 1 (r − W   = + + ···   (0) − εW (1) | (0) | (0) |3  |r − W |r − W |r − W       ) = Φ(0) (r, W (0) ) + ε∇Φ(1) (r, W (0) ) · ∇W (1) + · · · . Φ(r, W (1.47d) Rigorous mathematical analysis of such ε-perturbative expansion is left to our mathematically oriented readers, including the open problem in advanced calculus to find explicitly the Green function for a toroidal surface and the M¨ oebius surface band and its weakly random perturbations. An important mathematical question is the one about the wellposedness properties of the Dirichlet problem in relation to the uniform convergence of the datum. We have the following result in this direction: Lemma 1.2. If in Eq. (1.27) gn (r ) ∈ C 0 (∂Ω) converges uniformly for a function g(r ) ∈ C0 (∂Ω), then we have the uniform convergence of the solutions (1.32) associated with the sequence data {gn } to the solution associated with g(r ). ¯ be a compact in Ω such that dist(G, ¯ ∂Ω) = δ > 0. Since we Proof. Let G have the following estimate for the Green function 1 ≤ 1 (1.47e) sup sup |G(r, r )| ≤ r − r | 4πδ r ∈∂Ω r ∈∂Ω 4π| r ∈G; r∈G; and so on, sup |Un (r ) − U (r )| ≤

¯ r ∈G

Area(∂Ω) ||gn − g||C 0 (∂Ω) → 0. 4πδ

(1.47f)

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Finally, let us announce the combined problem of Poisson–Dirichlet in R and its associated integral representation. 3

Theorem 1.3. Let Ω ⊂ R3 be a region satisfying the hypothesis of the ¯ the solution of Gauss theorem. Let g(r ) and f (r ) be elements in C 1 (Ω); 2 1 ¯ this problem in C (Ω) ∩ C (Ω)

∆U (r ) = f (r ); r ∈ Ω, U (r )∂Ω = g(r )∂Ω ,

(1.48)

is (uniquely) given by the integral representation written below: U (r ) =

1 4π

d3r Ω

f (r ) |r − r |

+ v(r ).

(1.49)

Here the function v(r ) is the harmonic function related to the boundary data

v(r )∂Ω

∆ v(r ) = 0,

r ) 3 f ( + g( r ) = − d r . |r − r | Ω r ∈∂Ω

(1.50)

1.3. The Dirichlet Problem in Connected Planar Regions: A Conformal Transformation Method for Green Functions in String Theory Let Ω be a simply connected region in R2 and W ⊂ R2 another region with similar topological properties. A coordinate transformation (diffeomorphism) of W into Ω F

W ⊂ R2 −→

Ω ⊂ R2

(ξ, η)

(x(ξ, η), y(ξ, η))

(1.51)

is called conformal when one has the validity of the conditions in W :  ∂x(ξ, η) ∂y(ξ, η)   = ,  ∂ξ ∂η    ∂x(ξ, η) = − ∂y(ξ, η) . ∂η ∂ξ

(1.52)

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In this case one can see that z = x + iy = f −1 (ξ + iη) and f (z) is an analytic function in Ω. We have the following relation among the functional expressions of the operator Laplaceans among the two given domains W and Ω: 2 2 ∂y ∂x ¯ y) + , (1.53) ∆ξ,η U (ξ, η) = ∆x,y U(x, ∂ξ ∂η ε+iη=f (x+iy) ¯ where U (ξ(x, y), η(x, y)) ≡ U(x, y). In the case of the given domain transformation F possesing a unique extension as a diffeomorphism between the domain boundaries Γ1 = ∂W and Γ2 = ∂Ω, we have the following relationships among the Dirichlet problems in each region:   ∆ξ,η U (ξ, η) = 0 (ξ, η) ∈ W, (a) (1.54)  U (ξ, η) = f (ξ, η) , Γ1 Γ1  ¯ (x, y) = 0  ∆x,y U (b)

 U ¯ (x, y)

Γ2

(x, y) ∈ Ω,

= f¯(x, y)Γ .

(1.55)

2

Then, ¯ (x, y). U (ξ, η) = U

(1.56)

It is a deep theorem of B. Riemann in the theory of one-complex variables that given two simply connected bounded domains where the boundaries are curved with continuous curvature there is a conformal transformation of the given domains realizing a coordinate (sense preserving) of the boundaries.3 A proof of such a result can be envisaged along the following arguments: Firstly, one considers the universal domain B1 (0) = {(x, y) | x2 + y 2 < 1} as the domain Ω. Secondly, one considers a triangularization T(n) of the region W . It is possible to write explicitly a conformal transformation of T(n) into B1 (0) by means of the well-known Schwartz– Christoffell transformation of a polygon into the disc B1 (0) denoted by fn (z). One can show now that the family of functions {fn (z)} is a uniformly bounded (compact) set in the space H(Ω) (Holomorphic function in Ω). By choosing the family of triangulations Tn in such a way that Tn ⊂ Tn+1 ,

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one can expect that fn (z) converges into H(Ω) to a function f (z) ∈ H(Ω) carrying the conformal transformation of W into B1 (0). (Details of our suggested proof are left to our mathematically oriented readers again.) At this point we intend to present a Hilbert space approach to construct explicitly such canonical conformal mapping of the domain W into the disc B1 (0). Let H = {F (W, C), f : W → C, f is a holomorphic function in W }. We introduce the following Hilbert space inner product in this space of holomorphic (analytical) functions in W : 1 dz ∧ d¯ z f (z)g(z). (1.57) f, gc = 2i W We have the following straightforward result for two elements of the Hilbert space (H, , c ) obtained by a simple application of the Green ¯ ⊂ W: theorem in the plane for any domain W 1 ¯ g(z)h(z) dz = (g(z) h (z))dz ∧ d¯ z. (1.58) 2i ¯ =Γ1 ¯ ∂W W ¯ = Br (z0 ) in Eq. (1.58), one obtains If one considers h(z) = z −z0 and W that g(z) r2 2 g(z0 ). (1.59) g(z)(z − z0 )dz = r = 2πi |z−z0 |=r |z−z0 |=r z − z0 In other words, 2π |g(z0 )| ≤ 2 r

1 g(z) dxdy 2 |z−z0 |≤r

12

12 2π 1 2 |g (z)| dxdy dxdy r2 2 |z−z0 |≤r |z−z0 |≤r

π √ ≤ π · r ||g||2H(W ) r2

≤

π 3/2 ||g||2H(W ) ≤ c||g||2W (D) . (1.60) r By the Riesz theorem for representations of linear functionals in Hilbert spaces, the bounded (so continuous) linear evaluation functional on (H(D), , c ) = HD has the following representation: 1 g(z0 ) = dz ∧ d¯ z R(z, z0 )g(z) (1.61) 2i D ≤

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and H

R(z, z0 ) =

∞

ek (z)ek (z0 ),

(1.62)

k=1

where {ek (z)} denotes any orthonormal set of holomorphic functions in HD . Let f : W → B1 (0) be the aforementioned Riemann function mapping holomorphically the given bounded simple connected-smooth boundary domain in B1 (0). Since f (z) is univalent, we have that for f (z0 ) = 0 and f (z0 ) > 0 the following identity holds true: 1 g(z0 ) g(z) = for any g ∈ HD . (1.63) 2πi ∂W =Γ1 f (z) f (z0 ) Let Cr = {(f (z))(f (z)) = r < 1} be the curve in the complex plane. For this special contour, we have that (see Eq. (1.58)) 1 g(z) 1 dz = g(z)f¯(z) dz 2πi Cr f (z) 2πir2 Cr 1 g(z)f (z) dxdy. (1.64) = 2 πr (W \f −1 (Br (0)) Note that we have implicitly assumed that the domain W is a homotopical deformation of its boundary Γ1 ! As a consequence (by considering the limit of r → 1) 1 g(z0 ) = f (z0 ) g(z) f (z) dxdy . (1.65) π W In view of the uniqueness of the Reproducing Kernel Eq. (1.57), we have the explicit representation for the conformal transformation (f (z0 ) = 0 and f (z0 ) > 0) f (z) =

π R(z0 , z0 )

12

z

R(ζ, z0 )dζ .

(1.66)

z0

For instance, one could consider the Graham–Schmidt orthogonalization process applied to the functions {z n , z¯n }n∈Z+ and obtain Eq. (1.66) as an infinite sum of holomorphic functions, or use triangulations of the domain D in order to evaluate the functions en (z) in Eq. (1.62).

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Finally let us devise some useful formulae for the Dirichlet problem in planar “smooth” regions, useful in string path integral theory. Let U0 f −1 : B1 (0) → R be the associated harmonic function on the disc B1 (0) related to the Dirichlet problem in the region W . Note that by construction f (z0 ) = 0 and f (z0 ) > 0 for a given z0 ∈ W . As a consequence of the mean value for harmonic function, we have that 2π 1 U (z0 ) = (U0 f −1 )(0) = (U0 f −1 )(cos θ, sin θ)dθ 2π 0 f (z) 1 dz. (1.67) U (z) z0 = 2πi ∂W fz0 (z) From Eq. (1.67) we can write explicitly the Green functions G(z, z0 ) = (z) fz0 (z) for the given canonical regions. For instance, fz0 (z) = R(z − fz 0 z0 ) (R2 − z¯0 z) applies BR (0) conformally into B1 (0), leading to the Poisson formulae in the circle of radius R

2π R2 − r 2 1 dφ U (Reiφ ) U (r, θ) = . (1.68) 2π 0 R2 + r2 − 2Rr cos(φ − θ) The function fz0 (z) = (z−z0 ) (z+z0 ) applies conformally the right-half+ = {z = x + iy, x ≥ 0, −∞ < y < ∞} into B1 (0) and fz 0 (z0 ) = 0. plane As a consequence, one has the integral representation for this region x +∞ U (it) dt · (1.69) U (x, y) = π −∞ (t − y)2 + x2 The function fz0 (z) = (z 2 −z02 ) (z 2 −(¯ z0 )2 ) solves the Dirichlet problem in the first quadrant plane, leading to the integral representation below: ∞ t U (t) dt 4xy U (x, y) = π t4 − 2t2 (x2 − y 2 ) + (x2 − y )2 0 ∞ t U (it) dt + . (1.70a) t4 + 2t2 (x2 − y 2 ) + (x2 − y 2 ) 0 In general G(z, z ) =

fz0 (z ) − fz0 (z) 1 − fz0 (z) fz0 (z )

(1.70b)

with f (z0 ) = 0 is the explicit Green function associated with the Dirichlet problem in the planar region W .

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Just for completeness let us write the Green function of the Dirichlet 2 in polar coordinates problem in the SR 1 G((ρ, θ), (ρ , θ )) = − log[ρ2 + ρ2 − 2ρρ cos(θ − θ ) 4π ρ2 ρ2 + log R2 + − 2ρρ cos(θ − θ ) . (1.70c) R2 Here

  ∆ρ,θ G((ρ, θ), (ρ , θ ) = 0,

1 G((R, θ), (R, θ )) = + g|r − r |   2π

r =Reiθ r =Reiθ

.

(1.70d)

1.4. Hilbert Spaces Methods in the Poisson Problem Let us pass now to the formulation of the Poisson and Dirichlet problems in ¯ a compact set of RN . This formulation is the Hilbert spaces L2 (Ω), with Ω known in the mathematical literature as the Hilbert space weak formulation of the problem. of the vector space Cc1 (Ω) in relation to the Let H01 (Ω) be the completion d4 x ∇f ∇g; the weak formulation of Poisson

inner product f, gH 1 = problem reads as of

Ω

U, ϕH 1 =

dN x ∇U (x) · (∇ϕ)(x) Ω

dN x(f · ϕ)(x) ¯ = f, ϕL2

= Ω

∀ ϕ ∈ H01 (Ω),

(1.71)

where one must determine a U (x) ∈ H01 (Ω) for a given f ∈ L2 (Ω). That such (unique) U (x) exists is readily seen from the fact that the functional linear below is bounded in H01 (Ω): Lf : H01 (Ω → C) dN x(f · ϕ)(x) ¯ = Lf (ϕ) ϕ→

(1.72)

Ω

as a consequence of the Friedrichs inequality

N −1 N 2 N 2 ¯ d x|U | (x) ≤ diam(Ω) d x|∇U | (x) . Ω Ω ||U ||2L2 (Ω)

≤ C(Ω) ||U ||2H 1 (Ω) 0

(1.73a) (1.73b)

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By the Riesz representation theorem, there is a unique element U (x) ∈ such that it satisfies Eq. (1.72) and has the explicit expansion

H01 (Ω),

U (x)

H01 (Ω)

=

Lf (en )en (x)

n

=

dN y f¯(y)en (y) e¯n (x)

Ω

n

N

=

d y Ω

¯ e¯n (x)en (y) f (y) = dN y G(x, y)f¯(y).

n

(1.74)

Ω

Here, en (x) is an arbitrary orthonormal basis in H01 (Ω), which can be constructed from the application of the Grahm–Schmidt orthogonalization process to any linear independent set of Cc∞ (Ω), for instance, the set of the polynomials {x|n| , x|m| } with x ∈ RN . This method of determining weak solutions is easily generalized to the Poisson problem with variable coefficients — the Lax–Milgram theorem, a useful result in covariant euclidean path integrals in string theory. ¯ a compact set of RN . Let aij (x) and U (x) be functions in C(Ω), with Ω Let us consider the following Hilbert space: L2{a} (Ω) = topological closure of

∂ ∂ ¯ N ¯)(x) < ∞ · f ∈ C0 (Ω) dN x aij (x) i f · d x V (x)(f f f + ∂x ∂xj Ω Ω with aij (x) = aji (x), aij (x)λi λj ≥ a0 |λ|2 and V (x) > V0 > 0 .

(1.75)

We left as an exercise to our readers to mimic with some improvements the previous study to prove the existence and uniqueness of the Poisson problem in L2{a} (Ω) for f ∈ L2{a} (Ω) ⊇ H01 (Ω) and ϕ(x) ∈ H01 (Ω)

∂ ∂ d x aij (x) i U (x) j ϕ(x) + dN x V (x)U (x)ϕ(x) ¯ ∂x ∂x Ω Ω = dN x f (x)ϕ(x). ¯ (1.76)

N

Ω

It is another nontrivial problem to find conditions on the data function f (x) (see Theorem 1.1) in order to have the weak solution U (x) to belong

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¯ which will lead to the strong solution of the to the space C 2 (Ω) ∩ C 1 (Ω), Poisson problem in Ω −

N ∂ ∂ (x) U (x) + V (x)U (x) = f (x) x ∈ Ω. a ij ∂xj ∂xi i,j=1

(1.77)

Another important generalization of the Poisson problem is the class of semilinear Poisson problems defined by Lipschitzian nonlinearities as written below for Ω ⊂ R3 : (−∆U )(x) + F (U (x)) + V (x)U (x) = f (x).

(1.78)

Here f (x) ∈ L2 (Ω) and F (z) are real Lipschitzian functions (i.e. ∀(z1 , z2 ) ∈ R × R ⇒ |F (z1 ) − F (z2 )| ≤ C|z1 − z2 | for a uniform bound c). In order to show the existence and uniqueness through a construction technique, we rewrite Eq. (1.78) into the form of an integral equation (see Theorem 1.1)

F (U (x )) + V (x )U (x ) − f (x ) 3 1 d x = (T U )(x). U (x) = − 4π |x − x | (1.79) One can see easily that T defines an application with domain L2 (Ω) and range in L2 (Ω). Besides, T is a contraction application in L2 (Ω) for small domains, as we can see from the following estimate: ||T u − T v||2L2 (Ω) ≤

1 1 2 2 2 3 ||u − v|| + ||V || ) × d x (c L2 (Ω) L2 (Ω) 16π 2 |x − x |2 Ω

≤

C 2 + ||V ||2L2 (Ω) 16π 2

(4π diam(Ω))||u − v||2L2 (Ω) .

(1.80)

As a consequence of the Banach Fixed Point Theorem, we have that the sequence Un = T (Un+1 )

(1.81)

converges strongly to the solution in L2 (Ω) of Eq. (1.79). The reader should repeat the above-made analysis in the general case Ω ⊂ RN (see Eq. (1.23)).

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1.5. The Abstract Formulation of the Poisson Problem Let A be a linear operator defined in a Hilbert space H = (H, ,) (A : H → H, A(αU + βv) = αA(u) + βA(v)) such that its domain D(A) forms a (nonclosed) vectorial subspace dense in H. Note that D(A) has the explicit analytical representation (1.82) D(A) = u ∈ H | Au ∈ H ⇔ Au, AuH < ∞ . Let λ ∈ C such that there is a vector Uλ ∈ H, such that AUλ = λUλ . Such vector is called the eigenvector of A associated with the eigenvalue λ. The set of all these eigenvectors {Uλ } makes an invariant subspace for the operator A. We have the following results in operator theory in separable Hilbert spaces, useful to the Poisson problem solution. Theorem 1.4. If the operator A satisfies the symmetric condition (so A is called a symmetric operator) Au, vH = (u, Av)H (u, v ∈ D(A)), then all eigenvalues λ are real numbers. Proof. Let AUλ = λUλ . As a consequence of the symmetry property of the operator A, we have that ¯ λ ||2 . AUλ , Uλ H = λUλ , Uλ = λ||Uλ ||2 = Uλ , AUλ = λ||U

(1.83)

Theorem 1.5. If A is a symmetric operator, then different eigenvectors corresponding to different eigenvalues are orthogonals. So A has at most a countable number of eigenvalues in the separable H. Proof. Let AUλ1 = λ1 Uλ1 and AUλ2 = λ2 Uλ2 (λ1 = λ2 ). Thus we have the identity 0 = AUλ1 , Uλ2 H − Uλ1 , AUλ2 H = (λ1 − λ2 )Uλ1 , Uλ2 H ,

(1.84)

which means the result searched Uλ1 , Uλ2 H = 0.

(1.85)

We have our basic result in the theory of the Poisson problem in Hilbert spaces.

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Theorem 1.6. (Dirichlet–Riemann Variational formulation of Poisson problems.) Let A be a positive-definite symmetric operator in a given separable Hilbert space H = (H, , ). (Exists c ∈ R+ such that AU, U H ≥ c(U, U )H for any U ∈ D(A).) Let the functional given below for any F ∈ Range(A) be F(A) (ϕ) = Aϕ, ϕH − f, ϕH .

(1.86)

Then, the functional equation (1.86) has a minimum value at U ∈ D(A) such that A U = f,

(1.87)

the converse still holds true. Let us exemplify the above result for the Poisson problem in bounded open sets in RN with a Riemannian structure {gab (xa ); a, b = 1, . . . , n} (manifolds charts). Thus we consider the following covariant Hilbert space associated with the given metric structures

dN x g(x)f (x)f (x) , L2g (Ω) = closure of f ∈ Cc (Ω) | , ≡ ¯ W

(1.88a) 1 H0,g (Ω) = closure of

f ∈ C01 (Ω) | , =

¯ W

√ dN x g g ab ∂a f ∂b f , (1.88b)

p H0,g (Ω) = closure of f ∈ C0p (Ω) | , H 0 √ = dN x g(∇a1 . . . ∇a(p/2) f )(x) ¯ W

× {g a1 ,a(p/2+1) . . . g a(p/2) ap }(x)(∇a(p/2+1) . . . ∇ap f )(x). (1.88c) 1 (Ω), we consider the functional In the covariant Hilbert space H0,g (positive-definite) associated to the Laplace–Beltrami operator ∆g = √ − √1g ∂a ( g δ ab ∂b ) acting on real functions

F∆q (ϕ) =

dN x ¯ W

√ g ϕ(−∆g )ϕ (x) −

dN x ¯ W

√

g f (x)ϕ(x).

(1.89)

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1 ¯ ∈ H0,g The minimizing U (Ω) of the above-written functional is the solution of the covariant Poisson problem in the open domain W :

¯ )(x) = f (x) for x ∈ W. (−∆g U

(1.90)

Let us now analyze the eigenvalue problem in the Dirichlet problem in the Hilbert space L2 (Ω). We can accomplish such studies by considering the problem rewritten into the integral operator form 1 4π

µ Uλ (r ) =

d3r

Ω

Uλ (r ) = (T Uλ )(F ). |r − r |

(1.91)

First, let us see that the integral operator T has as its domain C 1 (Ω) ⊂ L (Ω) and as its range the functional space C 2 (Ω). However, it can be extended to the whole space L2 (Ω) by the Hahn–Banach theorem 2

T : L2 (Ω) → L2 (Ω)

(1.92)

since we have the estimate |(T U )(r )| ≤ ≤

1 4π

d3r

Ω

1 |r − r |2

12

×

12 d3r U 2 (r )

Ω

12 1 diam(Ω) ||U ||L2 (Ω) . 3

(1.93)

We can see that T is a symmetric bounded compact operator with a set of eigenvalues |µr | ≤ 4π 3 diam(Ω) and µn → 0 for n → ∞. 1.6. Potential Theory for the Wave Equation in R3 — Kirchhoff Potentials (Spherical Means) Let us start this section by defining the Poisson–Kirchhoff potential associated with the given C 2 (Ω) functions. The first potential associated with a certain f (r ) ∈ C 2 (Ω) is defined by the spherical means (r = (x, y, z) ∈ Ω) U

(1)

t (r, t, [f ]) = 4π

3 ( Sct r)

f (x + αct, y + βct, z + γct)dΩ.

(1.94)

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3 3 Here, Sct (r ) = ∂{Bct (r )} is the spherical surface centered at the point r = (x, y, z) with radius |r − r| = ct. The spherical parameter α, β, and γ are defined as usual as

α = cos ϕ sin θ, β = sin ϕ sin θ,

(1.95)

γ = cos θ, dΩ = sin θdθ · dϕ. The second potential is defined in an analogous way as 1 ∂ 1 (1) U (r, t, [g]) . U (2) (r, t, [g]) = 4π ∂t 2

(1.96)

We have the following elementary initial-value properties for the above wave potentials, which are the functions defined in the infinite domain R3 × R+ : U (1) (r, 0, [f ]) = 0, ∂ (1) U (r, 0, [f ]) = f (x, y, z) × lim ∂t t→0+

t 4π

3 ( Sct r)

dΩ

= f (r ), (1.97)

U (2) (r, 0, [g]) = g(r ), ∂ U2 (r, 0, [g]) = 0 (exercise). ∂t The main differentiability property of the above-written wave potentials is the following: ∆r U

(1)

t (r, t, [f ]) = 4π

2π

dϕ 0

π

sin θdθ∆r f (x + αct, y + βct, z + γct) .

0

(1.98) We should now evaluate the second-time derivative of the potential (¯ x = x + αct, y¯ = y + βct, z¯ = z + γct) ∂U (1) (r, t, [f ]) ∂t t U (1) (r, t, [f ]) + = t 4π

∂ ∂ ∂ 2 + cβ + cγ cα f (¯ x, y¯, z¯)d Ω 3 ( ∂x ¯ ∂ y¯ ∂ z¯ Sct r)

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=

1 U (1) (r, t, [f ]) + t 4πct

n·∇

∂ ∂ ∂ +β +γ f (¯ x, y¯, z¯)dA α 3 ( ∂x ¯ ∂ y¯ ∂ z¯ Sct r)

(Gauss theorem) 1 U (1) (r, t, [f ]) + = t 4πct U (1) (r, t, [f ]) 1 = + t 4πct since

3 ( Bct r)

(∆ f )(¯ x, y¯, z¯)d¯ x d¯ y d¯ z

ct

dρ 0

d A(∆f )(¯ x, y¯, z¯) 2

3 ( Sct r)

d¯ x d¯ y d¯ z = dρ d2 A, 0 ≤ ρ ≤ ct.

(1.99)

(1.100)

Since we have the usual Leibnitz rule for derivatives inside integrals ct d f (x) dx = f (ct) · c. (1.101) dt 0 We have the result for evaluation of the second-time derivative of Eq. (1.94): 1 ∂ 2 (1) ∂U (1) (r, t, [f ]) 1 · − 2 U (1) (r, t, [f ]) U ( r , t, [f ]) = 2 ∂t ∂t t t 1 2 + ×c× d A(∆f )(¯ x, y¯, z¯) 3 ( 4πct Sct r) 1 − 4πct2

ct

dρ 0

d A(∆f )(¯ x, y¯, z¯) . 2

(1.102)

Sρ3 ( r)

Using Eq. (1.99) again to simplify Eq. (1.102), yields 1 ∂ 2 U (1) (r, t, [f ]) 2 = d A(∆f )(¯ x , y ¯ , z ¯ ) ∂2t 4πt Sρ3 (r ) 1 2 2 =c t d Ω(∆f )(¯ x, y¯, z¯) = 2 ∆r U (1) (r, t, [f ]). 3 ( c Sct r) (1.103) As a result of the above differential calculus evaluations we have the analogous of Theorem 1.1 (the Cauchy problem for the wave equation).

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Theorem 1.7. The solution for the Cauchy (initial values) problem for the wave equation with “smooth” C 2 (Ω) data (including an external forcing F (r, t), continuous in the t-variable), is given by the wave potentials  2 ∂ U (r, t) 1   = 2 ∆r U (r, t) + F (r, t),  2  c  ∂ t (1.104) U ( r , 0) = f ( r ), r ∈ Ω,      Ut (r, 0) = g(r ), r ∈ Ω, U (r, t) = U (1) (r, t, [f ]) + U (2) (r, t, [g]) F (r , ct − |r − r |) 3 1 d + r . 4π |r − r | | r− r |≤ct In the case of Ω ⊂ R2 we have the Poisson wave potentials:

∂ 1 f (r ) 2

U (r, t) = d r ∂t 2π |r−r | 0, such that

∞

δ+

d W (t)f < ∞, dt H

−δ −

−∞

d W (t)f < ∞ dt H

(2.6)

for f ∈ H. Then W (t)f converges for t → ±∞. Proof. It is a single consequence of the estimate Wt1 f − Wt2 f H ≤

t1 t2

(t1 ,t2 )→+∞ d −−−−−−−→ 0. W (t)f ds ds H

(2.7)

We now have the basic result on the existence of wave operators (Cook– Kuroda–Jauch).1,3,4 Theorem 2.1. Let M be a dense subset of Dom(H) ∩ Dom(H0 ) with the following properties: (a) (H − H0 ) exp(−itH0 )f H is a continuous function in R. (b) (H − H0 ) exp(−itH0 )f H is integrable in [δ, ∞) (or in (−∞, −δ]). Then the wave operators are well defined. Proof. By the Stone theorem d (W (t)f ) = (eitH (H − H0 )e−itH0 )f. dt

(2.8)

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M¨ oller Wave Operators and Asymptotic Completeness

As a straightforward consequence of hypothesis (b), we have d W (t)f dt

H

= (H − H0 )e−itH0 f H ∈ L1 ([δ1 , ∞)).

(2.9)

Since M is dense in PAC (H)H and W (t) are uniformly bounded (W (t)operator = 1) we have the validity of the lemma of Cock–Kuroda– Jauch. Let us apply the above result to concrete examples. First, let the quantum mechanical potential V (x) ∈ L2loc (RN ) be such that there exists ε > 0 with N

V (x)(1 + |x|)−( 2 −1)+ε L2 (RN ) = Nv < ∞.

(2.10)

(The so-called short-range quantum mechanical scattering potential.) Additionally, let us suppose that H = − 12 ∆ + V (x) is a self-adjoint operator in L2 (RN ). Then there are wave operators associated with the pair (H, H0 ) in L2 (RN ). In order to show this result, let us remark that the set of functions   M = ψa (x) ∈ L2 (RN ) | ψa (x) = φ(x − a), for a ∈ RN   and φ(x) = 2

−N 2



N

xj e−|x|

j=1

2

  /4  

(2.11)

is dense in L2 (RN ) (Wiener theorem2 ) and that (exercise!) 

N

(exp(−it(−∆))ψa )(x) =

22

(1 + it)

3n 2



N

»

(xj − aj )e

−

(x−a)2 4(1+it)

–

.

(2.12)

j=1

We have thus the estimate for any δ |(exp(−it(−∆))ψa )(x)| ≤2

−N 2

3N −1−δ N |x − a|−( 2 −1)+δ x − a 2 |(x − a)|2 exp − . 1 + it |1 + it|1+δ 4(1 + t2 | (2.13)

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As a consequence and by choosing 0 < δ < estimate involving the scattering potential N

|(V exp(−it(−∆))ψa )(x)| ≤ Ca 2− 2

1 2,

we have the combined N

|V (x)|x − a|−( 2 −1)+δ | , |1 + it|1+δ

where t > 0 and Ca = supΩ∈R (2Ω) 1 Ω = |(x − a)|/(1 + t2 ) 2 . If we choose δ < min( 12 , ), we have

3N 2

−1−δ −Ω2

(V e−it(−∆) ψa )(x)L2 (RN ) ≤

e

(2.14)

L∞ (RN ) < ∞, here

NV , |1 + it|1+δ

which obviously leads to the searched integrability result ∞ ∞ dt V e−it(−∆) ψa L2 (RN ) ≤ NV 0 c 1 N F |x| < a|t| . | exp(−it(−∆))(−∆ + 1) f |(x) ≤ (1 + |x| + |t|)N +1 2 (2.18) This set is dense in L2 (RN ) since supp f˜(x) ⊂ (Ba (0))c . We have the identity V exp(−it(−∆))f L2 (RN ) 1 −M −iH0 t M ≤ V (H0 + i) F |x| ≤ a|t| e (H0 + i) f 2 L2 (RN ) 1 −M −iH0 t M + V (H0 + i) F |x| ≥ a|t| e (H0 + i) f . 2 L2 (RN ) (2.19)

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Since every compact operator is a bounded operator and the estimate equation (2.18) added to the Enss localizibility property equation (2.17), one can easily obtain the Cook–Kuroda–Jauch lemma validity for the existence of wave operator for Enss potentials. At this point of our exposition we call the readers’ attention to the following theorem which is easy to proof and is left to the reader’s completion.3 Theorem 2.2. Let us suppose that the wave operators W ± (H, H0 ) exist in H, then we have (a) W ± are partial isometries between HAC (H0 ) (the subspace of absolute continuity of H0 ) and the final range spaces H + = Range(W + (H, H0 )) (H − = Range(W − (H, H0 ))). (b) HW ± = W ± H0 . (c) H± ⊂ HAC (H). (d) The S-matrix operator S = (W + )∗ (W − ) : H → H. An important physical requirement on quantum mechanics is the requirement that W ± (H, H 0 ) should be unitary applications between the subspaces H(H0 ) and H(H), with the condition of H + = H − = HAC (H0 ), and in such a way that the operator S¯ = (W + )∗ W − must be a unitary operator between H− and H+ , and thus, insuring that the physical S-matrix operator S¯ is a unitary operator, the backbone of the Copenhagen School interpetration of the quantum phenomena. We intend to show the confirmation of this result in two situations. First, we show that it is possible to always have asymptotic completeness by reducing the physical Hilbert space of the scattering states. Second, we present mathematical details in next sections, the beautiful original Enss’s geometrical work on the subject.5 Our first claim is a simple result of ours obtained from a result of Halmos (exercise). Halmos Lemma. Let T be an isometry in H. Let us consider the closed subspace N = (Range T )⊥ . Then we have the following subspace decomposition of the original Hilbert space in terms of invariant subspace of the isometry T ∞ k H=M⊕ T (N ) , (2.20) k=0 k

with T denoting the kth power of T and most importantly, the composite operator TM = T PM (when P : H → M is the orthogonal projection onto the closed subspace M ) is a unitary operator TM : M → M .

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In our case M ⊂ H+ ∩ H− and SM = SPM is automatically a unitary operator in M .

2.2. Asymptotic Properties of States in the Continuous Spectra of the Enss Hamiltonian H0

z }| { Let H = (− 12 ∆) +V be a self-adjoint operator defined by an Enss potential

with M = 1 (see Eq. (2.13)). Then we have the result5 :

Enss Lemma. Let f ∈ Hc (H) be a state belonging to the continuity subspace of the Enss Hamiltonian. Then there is a sequence of times tn such that limn→∞ tn = +∞ (tn ≤ tn+1 , ∀n ∈ Z) and lim F (|x| < 13n)eiHtn f L2 (RN ) = 0,

(2.21a)

dt|F (|x| < n)e−iH(t+tn ) (H + i1)f L2 = 0.

(2.21b)

n→∞

n

lim

n→∞

−n

In order to show such result let us remark T that if a continuous function has an ergodic mean zero, i.e. limT →∞ T1 ( 0 g(z)dz) = 0, then there is a monotone sequence (tn ) such that limn→∞ g(tn ) = 0 with tn ≤ tn+1 and tn → ∞. Now it is a deep theorem that (see Chap. 8) 1 T dtf (|x| < n)e−iH(t) (H + i1)f L2 (RN ) = 0. (2.22) lim T →∞ T 0 So, the result of lemma is a simple consequence of the above-stated RAGE theorem applied to the z-functions below written together with Fubbini theorem in order to interchange an integration order n F (|x| < n)e−iH(t+z) (H + i1)f L2 dt, (2.23) W2n (z) = −n

W2n+1 (z) = F (|x| < n)e−iHz f L2 .

(2.24)

Let us now define the Enss’ scattering states for a given state f ∈ Hc (H) fn = exp(−iHtn )f.

(2.25)

We now have the following operatorial limit: lim ((H + z1)−1 − (H0 + z1)−1 )F (|x| > (13n)operator = 0.

n→∞

(2.26)

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The validity of Eq. (2.26) is a simple consequence of the continuity and integrability of the Enss function (see Eq. (2.13)) ((H + z1)−1 − (H0 + z1)−1 )F (|x| > (13n)operator ≤ V (H0 + z1)−1 F (|x| ≥ (13n)operator = hz (R)

(2.27)

since

14n

0 = lim

n→∞

13n

dRhz (R) ≥ lim nhz (13n). n→∞

(2.28)

Similar result is obtained through derivatives in relation to the zparameter. As the polynomials in the variables (x + i)−1 and (x − i)−1 are dense in the space of the continuous functions vanishing at ∞ (the Stone– Weierstrass theorem — Chap. 1), by a simple density argument we arrive at the following result for any function φ(x) ∈ C0 (R): lim (φ(H) − φ(H0 ))fn L2 (RN ) = 0.

n→∞

(2.29)

At this point we introduce the Enss scattering states fn ∈ Hc (H), the physical requirement that they are really physical by possessing finite (interacting) energy. This last condition is achieved by considering the energy cut-off new-states (0 ≤ a < b < ∞) 2 2 |b − a| ¯ (2.30) fn , fn = EH 73a , 4 where EH ([a, b]) denotes the spectral projection of the full Enss Hamiltonian H, associated with the spectral interval [a, b]. (Note that Hc (H) ⊂ R+ since (H + z1)−1 is a compact operator — see Appendix C.) ˜ Let us consider φ(x) ∈ C ∞ (R) such that φ(x) = 1 for x ∈ [73a2 , 2 (b − a) /4] and φ(x) ≡ 0 for r ∈ [72a2 , (b − a)2 /4]. We now consider the “physical free-energy packet” of scattering states 1 φn = ϕ˜ − ∆ f¯n . (2.31) 2 Let us point out that the Fourier transforms of the states φn (x) all have √ 2 (b−a) support in the region 2 · 72a2 ≤ |P | ≤ 2 2 ⇔ supp(F [φn ](p)) ⊆ {p ∈ RN | 12a ≤ |p| ≤ b − a}.

(2.32)

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At this point we remark that we need to further localize our scattering states by taking into account fully its quantum nature as given by the Heisenberg Principle.5 In order to achieve such a result, we introduce a function s(x) ∈ S(RN ) such that its Fourier transform has support in the ˜ = F [S](0) = 1. ball Ba (0) in RN with S(0) Let us define the quantum mechanical classical localization operators for a given Borelian A dyS(x − y)χA (y). (2.33) F0 (A)(x) = RN

At this point we can see that the family of functions {F0 (Aj )}j∈I is a unity decomposition if {Aj }j∈J is a disjoint decomposition of RN

F0 (Aj ) =

˜ S(y)dN y = S(0) = 1.

(2.34)

j∈I

Another important point to call the readers’ attention is the fact that the quantum localized states F0 (A)φn all have their momenta in the Ball with radius b. This result is easily obtained through the result that for m(A) < ∞, we have ˜ ⊆ Ba (0). supp F [F0 (A)](p) ⊆ supp S(p)

(2.35)

In the case of m(A) = +∞ where χA (x) ∈ / L1 (RN ), we left it for our readers by just considering the equivalent expression χA (x − y) . (2.36) F0 (A) = dN y · S(y)(|(x − y)2N + 1|) (|x − y|2N + 1) Let us now consider the best quantum mechanical spatial classical localization decomposition of RN in Balls and truncated cones (j) (j) {B12n (0), C12n }j∈I ,6 where the cones C12n = {x ∈ RN | |x| ≥ 12n, xej ≥ |x|; 2} have axis along the vector {ˆ ej }, set of unity vector to be suitably choosen. Then we have the following results: lim F0 (B12n )φn L2 = 0, n→∞ (j) lim fn − F0 (C12n )φn = 0. n→∞ 2 j∈I

L

(2.37a) (2.37b)

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The proof is achieved by using the estimates F0 (B12n )φn − F (|x| ≤ 13n)f¯n L2 ˜ ≤S(0)=1)

≤ ϕ(H ˜ 0 )f¯n − ϕ(H) ˜ f¯n L2 F0 (B12n )L2 + [(F0 (B12n ) − f (|x| ≤ 13n)]ϕ(H) ˜ f¯n L2 and

F0 (B12n (fn )) lim n→∞ j∈I

= 0.

(2.38a)

(2.38b)

L2

Let us finally consider the physical Enss localized scattering states by its classical motion direction. Let hj (p) ∈ C0∞ (RN ) be the functions satisfying the conditions hj (p) = 0

for p · eˆj ≤ −2a,

hj (p) + hj (−p) = 1

for |p| < b

(2.39) (j)

with eˆj denoting the vector in the direction of the cone’s axis C12n as previously stated. The Enss OUT and IN states are explicitly given by5 (j)

φn (j)OUT = F0 (C12n )hj (p)φn , (j)

φn (j)IN = F0 (C12n )hj (−p)φn .

(2.40a) (2.40b)

We now have the important technical result of free evolution of the above considered directional scattering states (for a detailed proof see Appendix A). Enss Theorem (Technical). We have the following result: ∞ lim dtF (|x| ≤ n + at)e−iH0 t (H0 + i1)φn (j)OUT = 0, n→∞

0

lim

n→∞

(2.41a)

0

−∞

dtF (|x| ≤ n − at)e−iH0 t (H0 + i1)φn (j)IN = 0.

(2.41b)

After displaying their geometrical properties of the Enss scattering states, we now pass to the proof of the asymptotic completeness of the Enss Hamiltonian in Sec. 2.3.

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2.3. The Enss Proof of the Non-Relativistic One-Body Quantum Mechanical Scattering Enss Theorem (Asymptotic completeness).5 Let H be the Enss Hamiltonian and its associated wave operators W ± (H, H0 ). Then, the subspaces H± ≡ Range(W ± ) coincide with the subspace of continuity Hc (H) of the Enss Hamiltonian. Proof. Let ψ ∈ M be one of the quantum mechanical localized state of the previous section. Note that M is dense in Hc (H). We now observe that (Eq. (2.37b)) IN OUT φn (j) + φn (j) lim ψn − n→∞ 2 j∈I j∈I L (j) lim ψn − F0 (C12n φn ) = 0. (2.42) n→∞ 2 j∈I

L

Let us suppose that there is a state ψ ∈ Hc (H) such that it is orthogonal to H+ = Range(W + ). We observe now that the estimate below holds true lim |ψn , ψn | IN OUT = lim ψn , ψn − φn (j) + φn (j) n→∞

n→∞

−

j∈I

j∈I

IN OUT φn (j) + φn (j) j∈I

j∈I

IN OUT ≤ lim ψn , ψn − φn (j) + φn (j) n→∞ j∈I

j∈I

IN OUT + lim ψn , φn (j) φn (j) + lim ψn , n→∞ n→∞ j∈I

j∈I

(2.43a) φn (j)IN + φn (j)OUT ψn ψn −

≤ lim

n→∞

j∈I

j∈I

(2.43b)

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(2.43c)

j∈I

+ OUT + lim ψn , W φn (j) . n→∞

(2.43d)

j∈I

Here, we have used the technical lemma (see Appendix B) lim (W + − 1)φn (j)OUT = 0,

(2.44a)

lim |φn , φn (j)IN = 0.

(2.44b)

n→∞

n→∞

As a consequence that ψ, W + w = 0, ∀w ∈ Dom(W + ), we have that the only remaining term in the estimate considered above is the following: IN OUT (2.45) φn (j) + φn (j) lim ψn , ψn − = 0. n→∞ j∈E

In view of Eq. (2.42), we have the contradiction 0 = ψ2 = lim |ψn , ψn | n→∞ IN OUT ≤ lim ψn , ψn − φn (j) + φn (j) =0 n→∞

(2.46)

j∈I

The proof of H− = Hc (H) is similar and only need to consider φn (j)IN in Eq. (2.44a). Details are left as an exercise. Note that this result implies that the singularly continuous spectrum of the Enss Hamiltonian is empty and all bound states are orthogonal to HAC (H).

References 1. S. T. Kuroda, On the existence and the unitary property of the scattering operator, Nuovo Cimento 12, 431–454 (1959). 2. N. Wiener, The Fourier Integral and Certain of its Applications (Cambridge University Press, NY, 1933). 3. W. O. Amrein, M. M. Jauch and B. L. Sinha, Scattering Theory in Quantum Mechanics (W.A. Benjamin, Ins., Massachussetts, 1977). 4. J. B. Dollard, Adiabatic switching in the Schr¨ odinger theory of scattering, J. Math. Phys. 802–810 (1966). 5. V. Enss, Asymptotic completeness for quantum mechanical potential scattering, Commun. Math. Phys. 61, 285–291 (1979).

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6. E. B. Davies, Quantum Theorem of Open Systems (Academic Press, New York, 1976). 7. S. Steinberg, Meromorphic families of compact operators, Arch. Rational Mech. Anal. 31 (1969).

Appendix A IN

Theorem A.1. Let us consider the Enss scatterbug states φn (j)OUT . Then we have the result ∞ dtF (|x| ≤ n + at)e−iH0 t (H0 + i1)φn (j)OUT = 0, (A.1) lim n→∞

0

lim

n→∞

0

∞

dtF (|x| ≤ n − at)e−iH0 t (H0 + i1)φn (j)IN = 0. (A.2)

Proof. Our aim is to reduce the above estimates to the one-dimensional case. Thus, let us analyze Eq. (A.1) by considering a new decomposition of the Enss scattering states φn (j)IN in the x-space RN . First, let us observe that the half-spaces x · F ≤ n + at with f = {fn }1≤k≤N ∈ RN and f = 1 contains the Ball Bn+at (0) = {x ∈ RN | |x| ≤ n + at}. We let the function hj (p) with {p ∈ RN | 12a ≤ |p| ≤ b−a} be decomposed into a finite number (j) (j) of summands ξk (p) ∈ C0∞ (RN ) in such a way that support {ξk (p)} ⊆ N {p ∈ R | P · fn ≥ 2a}. (j) Let us choose fk unity vectors with an angle of π/9 rads with the cones (j) vectors axis eˆj of C12n . (j) (j) As a consequence x · fk > 2n for x ∈ C12n . We remark that Eq. (A.1) will be a result of the validity of the following limit: ∞ (j) (j) (j) dtF (x · fk ≤ n + at)e−iH0 t (H0 + i1)F0 (C12n )ξk (p)φn = 0 lim n→∞

0

(A.3) since any state φn has its “momenta” support in the region 12a ≤ |p| ≤ b−a. Within this region one has decomposition of the Enss scattering state (j) (j) (j) φn (j)OUT = F0 (C12n )h0 (p)φn = F0 (C12n )ξk (p)φn . (A.4) k

Just in order to simplify the estimates which follows and choose the axis (k) x1 parallel to fj under consideration. Let us thus consider (j)

Fm = (H0 + i1)F0 (C12n ∩ {2n + m ≤ x1 ≤ 2n + m + 1}).

(A.5)

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Obviously (H0 +

(j) i1)F0 (C12n )

=

∞

Fm .

(A.6)

m=0

It will be necessary to implement the action of (H0 + i1) by means of a multiplication operator in p-space by means of a function with compact support. This can be implemented by noting that all the states F0 (A)φn , with A denoting a Borelian of RN , have p-momentum support in the region |p| ≤ b. Another point worth remarking is that the momentum component p1 (related to the x1 -axis) of the states (j)

(j)

(H0 + i1)F0 (C12n ∩ {2n + m ≤ x1 ≤ 2n + m + 1})ξk (p)φn

(A.7)

is in the interval a ≤ p1 ≤ b. Let us now pass to the estimates. First, the estimate equation (A.3) reduces to the one-dimensional case as a consequence of the dependence of solely the coordinate x1 into our sliced up states equation (A.7). We have thus reduced our analysis to the one-dimensional case ! ∞ 2 −i ∂ ∂ t 2x 1 dtF (x1 ≤ n + at)e φ(x1 , x ˆ) 2 (A.8) L

0

with φ(x, xˆ) denoting our new sliced up states. Let us define the following operator family of operators in L2 (RN ): 2

(Wζ φ) =

eix1 /2ζ (2πiζ)1/2

"

+∞

−∞

2 2$ % & # x1 iy1 iy 2 dy1 1 + 1 + ˆ) . 2 e−i ζ y1 φ(y1 , x 2ζ 2ζ (A.9)

It is a consequence of the support of the state φ(y1 , xˆ) that (Wt φ)(x) = 0 for x1 < at

(A.10)

by just considering Eq. (A.9) by means of the Fourier transform (Wt φ)(x) =

# 2 i ∂2 eix1 /2t 1 + 2t ∂ 2 ( xt1 ) (2πit)1/2 +

and Fy1 [φ](p1 , x ˆ) ≡ 0 for p ≤

i 2t

x1 ζ

2

% x1 ∂3 , x ˆ [φ] F y1 ∂ 3 ( xt1 ) t

< a.

(A.11)

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We now note the validity of the result written as # % 2 t −i ∂∂ 2x 2 (1 + t)(−x1 + 2at)2 1 e − W t φ (x)

L2

2 # iy12 % iy12 ∂ 2 ib21 /2t 2t + = (1 + t)t − i + 2a −1+ ˆ) e φ(y1 , x 2 ∂y1 2t 2 L 2 (1 + t)t ∂ ≤ |P1∗ (|y1 |)|(φ(y1 , x ˆ)) + |P2∗ (|y1 |)| φ(y1 , xˆ) t3 ∂y1 2 ∂ ∗ + |P3 (|y1 |)| 2 φ(y1 , x ˆ) (A.12) 2, ∂y1 L ˜) where P∗ (|y1 |), = 1, 2, 3 are polynomials satisfying the relations (x1 ≥ a ' P3∗ (|y1 |) = |y1 |6 48, 2|y1 |5 4a|y1 |6 + , 48 8 1 1 ∗ P1 (|y1 |) ≥ P |y1 |, ≥ P |y1 |, . a ˜ t P2∗ (|y1 |) =

(A.13)

We have this for x1 ≤ at the estimate % 2 # (iy12 ) 1 iy12 iy12 /2t (1 + t)t2 − i ∂ + 2a + · φ(y e − 1 + , x ˆ ) 1 ∂y1 2t (2t) 2 ∗ ∂ a +1 ˆ) + P2∗ (|y1 |) φ(y1 , x ˆ) ≤ P1 (|y1 |)φ(y1 , x a ˜ ∂y1 " ∂ ˆ) ∂ 2 φ(y1 , x ∗ = a P1∗ ∂ + P3∗ (|y1 |) + P ip1 2 ∂ 2 y1 a ˜ ∂p1 ∂p1 & ∂ + P3∗ (A.14) (−p1 )2 F [φ](p1 , pˆ) 2 ≤A ∂p1 L ˜ 1 , pˆ) ⊂ {p ∈ RN | |p| ≤ b}. since supp φ(p As a consequence (see Eqs. (A.10) and (A.14)) ! 2 − i ∂ 2 2t 2 ∂x1 φ(x1 , xˆ) ≤ A. (1 + t)(−x1 + 2at) F (x1 ≤ at)e The result of Eq. (A.15) is the following: 2 −i ∂ 2 2t ! F (x1 ≤ −v + at) e ∂x1 φ (x) ≤

A (1 + t)(u + at)2

(A.15)

(A.16)

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since for v > 0, (x1 ≤ −v + at) 1 1 ≤ · (−x1 + 2at)2 (v + at)2

(A.17)

As a consequence we have the following estimate for v = m + n and (j) ˆ) and x1 ∈ C12n ∩ {2n + m ≤ x1 ≤ φ(x) = Fm ζkj (p)φn (x1 + 2m + n, x 2n + m + 1} (Ref. 5) by using translational invariance x1 → x1 + v: (j)

F (x1 − 2n − m < −(n + m) + at)e−iH0 t Fm ζk (p)φn (x1 , x ˆ)L2 ≤

A . |(1 + t)|m + n + at|2 |

(A.18)

By noting again that ∞

(j)

m=0

=

F (x1 − 2n − m < −n − m + at)e−iH0 t Fn ξk (p)φn (x) ∞

(j)

m=0

F (x1 < n + at)e−iH0 t Fn ξk (p)φn (x).

We have (j)

F (x1 < n + at)e−iH0 t (H0 + i1)ξk (p)φn (x) ∞ (j) −iH0 t = F (x1 < n + at)e Fn ξk (p)φn (x) m=0

≤

∞ m=0

(j)

F (x1 + n + at)e−iH0 t Fn ξk (p)φn (x)

1 . ≤A |(1 + t)(m + n + at)2 | m=0

∞

We now left as an exercise to our diligent reader to establish the result5 ∞ (j) (j) (j) lim dtF (x · fk ≤ n + at)e−iH0 t (H0 + i1)F0 (C12n )ξk (p)φn n→∞ 0 ∞ ∞ 1 = 0. dt ≤ A lim n→∞ |(1 + t)(m + n + at)2 | m=0 0

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Appendix B Let us show the following technical result used in the proof of the Enss theorem. IN

Lemma B.1. Let W± be the Enss wave operators and φn (j)OUT considered in Sec. 2.2, then we have the result ( = L2 ). lim (W + − 1)φn (j)OUT = 0,

(B.1)

lim (W − − 1)φn (j)IN = 0.

(B.2)

n→∞

n→∞

Proof. Since t −iH0 ζ dζ(V e f ) (W (t) − W (0)f ≤

(B.3)

0

we have that ∞ −iH0 ζ OUT ≤ dζV e φn (j) (W − 1)φn (j) 0 ∞ −1 −iHo ζ OUT ≤ V (H0 + i1) op dζF (|x| ≤ n + aζ)e (H0 + i1)φn (j) +

OUT

+ 0

0

∞

−1

dζV (H0 + i1)

F (|x| ≥ n + aζ)op

× e−iH0 ζ (H0 + i1)φn (j)OUT L2 .

(B.4)

The first term in Eq. (B.4) goes to zero by the long estimate given in Appendix A. The second term can be estimated as follows (e−iH0 ζ (· · · ) ≤ (· · · )!) (H0 + i1)φn (j)OUT

∞

h(n + aζ)dζ 0

≤ (H0 + i1)φn (j)OUT

∞

n

h(R)dR .

(B.5)

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Since (H0 + i1)φn (j)OUT L2 = (p2 + i1)F (φn (j)OUT )(p)L2 ≤ (b2 + 1)F (φn (j)OUT )L2 < ∞

(B.6)

since F [φn (j)OUT ](p) has support in |p| ≤ b and obviously due to the fact that h(R) ∈ L1 (R), we have

∞

lim

n→∞

dR h(R) = 0.

(B.7)

n

We now give a proof of the following additional result in the Enss proof of asymptotic completeness presented in Sec. 2.3 lim |φn (j)IN , ψn | = 0.

(B.8)

n→∞

First, we remark that |φn (j)IN , ψn | ≤ |(W − φn (j)IN , e−iHtn ψ + (I − W − )φn (j)IN , e−iHtn ψ| (B.9) ≤ |W− e

iH0 tn

IN

−

IN

φn (j) , ψ| + (I − W )φn (j) ψ.

(B.10)

Here we have used the result (exercise) eiHt W− = W− eiH0 t .

(B.11)

As a consequence |φn (j)IN , ψn | ≤ (I − W − )φn (j)IN + |eiH0 tn φn (j)IN , (W − )∗ ψ| (B.12) since lim (1 − W − )φn (j)IN = 0

n→∞

(B.13)

we only need to prove that lim |eiH0 tn φn (j)IN , (W − )∗ ψ| = 0

n→∞

in order to show the validity of Eq. (B.8).

(B.14)

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Now, we have the set of estimates |eiH0 tn φn (j)IN , (W − )∗ ψ| = |(F (|x| ≤ n + atn ) + F (|x| ≥ n + atn ))eiH0 tn φn (j)IN , (W − )∗ ψ| ≤ {F (|x| ≤ n + atn )(W − )∗ ψF (|x| ≤ n + atn )eiH0 tn φn (j)IN +F (|x| ≥ n + atn )(W − )∗ ψF (|x| ≥ n + atn )eiH0 tn φn (j)IN ≤ (W − )∗ ψF (|x| ≤ n + atn )eiH0 tn φn (j)IN + F (|x| ≥ n + atn )(W − )∗ ψφn (j)IN .

(B.15)

At this point we observe that lim F (|x| ≤ n + atn )eiH0 tn φn (j)IN = 0

n→∞

by a similar analysis made in Appendix A.5

(B.16)

Appendix C In the appendix, we intend to show that compact resolvent perturbations of the free quantum dynamics H0 = − 12 ∆ does not modify the essential operator spectrum: defined as the union of the continuous spectrum and the accumulation points of the bound-states (point spectrum). Let us announce this deep result in its general terms. Theorem C.1. Let H = H0 + U and H0 self-adjoint operators acting on a separable Hilbert Space H. Let us suppose that all H0 spectrum is contained in positive real axis and such that (H0 +U +λ1)−1 −(H0 +λ1)−1 is a compact operator for some complex λ0 , with Im(λ0 ) = 0. Then the spectral part of the operator H contained in Real(λ) ≤ 0 is formed entirely by discrete points possesing zero as the only possible accumulation point for them. Proof. This result is obtained through an application of the Holomorphic Fredholm theorem to the following operatorial-valued holomorphic function: f (λ) = 1 − (λ − λ0 )[1 − (λ − λ0 )(H0 + λ1)−1 ]−1 × [(H0 + U + λ1)−1 − (H0 + λ1)−1 ] C(λ)-compact operator

(C.1)

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Since there is λ0 ∈ C\R+ such that f (x) is inversible (f (λ0 ) = 1!), we have that C(λ) has inverse in all C\R+ , with the exception of a discret set S ⊂ C\R+ . Since for λ with non-zero imaginary part C(λ) is always inversible, we have that S ⊂ R− . It is straightforward to see that S coincides with the negtive part of the spectrum of H. Just for completeness let us announce the holomorphic Fredholm theorem.7 Holomorphic Fredholm Theorem. Let Ω be a connected open set of C (a domain). Let f : Ω → (L(H, H), uniform ), a holomorphic bounded-operator valued function such that for each z ∈ Ω, f (z) is a compact operator. Then we have the Fredholm alternatives: (a) (1 − f (z))−1 does not exists for any z ∈ Ω or (b) (1 − f (z))−1 exists for z ∈ Ω\S, where S is a discret subset of Ω. In this case (1 − f (z))−1 is a meromorphic function in Ω, holomorphic in Ω\S and in z ∈ S, the equation f (z)ψ = ψ, has solely a finite number of linearly independent solutions in H.

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Chapter 3 On the Hilbert Space Integration Method for the Wave Equation and Some Applications to Wave Physics∗

3.1. Introduction One important topic of mathematical methods for wave propagation physics is to devise functional analytic techniques useful to implement approximatenumerical schemes.1 One important framework to implement such timeevolution studies is the famous Trotter–Kato–Feynman short-time method to approximate rigorously the full-wave evolution by means of a finite number of suitable short-time propagations — the well-known Feynman idea of a path-integral representation for the Cauchy problem.2 In this chapter we propose a pure Hilbert space analytical framework — the content of Secs. 3.2 and 3.3 — and suitable to generalize rigorously to the hyperbolic case, the powerful Trotter–Kato–Feynman formulae in Sec. 3.4. In Sec. 3.5, we present as a further example of usefulness of our abstract Hilbert space Techniques, a simple proof (by means of the use of the integral Cook–Kuroda criterium) of the existence of reduced (finite-time) scattering wave operators W± for elastic wave scattering by a compact supported inhomogenous medium (a “small” space-dependent Young medium elasticity modulus E(x) ∈ Cc∞ (R3 ), i.e. sup | E(x) E0 | 1 and ∇E(x) 3–7 sup | | 1). E0

In Sec. 3.6 and Appendix A, we address the very important problem of giving a complete proof for the asymptotic stability for systems in thermoelasticity in the relevant linearizing approximation for the nonlinear magnetoelasticity wave equations,8,9 i.e. a coupling between the divergences Lamé system of elastic vibrations and the convective parabolic equation for a constant external magnetic field. We give a (new) proof of stability for this system by producing an exponential uniform bound in time for the total energy of the linearized ∗ Author’s

original results. 58

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magnetoelastic model through the systematic use of simple estimates on the Trotter formula representations for our magnetoelastic wave propagators in our proposed Hilbert space abstract scheme. The resulting exponential bound solve an important long-standing problem on the structural form of the decay behavior of magnetoelastic systems in the presence of external constant magnetic fields. Finally, in Sec. 3.7, we use again Hilbert space techniques to produce a proof of existence and uniqueness of a semilinear problem of Abstract Klein Gordon propagation11 as another example of usefulness of our proposed abstract approach.

3.2. The Abstract Spectral Method — The Nondissipative Case Let A be a positive (essential) self-adjoint operator with domain D(A) acting on a separable Hilbert space H (0) with the inner product denoted by (, )(0) . Associated to the above abstract operator A, one can consider the wave equation (Cauchy problem) on H (0) × [0, T ] ∂2U = −AU + f, ∂2t

(3.1)

U (0) = U0 ;

(3.2)

Ut (0) = v0

with initial dates given in suitable subspaces of H (0) to be specified later and f ∈ L2 ([0, T ], H (0) ). It is an important problem in the mathematical physics associated with Eqs. (3.1) and (3.2) to determine exactly the subspaces where the initial date should belong in order to insure the existence of a global solution of Eqs. (3.1) and (3.2) in C ∞ ((0, ∞), H (0) ) when the time-interval [0, T ] of the external source f (x, t) is infinite (T = +∞) and besides, when there exist a natural extension of this global solution to the whole time interval (−∞, ∞), namely, U (x, t) ∈ C ∞ ((−∞, ∞); H (0) ). Let us present in this section a solution for this problem by means of a convenient generalization method exposed in Ref. 1 and based uniquely on the Stone spectral theorem for one-parameter unitary groups in Hilbert space.

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In order to implement this Hilbert space abstract method,1 we introduce the following Hilbert space: H (1) = {(ϕ, π) | ϕ ∈ D(A); π ∈ H (0) , ; },

(3.3a)

where the closure is taken in relation to the inner product below: (ϕ1 , π1 ); (ϕ2 , π2 ) = (Aϕ1 , ϕ2 )(0) + (π1 , π2 )(0) .

(3.3b)

Let us introduce the following operator LA on H (1) and defined by the rule LA (ϕ, π) = (π, Aϕ).

(3.4)

Note that Eq. (3.4) is well defined since we assume that the range of A is dense on H (0) . LA is a symmetric operator on H (1) since (ϕ1 , π1 ); LA (ϕ2 , π2 ) = (Aφ1 , π2 )(0) + (π1 , Aφ2 )(0) = LA (ϕ1 , π1 ), (ϕ2 , π2 ).

(3.5)

¯ A . We Let us consider the smallest closed extension of LA , denoted by L ¯ ¯ claim that its deficiency indices n+ (LA ) and n− (LA ) are identically zero and ¯ A ) ⊂ H (1) as a consequence leading to its self-adjointness on its domain D(L (0) which is explicitly given by the vectors (ϕ, π) such that (Aϕ, ϕ) < ∞ and π ∈ H (0) . Let (ϕ1 , π1 ) be a nonzero vector such that (ϕ1 , π1 ) ∈ (Range(i + LA))⊥ , or equivalently for any ϕ2 ∈ D(A) and π2 ∈ H (0) : (ϕ1 , π1 ); (i + LA )(ϕ2 , π2 )(1) = 0,

(3.6)

which is the same content of the equation below: −i(Aϕ1 , ϕ2 )(0) + (π1 , Aϕ2 )(0) + i(π1 , π2 )(0) + (Aϕ1 , π2 ) = 0,

(3.7)

and due to the fact that Range(A ± i) is dense on H (0) and since A is selfadjoint it leads to ϕ1 = 0 and π1 = 0. In the same way one can prove that ¯ A ) = 0. n− (L Collecting all the above exposed results, one has the following theorem: Theorem 3.1. Let the spectral theorem be applied to the one-parameter ¯ A ); one has that it satisfies the ¯ A ) acting on D(L group U (t) = exp(itL abstract wave equation (3.1) on the strong sense for dates U0 ∈ D(A) and v0 ∈ H (0) and leading to a global (time-reversing) solution on C ∞ ((−∞, ∞), H (0) ).

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Proof. It is straightforward that in components the wave equation associated with the one-parameter semigroup U (t)(ϕ0 , π0 ) = (ϕ(t), π(t)) is given by ∂ ϕ(t) = iπ(t), ∂t ∂ π(t) = i(Aφ)(t), ∂t

(3.8) (3.9)

which is the same as Eqs. (3.1) and (3.2) without the external term f (x, t): ∂ 2 ϕ(t) = −(Aϕ)(t) ∂ 2t

(3.10)

ϕ(0) = U0 ,

(3.11)

ϕt (0) = v0 .

(3.12)

with the initial date

It is worth noting that one has always a weak solution of Eqs. (3.1) and (3.2) with a nonzero external term t U0 0 ϕ ¯A ¯A itL i(t−s)L ds e + , (3.13) (t) = e −if (s) π −iϕ0 0 namely, ϕ(·, t) ∈ C((−∞, ∞); H (0) ), unless f (·, t) belongs to D(A)(f : (−∞, ∞) → D(A)), in which case it produces a global C ∞ timedifferentiable solution. At this point, we make one of the most important remark that comes from such a theorem applied to the subject of random wave propagation.2 It is important to know the correct mathematical meaning of considering the Cauchy problem for the wave equation with both initial conditions now being stochastic process with realizations on L2 (Ω) space of functions (see Appendix B). In this case, Eq. (3.13) in its weak-integral form should be considered the correct mathematical solution for this problem since one can take safely the initial dates (U0 , v0 ) both in the Hilbert space H (0) , which in the usual case of wave propagation is the L2 (Ω) functional Hilbert space (H (0) = L2 (Ω)). Let us point out an abstract version of the wave total energy conservation during the time evolution. First, we observe that for any −∞ < t < ∞ we have the identity Im{(π(t), Aϕ(t))(0) } ≡ 0,

(3.14)

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which is obtained as an immediate consequence of the unitary property of the semigroup U (t) on the two-component Hilbert space H (1) given by Eqs. (3.3a) and (3.3b): ¯ A )(ϕ0 , π0 ) ¯ A )(ϕ0 , π0 ); exp(itL exp(itL = (ϕ0 , π0 ); (ϕ0 , π0 ),

(3.15)

and by deriving the above equation in relation to time, it yields ¯ A (ϕ(t), π(t)), (ϕ(t), π(t)) iL ¯ A (ϕ(t), π(t)) = 0, + (ϕ(t), π(t)); iL

(3.16)

which is Eq. (3.14) when it is written in terms of components. At this point one can easily see that the energy functional given by E(t) = is such that

1 1 (π(t), π(t))(0) + (φ(t), (Aφ)(t)) 2 2

dE(t) 1 1 = (i(Aφ)(t), π(t)) + (π(t), i(Aφ)(t)) 0 dt 2 2 1 1 + (iπ(t), (Aφ)(t)) + ((Aφ)(t), iπ(t)) 0 2 2 = 0.

(3.17)

(3.18)

As a consequence, we have the energy abstract conservation law 1 1 (v0 , v0 )(0) + (U0 , AU0 )(0) . (3.19) 2 2 Finally, we remark that by a straightforward application of the Banach fixed-point theorem within the context of our operatorial scheme, one has the existence and uniqueness of the nonlinear abstract wave equation initial value problem on H (1) : E(t) =

∂ 2U = −AU + F (U ), ∂2t U (0) = U0 ; Ut (0) = v0 ,

(3.20)

for F (x) denoting a Lipschitzian function on the real line (|F (x) − F (y)| ≤ C|x − y|). This result is the consequence of the fact that the application T : C([0, T ], H (1) ) → C([0, T ]; H (1) ) t ¯ ds{ei(t−s)LA (0, F (ϕ))} T (ϕ, π) = 0

(3.21)

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is a contraction on H (1) , i.e. ((Aϕ, ϕ)H ≥ C0 (ϕ, ϕ)H ): sup ||(0, F (ϕ1 (s)) − F (ϕ2 (s))||H (1)

0≤s≤T

≤ sup {(A0, 0)(0) + (F (ϕ1 (s)) − F (ϕ2 (s)), F (ϕ1 (s) − F (ϕ2 (s)))(0) } 0≤s≤T

≤ C sup (ϕ1 (s) − ϕ2 (s), ϕ1 (s) − ϕ2 (s))(0) 0≤s≤T

≤ CC0 sup {||(ϕ1 , π1 ) − (ϕ2 , π2 )||H (1) (s)},

(3.22)

0≤s≤T

which after successive iterations produces bounds leading to the global weak solutions of Eq. (3.20) in C((−∞, ∞), H (1) ), a nontrivial result easily obtained in this abstract spectral framework (see Sec. 3.7 for further studies on this problem). 3.3. The Abstract Spectral Method — The Dissipative Case In this section we generalize the Hilbert space method of Sec. 3.2 to the case of a wave equation in the presence of damping terms, a very important problem on statistical continuum mechanics.4 We, thus, consider the abstract problem in the notation of Sec. 3.2: ∂U ∂2U + B + (AU )(t) = 0, (3.23) ∂2t ∂t U (0) = U0 ∈ D(A),

(3.24)

Ut (0) = v0 ∈ H (0) ,

(3.25)

where A and B are essential positive self-adjoint operators such that D(B) ⊃ D(A). Note that we can write Eqs. (3.23)–(3.25) in the two-component Hilbert space H (1) of Sec. 3.2 as d U U 0 −1 =− . (3.26) A B dt π π Let us observe that the operator Aˆ =

0 −1 A B

is positive on H (1) as a

consequence of the operator B positive definiteness, namely, ˆ 1 , π1 ), (U1 , π1 ) = (Bπ1 , π1 )(0) > 0. A(U

(3.27)

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Since ∃ λ ∈ R ≥ 1 such that λ1 + Aˆ is onto and bijective on H (1) (a topological isomorphism), one can apply, thus, the well-known Hille– Yoshida theorem1 to the operator Aˆ in order to consider a contractive C0 semigroup on H (1) in the interval [0, ∞) possessing Aˆ as its generator, and naturally producing the unique global weak solution of the abstract wave equation with damping on C ∞ ([0, ∞), H (1) ) if the initial conditions are chosen to be in H (1) .

U (x, t) dU dt (x, t)

U0 −1 B v0

0 = exp −t A

(3.28)

as much as in our study presented in Sec. 3.2. For example, let us consider A as a strongly positive elliptic operator arding of order 2m on a suitable domain Ω ⊂ RN and satisfying the G¨ 3 condition there.

A=

(−1)|α| Dxα (aαβ (x)Dxβ ,

(3.29)

|a|≤m |β|≤m

D(A) = H 2m (Ω) ∩ H0m (Ω), Real(Af, f )L2 (Ω) ≥ C0 ||f ||H 2m (Ω) .

(3.30) (3.31)

The damping operator B is taken to be explicitly a strictly positive function ν(x) of C ∞ (Ω). As a result of this section, we have weak global solution on ∞ C ([0, ∞), H 2m (Ω) ∩ H0m (Ω)) of the damped wave equation problem in Ω with f (x, t) ∈ L∞ ([0, ∞), L2 (Ω)).4 dU d2 U = −(AU )(x, t) − ν(x) (x, t) + f (x, t) d2 t dt U (x, 0) = f (x) ∈ H 2m (Ω) ∩ H0m (Ω) 2

Ut (x, 0) = g(x) ∈ L (Ω).

(3.32) (3.33) (3.34)

Finally, let us show that the operator on D(A) ⊕ H (0) = H (1) : Aˆ =

0 A

−I B

(3.35)

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is a topological isomorphism from its (dense) domain to the complete Hilbert space H (1) as a consequence of the fact that the self-adjoint operator A + B on D(A) satisfies the Hille–Yoshida criterion: For any real number λ > 0, the equation (A + B + λ1)y = f has a unique solution on D(A) for each f ∈ H (0) . The same Hille–Yoshida criterion applied to the full twocomponent operator Eq. (3.35), shows us that for each (f 1 , f 2 ) ∈ H (1) , one can straightforwardly find (U 1 , U 2 ) ∈ D(A) × D(A) with λ = 1, namely, U 1 = (f 1 + U2 ) and (A + B + 1)U 2 = f 2 − Af 1 ∈ H (0) (Af 1 ∈ H (0) , since f 1 ∈ D(A)!). The uniqueness of (U 1 , U 2 ) is a direct consequence of the strict positivity of the operator A + B on D(A).

3.4. The Wave Equation “Path-Integral” Propagator In this section of a more physicist-oriented nature, we consider the classical unsolved problem of writing a rigorous Trotter like path-integral formulae for the Cauchy problem on R3 associated with the elastic vibrations of an infinite length string with a specific mass (ρ(x)−ρ0 ) ∈ Cc∞ (R3 ) and a Young elasticity modulus (E(x) − E0 ) ∈ Cc∞ (R3 ). The general wave evolution equation governing such vibrations is given by3,4 ∂ ∂t

∂U (x, t) ρ(x) = ∇(E(x)∇ · U (x, t)), ∂t U (x, 0) = f (x) ∈ H 2 (Ω), 2

Ut (x, 0) = g(x) ∈ L (Ω).

(3.36) (3.37) (3.38)

Just for simplicity of our mathematical arguments, we consider the case of a string with a constant specific mass ρ(x) = ρ0 in our next discussions. Firstly, let us show that the operator A=

E0 ∆+∇ ρ0

(E(x) − E0 ) ∇ = A(0) + B ρ0

(3.39)

is a self-adjoint operator on the domain H 2 (Ω). This result is an immediate of the fact that the “perturbation” operator B = consequence E(x)−E0 ∇ is A(0) -bounded with A(0) -bound less than 1. As a conse∇ ρ0 quence, one can apply straightforwardly the Kato–Rellich theorem to guarantee the self-adjointness of A on H 2 (Ω). The A(0) -boundedness of B is a

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result of the following estimates for ϕ ∈ H 2 (Ω) (where E(x) E0 and |∇E(x)| E0 ):

∇ E(x) − E0 ∇ ϕ

2 ρ0 L (Ω)

|E(x) − E0 | ∇E(x)

E0 ∆ϕ ≤ sup , γ(Ω)

2 , (3.40)

E0 E0 ρ0 x∈Ω L (Ω) where γ(Ω) is the universal constant on the Poincaré relation for function ϕ(x) on H 2 (Ω) ||∇ϕ||L2 (Ω) ≤ γ(Ω)||∆ϕ||L2 (Ω) .

(3.41)

By the direct application of Theorem 3.1, we have the rigorous solution for the Cauchy problem Eqs. (3.36)–(3.38) if f (x) ∈ H 2 (Ω) and g(x) ∈ L2 (Ω) for any −∞ ≤ t ≤ +∞, namely,     0 1  f (x) U (x, t)  . (3.42) = exp it  E(x) ∂U  (x, t) 0  g(x) ∇ ρ ∇ ∂t 0

At this point, we apply the Trotter–Kato theorem to rewrite Eq. (3.42) in the short-time propagation form n 0 1 U (x, t) t f (x) exp i = Strong limit . ∂U(x,t) 0 ∇ E(x) g(x) n (H 2 (Ω)⊕L2 (Ω)) n→∞ ρ0 ∇ ∂t (3.43) However, for short-time propagation, we have the usual asymptotic result        0 1 f (x) lim exp iε  ∂ E(x) ∂  g  ε→0+  0 ∂x ρ ∂x 0

+∞

dpeipx

= lim

ε→0+

−∞

          

0 2 B 6 @iε4

e

0 E(x) − ρ0 p2 +

     5A E (x)  ˜ (ip) 0 f ρ0 (p)  g˜     31

17C

(3.44)

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with f˜(p) and g˜(p) denoting the Fourier transforms of the functions f (x)

p for and g(x), respectively, and E (x)p means the scalar product ∇E(x)

short. It is worth calling to the reader’s attention that ! " 0 1 f˜(p) exp iε p2 + Eρ(x) ip 0 − E(x) g˜(p) ρ0 0 12 ˜ f (p) E(x) 2 E (x) p + ip = cos ε − ρ0 ρ0 g˜(p) 12   E (x) 2 p + ip −i sin ε − E(x) iε˜ g(p) ρ0 ρ0 (1/2)  . # p2 + Eρ(x) ip ε − E(x) f˜(p) 2 + E (x) ip − E(x) p ρ 0 0 ρ0 ρ0 (3.45) As a consequence of the above formulae we have our discretized pathintegral representation for the initial-value Green function of Eqs. (3.36)– (3.38), i.e. by writing the evolution operator as an integral operatorFeynman Propagator  2 3   0 1   6  7 6 7    5  it4 ∂ E(x) ∂ 0  f ∂x ρ0 ∂x e   g           +∞ G11 (x, x , t) G12 (x, x , t) f (x ) = dx (3.46) g(x ) G21 (x, x , t) G22 (x, x , t) −∞ we have (for A = 1, 2; B = 1, 2) the “path-integral” product representation n +∞ +∞ 1 dp1 · · · dpn dxn−1 · · · GAB (x, x , t) = strong − lim n→∞ 2π −∞ −∞    +∞ n $ t  GAj Aj+1 xj , pj ; dx1 ×  ×  n −∞ j=1

× exp(ipj (xj − xj−1 ),

(3.47)

where GAj Bj (xj , pj , nt ) is the 2 × 2 matrix on the integrand of Eq. (3.31) for x = xj and p = pj . Note that xn = x and x0 = x .

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3.5. On The Existence of Wave-Scattering Operators In this section we give a proof for the existence of somewhat reduced wave˜ H ˜ 0) scattering operators associated to the following wave dynamics pair (H, (1) 2 2 N 7 on the H = H (Ω) ⊕ L (Ω) (with Ω = R , N ≥ 3) : ˜0 = H ˜ = H

0 E0 ρ0

1 ∆ 0

0

∇·

E(x) ρ0

,

(3.48)

1 ∇ 0

.

(3.49)

We need, thus, to show the strong convergence on H (1) of the following family of one-parameter evolution wave operators (s = strong) for T arbitrary (but finite): ˜ H ˜ 0 ) = S − lim (exp(itH) ˜ exp(−itH ˜ 0 ). W ± (H, t→T

(3.50)

Let us remark that the existence of the strong limit of Eq. (3.50) is a direct consequence of the validity of the famous Cook–Kuroda–Jauch ˜ H ˜ 0 ) in a dense subset of the integral criterion6 applied to the pair (H, ˜ domain of H. This integral criterion means that there exists a δ > 0, such that the following integral is finite for any (f, g) belonging to a dense set of H (1) , i.e. δ

T

˜0t f iH ˜ ˜

dt (H − H0 )e g

H (1)

< ∞.

(3.51)

˜ Let us denote E(x) = (E(x) − E0 ) ∈ Cc∞ (RN ), the scattering wave potential in what follows. We can, thus, rewrite Eq. (3.51) in the following explicit form: δ

T

˜ dt((−∇(E(x)∇)g(x, t), g(x, t)))L2 (RN ) < ∞

(3.52)

or

T δ

˜ ˜ dt((+∇(E(x)∇)f (x, t), +∇(E(x)∇)f (x, t)))L2 (RN ) < ∞,

(3.53)

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where the pair of functions f (x, t) and g(x, t) satisfy the free wave equation d2 f (x, t) = d2 t

E0 ∆ f (x, t), ρ

f (x, 0) = f ∧ (x), ∧

ft (x, 0) = g (x),

(3.54) (3.55) (3.56)

and g(x, t) =

d f (x, t). dt

(3.57)

Let us choose the initial dates f ∧ (x) and g ∧ (x) as the Fourier transforms of L2 (RN ) functions polynomial-exponentially bounded ( ∈ Z+ and Λ ∈ R+ ):

∧

f (x) = RN

dN keikx f˜Λ (k);

|f˜Λ (k)| ≤ M e−Λ|k| |k| ,

(3.58)

dn keikx g˜Λ (k);

|˜ gΛ (k)| ≤ M e−Λ|k| |k| .

(3.59)

∧

g (x) = RN

Note that we have an explicit representation in momentum space for the free waves f (x, t) and g(x, t), namely, g˜Λ (|k|) sin(|k|t), f˜(k, t) = f˜Λ (|k|) cos(|k|t) + |k| ∂ g˜(k, t) = (f˜(k, t)). ∂t

(3.60) (3.61)

At this point we observe the estimates for any δ > 0:

T

δ

˜ ˜ · ∇f, E∆f ˜ ˜ · ∇f )L2 (RN ) (t)| dt|(E∆f + ∇E + ∇E

T

≤

dt δ

2 4˜ ˜ sup |E(x)| (k f (k, t), f˜(k, t))L2 (RN )

x∈RN

2 ˜ ˜ E(x)))(k + sup (max(∇i E(x), ki f˜(k, t), f˜(k, t))L2 (RN ) x∈RN

˜ ˜ ˜ ˜ + sup max(∇i E(x), ∇j E(x))(ki kj f (k, t), f (k, t))L2 (RN ) . x∈RN

(3.62)

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By using the explicit representations Eqs. (3.60) and (3.61) and the simple estimate below for α > 0 with t ∈ [δ, T ) and δ > 0 (with T < ∞)

∞

I(t) =

d|k||k|α+ cos2 (|k|t)e−Λ|k|

0

≤

1 tα+1+

×

0

∞

dU U α+ e−

ΛU T

≤

¯ M t1+α

∈ L1 ([δ, ∞)),

(3.63)

one can see that by just choosing δ > 0, the existence of the reduced (T < ∞) wave operators of our pair Eqs. (3.48)–(3.49) is a direct consequence of the application of the Cook–Kuroda–Jauch criterion,6 since the set of functions polynomial-exponentially bounded is dense on C0 (RN ) (continuous functions vanishing at the infinite), and so, dense in L2 (RN ).7 The complete wave operator (with T → +∞) can be obtained by means of Zorn theorem applied to the chain of reduced wave operators {W± (T )} ordered by inclusion, i.e. T1 < T2 ⇒ W± (T1 ) ⊂ W± (T2 ). 3.6. Exponential Stability in Two-Dimensional Magneto-Elasticity: A Proof on a Dissipative Medium One interesting question in the magneto-elasticity property of an isotropic (uncompressible) medium vibration interacting with an external magnetic is whether the energy of the total system should decay with an exponential uniform bound as time reaches infinity.8 We aim in this section to give a rigorous proof of this exponential bound for a model of imaginary medium electric conductivity as an interesting example of abstract techniques developed in the previous sections. (See Appendix A for a complementary analysis on this problem.) The governing differential equations for the electric–magnetic medium

(x, t) depending on the time displacement vector U = (U 1 , U 2 , 0) ≡ U variable t ∈ [0, ∞) with x ∈ Ω and the intrinsic two-dimensional magnetic field h(x, t) = (h1 , h2 , 0) are given by

t) ∂ 2 U(x,

× h](x, t) × B),

= ∆U(x, t) + ([∇ 2 ∂ t ∂ h(x, t)

× ∂U (x, t) × B

= ∆ h(x, t) + β ∇ . β ∂t ∂t

(3.64) (3.65)

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= (B, 0, 0) is a constant vector external magnetic field, and β Here, B is the (constant) medium electric conductivity. Additionally, one has the initial conditions

(0, x) = U

0 (x); U

t (0, x) = U

1 (x); U

h(0, x) = h0 (x)

(3.66)

and the (physical) Dirichlet-type boundary conditions with n(x) being the normal of the boundary of the medium Ω9 :

( x, t)|Ω = 0; U

× h))( x, t)|Ω = 0. ( n × (∇

(3.67)

Let us follow the previous studies of Sec. 3.1 to consider the contraction semigroup defined by the following essentially self-adjoint operators in the H-energy space: ˆ Ω = {(U

, π , h) ∈ (L2 (Ω))3 | (∇U

, ∇U)

L2 (Ω) H + ( π , π )L2 (Ω) + ( h, h)L2 (Ω) < ∞}, namely, 

0 L0 = i∆ 0 and

 0  V = 0 0

i 0 0

0 0

× [( ) × B])

−(∇

 0 0  −∆

(3.68)

0

×( [∇



 )] × B .

(3.69)

0

ˆ Ω (see We, thus, have the contractive C0 -semigroup acting on H Appendix A for technical proof details):

, π , h) = exp(t(+iL0 − V ))(U

, π , h)(0). Tt (U

(3.70)

It is worth to call the reader’s attention that we have proved the essential self-adjointness of the operator V by using explicitly the boundary conditions on the relationships below:

× ( h)] × B)

= (B, B(∂1 h2 − ∂2 h1 )), ([∇

× ( h)] × B) = (−B∂2 π 2 , B∂1 π 2 , 0). ([∇

(3.71) (3.72)

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By standard theorems on contraction semigroup theory,8 one has that the left-hand side of Eq. (3.70) satisfies the magneto-elasticity equations in √ the strong sense with β = i = −1, a pure imaginary electric conductivity medium, physically meaning that the medium has “electromagnetic dissipation” and in the context that the initial values belong to the subspace ˆΩ. (H 2 (Ω) ∩ H01 (Ω))2 ⊕ (L2 (Ω))2 ⊕ (L2 (Ω))2 ⊂ H Let us note that Eq. (3.70) still produces a weak integral solution on ˆ Ω = (H01 (Ω))2 ⊕ (L2 (Ω))2 ⊕ (L2 (Ω))2 , the result the full Hilbert space H suitable when one has initial random conditions sampled on L2 (Ω) by the Minlos theorem (see Appendix B). ˆ Ω we have the following energy estimate:

0 , π0 , h0 ) ∈ H Thus, for any (U

, h)|| ˜

, ∂t U ||(U HΩ

2 + | h|2 )(x, t)

|2 + |∂t U| ≡ dx(|∇U Ω

h(0))|| ˜

(0), ∂t U(0), ≤ ||(exp(t(iL0 − V )))(U HΩ

n

it t

L0 exp − V ≤ S − lim (U0 , U1 , h0 ) exp

˜ n→∞ n n HΩ

(0), h(0)|| ˜ ,

0 , ∂t U ≤ exp(−tω(V ))||U HΩ

(3.73)

where ω(V ) is the infimum of the spectrum of the self-adjoint operator V ˆ on the space H. In order to determine the spectrum of the self-adjoint operator V , which is a discrete set since Ω is a compact region of R2 , we consider the associated V -eigenfunctions problem

= λn πn , [∇ × ( hn )] × B

(3.74)

= λn hn , −∇ × [( πn ) × B]

(3.75)

which, by its turn, leads to the usual spectral problem for the Laplacean with the usual Dirichlet boundary conditions on Ω: ∆π2n = − π2n (x)|∂Ω = 0.

λ2n B

π2n ,

(3.76) (3.77)

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As a result of Eqs. (3.75) and (3.76), one finally gets the exponential bound for the total magneto-elastic energy equation (3.73):

, h)|| ˜ (t)

, ∂t U ||(U HΩ ( ' )

(0), h(0)|| ˜ ,

0 , ∂t U ≤ exp − t Bλ0 (Ω) ||(U HΩ

(3.78)

where λ0 (Ω) = inf{spec(−∆)} em H 2 (Ω) ∩ H01 (Ω).

3.7. An Abstract Semilinear Klein Gordon Wave Equation — Existence and Uniqueness Let A be a positive self-adjoint operator as in Sec. 3.2 and (B1 , B2 ) a pair of bounded operators with domain containing D(A) (the A-operator domain). We can, thus, associate to these operatorial objects the following semilinear Klein–Gordon equation with initial dates (zero time field Hilbert-valued configurations): ∂U ∂2U = −AU − B1 + B2 U + F (U, t) ∂2t ∂t U (0) = U0 ∈ D(A)

(3.79)

Ut (0) = V0 ∈ H 0 . Here, F (U, t) is a Lipschitzian function on the Hilbert space H (0) with a Lipschitz-bounded being a function L(t) on Lq (R+ ) = {set of mensurable )1/q '* ∞ functions f (t) on R+ such that ||f ||Lq = 0 ds|f (s)|q < ∞ and q ≥ 1}, namely, |F (U1 , t) − F (U2 , t)|H (0) ≤ L(t)|U1 − U2 |H (0) ,

(3.80a)

F (0, t) ≡ 0.

(3.80b)

It is an important problem in the mathematical physics associated with Eq. (3.79) to find a functional space, where one can show the existence and uniqueness of the Cauchy problem for the global time interval [0, ∞).11 Let us address this problem through the abstract techniques developed in the previous section of this work and the Banach fixed-point theorem. First, let us write Eq. (3.79) in the integral operatorial form t U 0 ˆ ) U0 ˆ) t(iLA +V (t−s)(iLA +V dse + (t) = e , (3.81) 1 π V0 0 i F (U, s)

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where Vˆ is the bounded operator with operatorial norm µ acting on the two-component space H (1) by means of the rule: Vˆ (U, π) = (V1 U, V2 π),

(3.82)

where the (bounded) component operators are written explicitly in terms of the “damping” operator B1 and “mass” operator B2 , as given below: V1 = B1 − V2 , 2V2 = −B1 ±

(B12

(3.83) 1/2

− 4B2 )

.

(3.84)

Let us introduce the following exponential weighted space of H (1) -valued continuous functions: + Ck ([0, ∞), H (1) ) = g : R+ → H (1) , with K > 0 , (k) and ||g||H (1) = sup (e−kt ||g(t)||H (1) ) < ∞ . (3.85) 0≤t 0. 1 By choosing Eq. (4.14) p big enough such that ap ≡ γ(Ω) − 2p > 0, we have that 1 d 1 (U (n) 2L2 (Ω) ) + ap U (n) 2L2 (Ω) ≤ pf 2L2(Ω) (t), 2 dt 2

(4.15)

or by the Gronwall lemma, that there is a uniform constant M [even for T = +∞ if f ∈ L2 ([0, ∞], L2 (Ω))] such that sup (U (n) (t)2L2 (Ω) )

0≤t≤T

≤ sup

0≤t≤T

e−ap T

0

T

dsf 2L2(Ω) (s)e+ap s + g2L2(Ω)

= M. (4.16)

Note that the Galerkin system of (nonlinear) ordinary differential equa(n) tions for {Ui (t)} has a unique global solution on the interval [0, T ] as a n (n) consequence of Eq. (4.16) — which means that i=1 (Ui (t))2 ≤ M — and the Peano–Caratheodory theorem since f (t, ·) ∈ L∞ ([0, T ]). As a consequence of the Banach–Alaoglu theorem applied to the bounded set {U (n) } on L∞ ([0, T ], L2 (Ω)), there is a subsequence weak-star ¯ x) ∈ L∞ ([0, T ], L2 (Ω)), and this convergent to an element (function) U(t, subsequence will still be denoted by {U (n) (x, t)} in the analysis that follows. An important remark is at this point. Since F (x) is a Lipschitzian function on the closed interval where it is defined we have that the set of functions {F (U (n) )} is a bounded set on L∞ ([0, T ], L2(Ω)) if the set {U (n) } has this property. As a consequence of this remark and of a priori inequalities given in detail in Appendix A, we can apply the famous Aubin–Lion theorem1 to insure the strong convergence of the sequence {U (n) (x, t)} to the function ¯ (x, t). As a straightforward consequence, our special nonlinearity on U Eq. (4.1), namely F (x) is a Lipschitzian function; we obtain, thus, the

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strong convergence of the sequence {F (U (n) (x, t)} to the L2 (Ω)-function ¯ (x, t)). F (U As a consequence of the above-made comments and by noting that L2 (Ω) is continuously immersed in H −2 (Ω) and the weak-star convergence definition, we have the weak equation below in the test space function of v(x, t) ∈ C0∞ ([0, T ], H 2 (Ω) ∩ H01 (Ω)):

T (n) dv (n) (n) lim dt −U , + (AU , v)L2 (Ω) − (F (U ), ∆v)L2 (Ω) n→∞ 0 dt L2 (Ω) = lim

n→∞

0

T

dt(f (n) , v)L2 (Ω) ,

(4.17)

or by passing the weak-star limit T dv(x, t) ¯ ¯ (x, t), (Av)(x, t))L2 (Ω) dt −U (x, t), + (U dt 0 L2 (Ω) ¯ (x, t)), ∆v(x, t))L2 (Ω) = − (F (U

T

0

dt(f (x, t), v(x, t))L2 (Ω) . (4.18)

This concludes the existence proof of Theorem 4.1, since v(0, x) = v(T, x) = 0. Let us apply this to a concrete problem of random exponential nonlinear diffusion on a cube [0, L]3 ⊂ R3 , mainly for the explanation of the abovewritten abstract results: k0 ∂U (x, t) k0 =∇ + e−U(x,t) ∇U (x, t) + ∆U (x, t) (4.19) ∂t 2 2 with U (x, 0) = g(x) ∈ L2 (Ω), U (x, t)x∈[0,L]3 ≡ 0,

(4.20) (4.21)

where g(x) are the samples of a stochastic Gaussian process on Ω with correlation function defining an operator of Trace class on L2 (Ω), namely, E{g(x)g(y)} = K(x, y) ∈ (L2 (Ω)) (4.22) 1

or in terms of the spectral set associated with the Laplacian, i.e. g(x) = lim

n→∞

n i=0

(n)

gi ϕi (x)

(4.23)

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Nonlinear Diffusion and Wave-Damped Propagation

with

E{gi gk } = Kik =

Ω×Ω

dxdyK(x, y)ϕi (x)ϕk (y).

(4.24)

As a consequence of Theorem 4.1, an explicit solution of Eq. (4.19) must be taken in L∞ ([0, ∞), L2 (Ω)) in the weak sense of Eq. (4.18) and given thus by the Trigonometrical series sequence of functions: U (x, t, [g]) = n lim →∞ 1 n2 →∞ n3 →∞

 ,n2 ,n3 n1 

di,k,j (t, [g ]) sin

i,j,k=1

  jπ kπ iπ x sin y sin z ,  L L L (4.25)

where the set of absolutely continuous function {di,j,k (t, [g0 ])} on [0, T ] satisfies the Galerkin set of ordinary differential equations with “random” Gaussian initial conditions iπ jπ kπ = g . di,j,k (0, [g ]) = g(x), sin x sin y sin z L L L L2 (Ω) (4.26) Let us now comment on the uniqueness of solution problem for the nonlinear initial-value diffusion equations (4.1)–(4.3). In the case under study, the uniqueness comes from the following technical lemma. ¯(2) in L∞ ([0, T ] × L2 (Ω)) are two functions ¯(1) and U Lemma 4.1. If U satisfying the weak relationship, for any v ∈ C0∞ ([0, T ], H 2 (Ω) ∩ H01 (Ω))

T

¯(1) − U ¯(2) , − ∂v ¯(1) − U ¯(2) , Av)L2 (Ω) U + (U ∂t L2 (Ω) ¯ ¯ − (F (U(1) ) − F (U(2) ); ∆v)L2 (Ω) ≡ 0, (4.27a)

dt 0

¯(2) a.e. in L∞ ([0, T ] × L2 (Ω)).5 ¯(1) = U then U Let us notice that Eq. (4.27a) is a weak statement on the assumption of vanishing initial values and it is a consequence of ¯(1) (x, t) − U ¯(2) (x, t)|2 = 0. d3 x|U (4.27b) lim supp ess t→0+

Ω

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The proof of Eq. (4.27) is a direct consequence of the fact that F (x) is a Lipschitz function satisfying an inequality of the form ¯(2) )|(x, t) ≤ ¯(1) − U ¯(2) |(x, t) (4.27c) ¯(1) ) − F (U sup F (x) |U |F (U −∞<x ν1 and the previous bounds Eq. (4.39) that there is a positive constant B such that

T

0

dsU˙ (n) (s)2L2 (Ω) ≤ B = M T,

(4.49)

which by its turn yields that dUndt(t,x) is weakly convergent on P¯ (x, t) in L2 ([0, T ], L2(Ω)). As a consequence of general theorems of function convergence on the space of integrable functions (Aubin–Lion theorem1 again), one has that ¯ (x, t) almost everywhere on Ω, P¯ (x, t) is the time-derivative of the function U (n) since it is expected that U (x, t) should be a strongly convergent sequence ¯ (x, t) on the separable and reflexive Banach space L∞ ([0, T ] × L2 (Ω)).1 to U (See Appendix B for mathematical details.) 4.4. A Path-Integral Solution for the Parabolic Nonlinear Diffusion Let us start the physicist-oriented section of our note by writing the abstract scalar parabolic equations (4.1)–(4.3) in the integral (weak) form U (t) = e−At U (0) +

t

ds e−(t−s)A ∆F (U (s)),

(4.50)

0

where F (x) is a general nonlinear scalar functional such that F : L2 (Ω) → H 2 (Ω). Let us suppose that the initial conditions for Eq. (4.1) are samples of Gaussian stochastic processes belonging to the L2 (Ω) space. For instance, this mathematical fact is always true when the processes correlation function defines an integral operator of the trace class on L2 (Ω), E{U (0, x)U (0, y)} = K(x, y), where

(4.51)

Ω

dxK(x, x) < ∞.

(4.52)

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At a formal path-integral method , one aims to compute the initial condition average of arbitrary product of the function U (x, t) ≡ Ux (t) for arbitrary space–time points. This task is achieved by considering the associated characteristic functional for Eq. (4.51), namely, T 1 E exp i dtj(t), U (t) L2 (Ω) . (4.53) Z[j(t)] = Z(0) 0 In order to write a path-integral representation for Eq. (4.53), we follow our previous studies on the subject6 by realizing the random initial conditions U (0) = U0 average as a Gaussian functional integral on L2 (Ω), namely,     1  Z[f (t)] = dµ[U0 ] × dU (x, t) Z(0)  L2 (Ω) L2 ([0,T ],L2 (Ω)) x∈Ω,t∈[0,T ]

t × δ (F ) U (t) − e−At U0 + ds e−(t−s)A ∆F (U (s)) 0

× exp i

0

∞

dtj(t), U (t) L2 (Ω)

  

,

(4.54)

where formally 1 −1 dµ[U0 ] = (dU0 (x)) × exp − dxdyU0 (x)K (x, y)U0 (y) . 2 Ω x∈Ω

(4.55) By using the well-known Fourier functional integral representation for the delta-functional inside the identity equation (4.54)6 (see this old idea in A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Vol. 2 (MIT Press, Cambridge, 1971),6 one can rewrite Eq. (4.54) in the following form: 1 dλ(t) Z[j(t)] = (dU (x, t)) × Z(0) L∞ ([0,T ],L2 (Ω)) L2 ([0,T ],L2 (Ω)) ! " t t −(t−s)A × exp i ds e ∆F (U )(s) λ(t), U (t) + 0

0

L2 (Ω)

$ 1 t # ds λ(s)((e−As K −1 e+As )λ)(s) L2 (Ω) . × exp − 2 0

(4.56)

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After evaluating the Gaussian cylindrical measure associated with the Lagrange multiplier fields λ(x, t), one obtains our formal path-integral representation, however, with a weight well-defined mathematically as showed in Sec. 4.2, as the main conclusion of Secs. 4.2–4.4 1 Z[j(t)] = Z(0)

dU (x, t)

L∞ ((0,T ),L2 (Ω))

T 1 T ds ds × exp − 2 0 0 ! ∂U + AU − ∆F (U ) (s); × ∂s " ∂U × + AU − ∆F (U ) (s ) ∂s L2 (Ω) T

× exp i

dsj(s), U (s) L2 (Ω)

0

.

(4.57)

It is worth rewriting the result equation (4.57) in the usual physicist notation of Feynman path-integrals6 1 Z[f (x, t)] = Z(0)

DF [U (x, s)]

L∞ ((0,T ),L2 (Ω))

∂U + AU − ∆F (U ) (x, s) ∂t 0 0 Ω ∂U × + AU − ∆F (U ) (x, s ) ∂t T × exp i ds dx j(s, x)U (s, x) . (4.58) × exp −

1 2

0

T

ds

T

ds

dx

Ω

4.5. Random Anomalous Diffusion, A Semigroup Approach Recently, it has been an important issue on mathematical physics of diffusion on random porous medium, the study of the anomalous diffusion equation on the full space RD but under the presence of a positive random

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97

potential as modeled by the parabolic quasilinear equation (0 < t < ∞) as written below7 : ∂U (x, t) = −D0 (−∆)α U (x, t) + V (x)U (x, t) + F (U (x, t)), (4.59a) ∂t U (x, 0) = f (x),

(4.59b)

where U (x, t) is the diffusion field, D0 is the medium diffusibility constant, V (x) denotes the stochastic samples (positive functions) associated with a general random field process with realizations on the space of squareintegrable functions (by the Minlo’s theorem — see Appendix D), F (x) denotes the problem’s nonlinearity represented by a Lipschitzian function as in the previous sections and, finally, (−∆)α represents the effect of the anomalous diffusion of the field U (x, t) on the ambient RD where the diffusion takes place and it is represented here by a fractional power of the Laplacean operator. The constant α here is called the anomalous diffusion exponent. In order to give a precise mathematical meaning we will present a different mathematical scheme for the previous sections for Eq. (4.59). Let us, thus, proceed for a moment by rewriting it formally in the weak-integral form for those initial-values f (x) ∈ L2 (RD ) as in Eq. (4.54), namely, U (t, [V ]) = exp[−t(D0 (−∆)α + V )]f t +g ds{exp[−(t − s)(D0 (−∆)α + V )]}F (U (s)),

(4.60)

0

where we have used a notation emphasizing the stochastic variable nature of the diffusion field U (x, t, [V ]) as a functional of the samples V (x) ∈ L2 (RD ) associated with our random diffusion potential. Physical quantities are functionals of the diffusion field U (t, [V ]), and after its determination one should average over these random potential samples. The whole averaging information is contained in the space–time characteristic functional as pointed out in Sec. 4.4: ∞ dµ[V ] × exp i dx dt j(x, t) · U (x, t, [V ]) . Z[j(x, t)] = L2 (RD )

RD

0

(4.61) Here, dµ[V ] means the random potential cylindrical measure associated with the V (x)-characteristic functional dµ[V ] exp i(h, v L2 (RD ) ). (4.62) Z[h(x)] = L2 (RD )

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Note that we do not suppose the Gaussianity of the field statistics in Eq. (4.62). In order to write a functional integral representation for the characteristic functional equation (4.61) as much as similar representation obtained in Sec. 4.4, let us rewrite Eq. (4.61) into the weak-integral form as done in Eq. (4.56): dµ[V ] DF [λ] DF [U ] Z[j(x, t)] = L2 (RD )

L2 (RD )

L2 (RD )

× exp i{(λ, U + I(F (U )) + e−tA f )L2 (RD ) }

(4.63)

where I(F (U )) and e−tA denote the objects on the right-hand side of Eq. (4.60) and the (formal at this point of our study) contractive generator semigroup below: A = D0 (−∆)α + V.

(4.64)

At this point one could proceed in a physicist way by rewriting Eq. (4.63) as a path-integral associated with the dynamics of three fields (the wellknown Martin–Siggin–Rose component path-integral — Ref. 6): Z[j(x, t)] dµ[V ] = L2 (RD )



DF [λ]

L2 (RD )

DF [U ] L2 (RD )

 0

   × exp i  [λ, U ], 1 ∂ − − D0 (−∆)α 2 ∂t

1 2

  ∂ − D0 (−∆)α   ∂t  λ U 0 L2 (RD )

× exp i g(λ, F (U ))L2 (RD )

0 × exp i [λ, U ], V

V 0

λ . U L2 (RD )

(4.65)

It is worth remarking that in the usual case of the cylindrical measures dµ[V ] is a purely Gaussian measure, formally written in the physicist notation as 1 F −1 dx dyV (x)K (x, y)V (y) dµ[V ] = D [V ] exp − 2 L2 (RD ) L2 (RD ) (4.66)

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99

with K(x, y) defining an integral operator of the trace class on L2 (RD ), one can easily make a further simplification by integrating out exactly the V (x)Gaussian functional integral and producing the effective quartic interaction term as written below:

0 V λ dµ[V ] exp i [λ, U ], V 0 U L2 (RD ) L2 (RD ) ∞ ∞ 1 = exp − dx dt dy ds 2 RD 0 0 RD

× (λ(x, t)U (x, t))K(x, y)(λ(y, s)U (y, s)) .

(4.67)

Let us call the reader’s attention that at this point one can straightforwardly implement the usual Feynman–Wild–Martin–Siggia–Rosen diagramatics with the free propagator given explicitly in the momentum space by6 [∂t − D0 (−∆)α ]−1 ↔

1 iw − D0 (|k|2α )

.

(4.68)

It is important to remark that all the above analyses are still somewhat formal at this point of our study since it is based strongly on the hypothesis that the operator A as given by Eq. (4.64) is a C0 -generator of a contractive semigroup on L2 (RD ), which straightforwardly leads to the existence and uniqueness of global solution for Eq. (4.59). Let us give a rigorous proof of ours of such a self-adjointness result, which by its turn will provide a strong connection between the parameter α of the Laplacean power, related to the anomalous diffusion coefficient, and the underline dimension D of the space where the anomalous diffusion is taking place. Let us first recall the famous Kato–Rellich theorem on self-adjointness perturbative of self-adjoint operators. Theorem of Kato–Rellich. Suppose that A is a self-adjoint operator on L2 (RD ), B is a symmetric operator with Dom(B) ⊃ Dom(A), such that (0 ≤ a < 1), BϕL2 (RD ) ≤ aAϕL2 (RD ) + bϕL2 (RD ) ,

(4.69)

then, A + B is self-adjoint on Dom(A) and essentially self-adjoint on any core of A.

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In order to apply this powerful theorem to our general case under study, let us recall that the fractional power of the Laplacean operator, (−∆)α , has as an operator domain the (Sobolev) space of all square-integrable functions ϕ(x) such that its Fourier transform ϕ(k) ˜ satisfies the condition ˜ ∈ L2 (RD ) and as a core domain the Schwartz space S(RD ).8 |k|α ϕ(k) As a consequence, we have the straightforward estimate for ϕ ∈ S(RD ) as 1 (k 2α + 1)ϕ ˜ L2 (RD ) ϕ ˜ L2 (RD ) ≤ 1 + k 2α L2 (RD ) ≤ C(α, D)(k 2α ϕ ˆ L2 (RD ) + ϕ ˆ L2 (RD ) ), where the finite constant

C(α, D) = RD

dk < ∞, (1 + k 2α )2

(4.70)

(4.71)

if α > D 4 , a condition relating the “anomalous” diffusion coefficient α and the intrinsic space–time dimensionality D as said before in the introduction. Note that for the physical case of D = 3, we have that α = 34 +ε with ε > 0. Let us rescale the function ϕ(k) ˜ (r > 0) (for β > 0 arbitrary) ˜ ϕ˜r (k) = rβ ϕ(rk). We thus have

(4.72)

ϕ˜r L1 (RD ) =

dkrβ ϕ(rk) ˜ RD

˜ L1 (RD ) = rβ−D ϕ

(4.73)

and ϕ˜r L2 (RD ) = r(2β−D)/2 ϕ ˜ L2 (RD )

(4.74)

˜ L2 (RD ) . K 2α ϕ˜r L2 (RD ) = r(2β−D−4α)/2 K 2α ϕ

(4.75)

together with

Let us substitute Eqs. (4.73)–(4.75) into Eq. (4.70). We arrive at the estimate ϕ˜r L1 (RD ) ≤ C(α, D) r(2β−D−4α)/2 K 2α ϕ ˜ L2 (RD ) . + r(2β−D)/2 ϕ ˜ L2 (RD ) (4.76)

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or equivalently / D ϕ ˜ L1 (RD ) ≤ C(α, D) r 2 −2α K 2α ϕ ˜ L2 (RD ) 0 D + r 2 ϕ ˜ L2 (RD ) .

(4.77)

Let us now estimate the function V ϕ on the space L2 (RD ): V ϕL2 (RD ) ≤ V L2 (RD ) ϕL∞ (RD ) ≤ V L2 (RD ) ϕ ˜ L1 (RD ) D

≤ (V L2 (RD ) C(α, D) · r 2 −2α )K 2α ϕ ˜ L2 (RD ) D

+ (V L2 (RD ) C(α, D) · r 2 )ϕ ˜ L2 (RD ) .

(4.78)

Now one can just choose r such that V L2 (RD ) × C(α, D) × r(D−4α)/2 < 1,

(4.79)

in order to obtain the validity of the bound a < 1 on Eq. (4.69) and, thus, concluding that (−∆)α +V is an essentially self-adjoint operator on S(RD ). So, its closure on L2 (RD ) produces the operator used in the above-exposed semigroup C0 -construction for V (x) being a positive function almost everywhere. This result of ours is a substantial generalization of Theorem X.15 presented in Ref. 8. Note that the V (x) sample positivity is always the physical case of random porosity in a Porous medium. In such a case, the random potential is of the exponential form V (x) = V0 (exp{ga(x)}), where g is a positive small parameter and V0 is a background porosity term. The effective cylindrical measure on the random porosity parameters is written as (see Eq. (4.66)) 1 F ga(x) −1 ga(y) ∼ dx dye K (x, y)e dµ[V ] = D [a(x)] exp − . g1 2 RD RD (4.80) The above-exposed C0 -semigroup study complements the study on purely nonlinear diffusion made in the previous sections purely by compacity arguments. Finally, let us briefly sketch on the important case of wave propagation U (x, t) in a random medium described by a small stochastic damping pos2 itive L2 (RD ) function ν(x), such that ν 4(x) is a Gaussian process sampled in the L2 (RD ) space. The hyperbolic linear equation governing such physical

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wave propagation is given by (see Sec. 4.3) ∂U (x, t) ∂ 2 (U (x, t)) + ν(x) = −(−∆)α U (x, t), ∂2t ∂t

(4.81)

U (x, 0) = f (x) ∈ L2 (RD ),

(4.82)

2

Ut (x, 0) = g(x) ∈ L (R ), where

E

ν 2 (x) ν 2 (y) 4 4

D

= K(x, y) ∈

(L2 (RD )),

(4.83)

(4.84)

1

One can see that after the variable change U (x, t, [ν]) = e−

ν(x) 2 t

· Φ(x, t, [ν]),

(4.85)

the damped-stochastic wave equation takes the suitable form of a wave in the presence of a random potential 2 ν (x) ∂ 2 Φ(x, t) α Φ(x, t) + = −(−∆) Φ(x, t), ∂2t 4 Φ(x, 0) = f (x),

(4.86)

ν(x) f (x) + g(x) = g¯(x). 2 Equation (4.86) has an operatorial scalar weak-integral solution for t > 0 of the following form: √ √ sin(t A) √ Φ(t) = cos(t A)f + g¯, (4.87) A where the nonpositive self-adjoint operator A is given explicitly by Φt (x, 0) =

ν 2 (x) . (4.88) 4 As a conclusion, one can see that anomalous diffusion can be handled mathematically in the framework of our previous technique of path-integrals with constraints. A = (−∆)α −

References 1. J. Wloka, Partial Differential Equations (Cambridge University Press, 1987). 2. Y. Yamasaki, Measures on Infinite Dimensional Spaces, Series in Pure Mathematics, Vol. 5 (World Scientific, 1985); B. Simon, Functional Integration and Quantum Physics (Academic Press, 1979); J. Glimn and A. Jaffe, Quantum

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4. 5.

6.

7.

8.

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Physics — A Functional Integral Point of View (Springer Verlag, 1981); Xia Dao Xing, Measure and Integration Theory on Infinite Dimensional Space (Academic Press, 1972); L. Schwartz, Random Measures on Arbitrary Topological Space and Cylindrical Measures (Tata Institute, Oxford University Press, 1973); Ya G. Sinai, The Theory of Phase Transitions: Rigorous Results (London, Pergamon Press, 1981). O. A. Ladyzenskaja, V. A. Solonninkov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type (Amer. Math. Soc. Transl., A.M.S., Providence, RI, 1968). S. G. Mikhlin, Mathematical Physics, An Advanced Course, Series in Applied Mathematics and Mechanics (North-Holland, 1970). L. C. L. Botelho, J. Phys. A: Math. Gen., 34, L131–L137 (2001); Mod. Phys. Lett. 16B(21), 793–8-6, (2002); A. Bensoussan and R. Teman, J. Funct. Anal. 13, 195–222 (1973); A. Friedman, Variational Principles and Free-Boundary Problems, Pure & Applied Mathematics (John-Wiley & Sons, NY, 1982). L. C. L. Botelho, Il Nuovo Cimento, 117B(1), 15 (2002); J. Math. Phys. 42, 1682 (2001); A. S. Monin and A. M. Yaglon, Statistical Fluid Mechanics, Vol. 2 (MIT Press, Cambridge, 1971); G. Rosen, J. Math. Phys. 12, 812 (1971); L. C. L. Botelho, Mod. Phys. Lett. 13B, 317 (1999); A. A. Migdal, Int. J. Mod. Phys. A 9, 1197 (1994); V. Gurarie and A. Migdal, Phys. Rev. E 54, 4908 (1996); U. Frisch, Turbulence (Cambridge Press, Cambridge, 1996); W. D. Mc-Comb, The Physics of Fluid Turbulence (Oxford University, Oxford, 1990). L. C. L. Botelho, Mod. Phys. Lett. B 13, 363 (1999). L. C. L. Botelho, Nuov. Cimento 118B, 383 (2004); S. I. Denisov and W. Horsthemke, Phys. Lett. A 282(6), 367 (2001); L. C. L. Botelho, Int. J. Mod. Phys. B 13, 1663 (1999); L. C. L. Botelho, Mod. Phys. Lett. B 12, 301 (1998); J. C. Cresson and M. L. Lyra, J. Phys. Condens. Matter 8(7), L83 (1996); L. C. L. Botelho, Mod. Phys. Lett. B 12, 569 (1998); L. C. L. Botelho, Mod. Phys. Lett. B 12, 1191 (1998); L. C. L. Botelho, Mod. Phys. Lett. B 13, 203 (1999); L. C. L. Botelho, Int. J. Mod. Phys. B 12, 2857 (1998); L. C. L. Botelho, Phys. Rev. 58E, 1141 (1998). M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol 2 (Academic Press, New York, 1980).

Appendix A In this mathematical appendix, we intend to give a rigorous mathematical proof of the result used on Sec. 4.2 about parabolic nonlinear diffusion in relation to the weak continuity of the nonlinearity ∆F (U ) used in the passage of Eq. (4.17) for Eq. (4.18). First, we consider the physical hypothesis on the nonlinearity of the parabolic equation (4.1) that the Laplacean operator has a cutoff in its

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spectral range, namely, ∆F (U (x, t)) → ∆(Λ) F (U (x, t)),

(A.1)

(Λ)

where the regularized Laplacean ∆ means the bounded operator of norm Λ, i.e. in terms of the spectral theorem of the Laplacean +∞ λ dEs (λ), (A.2) ∆= −∞

it is given by ∆(Λ) =

|λ| 1: 1 T 1 1 dUn (t) 12 1 dt 1 ≤ Real((AUn (t), Un (t)) − (AUn (0), Un (0))) 1 dt 1 2 0

L (Ω)

T

+ 0

1 1 1 (Λ) dUn 1 1 1 dt 1∆ F (Un (t)) · dt 1

L2 (Ω)

≤0+0+a 1 + a In other words,

T

0

0

T

2 ∆F (Un (t))L2 (Ω)

1 1 1 dUn 12 1 dt 1 . 1 dt 1 2 L (Ω)

dt

(A.5)

1 T 1 1 dUn 12 1 1 dt 1 1− 1 dt 1 2 a 0 L (Ω) ≤ caΛ2 · 2

T

0

≤ CaΛ T M,

Un (t)2L2 (Ω) (A.6)

where c = supa≤x≤b |F (x)|, and M is the bound given by Eq. (4.16).

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Now, by a direct application of the well-known Aubin–Lion theorem,1 one has that the sequence {Un } is a compact set on L2 (Ω). So, it con¯ on L2 (Ω) as a consequence of the Lipschitzian property verges strongly to U of F (x). ¯ ), since for each We have either the strong convergence of F (Un ) on F (U t we have that the inequality below holds true: Ω

¯ )|2 2 (t) dx|F (Un ) − F (U L (Ω) 2 2 ¯ (F (x)) dx|Un − U|L2 (Ω) (t) → 0.

≤

sup

−∞<x 0: vi (x, t) = ϕi (x) χ[0,ε] (t)/ε , where ϕi (x) is a member of the spectral set associated with the self-adjoint operator A : H 2 (Ω) ∩ H01 (Ω) → L2 (Ω). The notation χ[a,b] (t) denotes the characteristic function of the interval [a, b]. Since Um (x, t) converges to U (x, t) in the weak-star topology of L∞ ((0, T ), L2 (Ω)) and D((0, T ), L2 (Ω)) ⊂ L1 ((0, T ), L2 (Ω)), we have the (ε) relation below for each vi (x, t) that holds true

T

0

(ε)

dtUm (x, t), vi (x, t) L2 (Ω) →

T

(ε)

dtU (x, t), vi (x, t) L2 (Ω) ,

0

(B.1)

or equivalently 1 ε

0

ε

dtUm (x, t), ϕi (x) L2 (Ω) →

1 ε

0

ε

dtU (x, t), ϕi (x) L2 (Ω) .

(B.2)

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By means of the mean-value theorem applied to both sides of Eq. (A.2), and taking the limit ε → 0, we have for each i ∈ Z that Um (x, 0), ϕi (x) L2 (Ω) → U (x, 0), ϕi (x) L2 (Ω)

(B.3)

as a direct consequence of our choice of the Galerkin elements Eqs. (4.6) whose functional form does not change on the function of the order m ∈ Z. As a consequence of Eqs. (4.5) and (4.6), it yields the result g(x), ϕi L2 (Ω) = U (x, 0), ϕi (x) L2 (Ω) ,

(B.4)

or by means of Parserval–Fourier theorem, lim U (x, t) − g(x)L2 (Ω) = 0

t→0

(B.5)

result, which by its turn, means that the function U (x, t) obtained by means of our compacity technique satisfies the initial-condition Eq. (4.2) in the L2 (Ω)-mean sense.

Appendix C ¯ Let us show that the functions U(x, t) ∈ L∞ ([0, T ], L2(Ω)) given by ∞ 2 ¯ Eq. (4.41) and P (x, t) ∈ L ([0, T ], L (Ω)) — Eq. (4.42) are coincident as elements of the above-written functional space of L2 (Ω) — valued essential bounded functions on (0, T ). To verify such a result, let us call the reader’s attention that since Um (x, t) is weakly convergent to U (x, t) in L∞ ([0, T ], L2 (Ω)), we have that the set {Um (x, t)} is convergent to the function U (x, t) as a Schwartz distribution on L2 (Ω) since D([0, T ], L2 (Ω)) ⊂ L1 ([0, T ], L2 (Ω)). This means that Um (x, t) → U (x, t)

in D ([0, T ], L2 (Ω)).

(C.1)

Analogous result holds true for the time-derivative of the above-written equation as a result of U (x, t) being a function ∂ ∂ Um (x, t) → U (x, t) in D ([0, T ], L2 (Ω)). (C.2) ∂t ∂t On the other side, the set ∂Um∂t(x,t) converges weakly in L1 ([0, T ], L2 (Ω)) to P¯ (x, t) ∈ L∞ ([0, T ], L2(Ω)), which, by its turn, means that ∂Um (x, t) → P¯ (x, t) ∂t

in D ([0, T ], L2 (Ω)).

(C.3)

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By the uniqueness of the limit on the distributional space and P¯ (x, t) as eleD ([0, T ], L2 (Ω)), we have the coincidence of ∂U(x,t) ∂t 2 ¯ so by general ments of D ([0, T ], L (Ω)). However, P (x, t) is a function; must be a function theorems on Schwartz distribution theory ∂U(x,t) ∂t

either since L2 (Ω) is a separable Hilbert space. As a consequence we have = P¯ (x, t) as elements of L∞ ([0, T ], L2(Ω)), which is the result that ∂U(x,t) ∂t searched: ∂U (x, t) = P¯ (x, t) ∂t

a.e. in ([0, T ] × Ω.

(C.4)

Appendix D. Probability Theory in Terms of Functional Integrals and the Minlos Theorem — An Overview In this complementary appendix, we discuss briefly for the mathematics oriented reader the mathematical basis and concepts of functional integrals formulation for stochastic process and the important Minlos theorem on Hilbert space support of probabilistic measures. The first basic notion of Kolmogorov’s probability theory framework is to postulate a convenient topological space (Polish spaces) Ω formed by the phenomena random events. The chosen topology of Ω should possess a rich set of nontrivial compact subsets. After that, we consider the σ-algebra generated by all open sets of this — the so-called topological sample space, denoted by FΩ . Thirdly, one introduces a regular measure dµ on this set algebra FΩ , assigning values in [0, 1] to each member of FΩ . The (abstract) triplet (Ω, FΩ , dµ) is thus called a probability space in the Kolmogorov–Schwartz scheme.10 Let {Xα }α∈Λ be a set of measurable real functions on Ω, which may be taken as continuous injective functions of compact support on Ω (without loss of generality). It is a standard result that one can “immerse” (represent) the abstract space Ω on the “concrete” infinite product compact space R∞ = 2 ˙ ˙ α∈Λ (Rα ), where R is a compactified copy of the real line. In order to achieve such result we consider the following injection of Ω in R∞ defined by the family {Xα }α∈Λ : I : Ω → R∞ ω → {Xα (ω)}α∈Λ .

(D.1)

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A new σ-algebra of events on Ω can be induced on Ω and affiliated to the family {Xα }α∈Λ . It is the σ-algebra generated by all the “finite-dimensional cylinders” subsets, which are explicitly given in its generic formulas by CΛfin = ω ∈ Ω | Λfin is a finite subset of Λ and [(Xα )α∈Λfin (Ω)] ⊂ [aα , bα ] . α∈Λfin

The measure restriction of the initial measure µ on this cylinder sets of Ω will be still denoted by µ on what follows. Now it is a basic theorem of 2 ˙ α , which can Probability theory that µ induces a measure ν (∞) on α∈Λ (R) be identified with the space of all functions of the index set Λ to the real ˙ F (Λ, R) with the topology of pontual convergence. compactified line R; At this point, it is straightforward to see that the average of any Borelian (mensurable) function G(ω) (a random variable) is given by the following functional integral on the functional space F (Λ, R): dµ(ω)G(ω) = dν (∞) (f ) · G(I −1 (f )). (D.2) Ω

F (Λ,R)

It is worth recalling that the real support of the measure dν (∞) (f ) in most practical cases is not the whole space F (Λ, R) but only a functional subspace of F (Λ, R). For instance, in the most practical cases Λ is the 2 index set of an algebraic vectorial base of a vector space E, and α∈Λ (R)α (without the compactification) is the space of all sequences with only a finite number of nonzero entries as it is necessary to consider in the case of the famous extension theorem of Kolmogorov. It turns out that one can consider the support of the probability measure as the set of all linear functionals on E, the so-called algebraic dual of E. This result is a direct consequence of considering the set of random variables given by the family {e∗α }α∈Λ . For applications, one is naturally lead to consider the probabilistic object called characteristic functional associated to the measure dν (∞) (f ) when F (Λ, R) is identified with the vector space of all algebraic linear functionals E alg of a given vector space E as pointed out above, namely, dν (∞) (f ) exp(if (j)), (D.3) Z[j] = E alg

where j are elements of E written as j = α∈Λfin xα eα with {eα }α∈Λ denoting a given Hammel (vectorial) basis of E.

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One can show that if there is a subspace H of E with a norm H coming from an inner product (, )H such that dν (∞) (f ) · f 2H < ∞ (D.4) E∩H

one can show that the support of the measure of exactly the Hilbert space (H, (, )H ) is the famous Minlos theorem. Another important theorem (the famous Wiener theorem) is when we have the following condition for the family of random variables with the index set Λ being a real interval of the form Λ = [a, b] dν (∞) (f )|Xt+h (f ) − Xt (f )|p ≤ c|h|1+ε (D.5) E

for p, ε, c positive constants. We have that it is possible to choose as the support of the functional measure dν (∞) (·) the space of the real continuous function on Λ the well-known famous Brownian path space ˜ (x)). dµ(ω)G(ω) = dν (∞) (f (x))G(f (D.6) Ω

C([a,b],R)

For rigorous proofs and precise formulations of the above-sketched deep mathematical results, the interested reader should consult the basic mathematical book on Random Measures on Arbitrary Topological Spaces and Cylindrical Measures by L. Schwartz (Tata Institute, Oxford University Press, 1973).10 Let us now finally give a concrete example of such results considering a generating functional Z[J(x)] of the exponential quadratic functional form on L2 (RN ): 1 Z[J(x)] = exp − dN x dN yj(x)K(x, y)j(y) (D.7) 2 RN RN and associate to a Gaussian random field with two-point correlation function given by the kernel of Eq. (D.7): E{v(x)v(y)} = K(x, y).

(D.8)

In the3case of the above-written kernel being a class-trace operator on L (RN ) ( RN dN xK(x, x) < ∞), one can represent Eq. (B.7) by means of a Gaussian measure with the support of the Hilbert space L2 (RN ). This result was called on for the reader’s attention for the bulk of this paper. 2

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Chapter 5 Domains of Bosonic Functional Integrals and Some Applications to the Mathematical Physics of Path-Integrals and String Theory∗

5.1. Introduction Since the result of R. P. Feynman on representing the initial value solution of Schrodinger equation by means of an analytically time continued integration on an infinite-dimensional space of functions, the subject of Euclidean functional integrals representations for quantum systems has become the mathematical-operational framework to analyze quantum phenomena and stochastic systems as shown in the previous decades of research on theoretical physics.1−3 One of the most important open problem in the mathematical theory of Euclidean functional integrals is related to the implementation of sound mathematical approximations to these infinite-dimensional integrals by means of finite-dimensional approximations outside of the always used [computer oriented] space-time lattice approximations (see Refs. 2 and 3 — Chap. 9). As a first step to tackle the above-cited problem, there is a need to mathematically characterize the functional domain where these functional integrals are defined. The purpose of this chapter is to present, in Sec. 5.2, the formulation of Euclidean quantum field theories as functional Fourier transforms by means of the Bochner–Martin–Kolmogorov theorem for topological vector spaces (Refs. 4 and 5 — Theorem 4.35) and suitable to define and analyze rigorously the functional integrals by means of the well-known Minlos theorem (Ref. 5, Theorem 4.31 and Ref. 6, Part 2) and presented in full details in Sec. 5.3. In Sec. 5.4, we present new results on the difficult problem of defining rigorously infinite-dimensional quantum field path-integrals in general space times Ω ⊂ Rν (ν = 2, 4, . . . ) by means of the analytical regularization scheme. ∗ Author’s


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5.2. The Euclidean Schwinger Generating Functional as a Functional Fourier Transform The basic object in a scalar Euclidean quantum field theory in RD is the Schwinger generating functional (see Refs. 1 and 3). Z[j(x)] = ΩV AC exp i dD xj(x, it)φ(m) (x, it) ΩV AC , (5.1) where φ(m) (x, it) is the supposed self-adjoint Minkowski quantum field analytically continued to imaginary time and j(x) = j(x, it) is a set of functions belonging to a given topological vector space of functions denoted by E which topology is not specified yet and will be called the Schwinger classical field source space. It is important to remark that {φm (x, it)} is a commuting algebra of self-adjoints operators as Symanzik has pointed out.7 In order to write Eq. (5.1) as an integral over the space E alg of all linear functionals on the Schwinger source space E (the so-called algebraic dual of E), we take the following procedure, different from the usual abstract approach (as given — for instance — in the proof of IV-11-2), by making the hypothesis that the restriction of the Schwinger generating functional equation (5.1) to any finite-dimensional RN of E is the Fourier transform of a positive continuous function, namely Z

N

α=1

Cα ¯jα (x)

=

exp i

RN

N

Cα Pα

g˜(P1 , . . . , PN )dP1 , . . . , dPN .

α=1

(5.2) Here, {jα (x)}α=1,...,N is a fixed vectorial basis of the given finitedimensional subspace (isomorphic to RN ) of E. As a consequence of the above-made hypothesis (based physically on the renormalizability and unitary of the associated quantum field theory), one can apply the Bochner–Martin–Kolmogorov theorem (Ref. 5, Theorem 4.35) to write Eq. (5.1) as a functional Fourier transform on the space E alg (see Appendix A) exp(ih(j(x))dµ(h), (5.3) Z[j(x)] = E alg

where dµ(h) is the Kolmogorov cylindrical measure on E alg = Πλ∈A (Rλ ) with A denoting the index set of the fixed Hamel vectorial basis used in

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Eq. (5.2) and h(j(x)) is the action of the given linear (algebric) functional (belonging to E alg ) on the element j(x) ∈ E. At this point, we relate the mathematically non-rigorous physicist point of view to the Kolmogorov measure dµ(h) (Eq. (5.3)) over the algebraic linear functions on the Schwinger source space. It is formally given by the famous Feynman formulae when one identifies the action of h on E by means of an “integral” average dxD j(x)h(x).

h(j) =

(5.4)

RD

Formally, we have the equation dµ(h) =

dh(x) exp{−S(h(x))},

(5.5)

x∈RD

where S is the classical action of the classical field theory under quantization, but with the necessary coupling constant renormalizations needed to make the associated quantum field theory well-defined. Let us outline these proposed steps on a λφ4 -field theory on R4 . At first we will introduce the massive free field theory generating functional directly into the infinite volume space R4 . 1 Z[j(x)] = exp − d4 xd4 x j(x)((−∆)α + m2 )−1 (x, x )j(x ) , 2

(5.6)

where the free field propagator is given by ((−∆)α + m2 )−1 (x, x ) =

d4 k

eik(x−x ) , k 2α + m2

(5.7)

with α a regularizing parameter with α > 1. As the source space, we will consider the vector space of all real sequences on Πλ∈(−∞,∞) (R)λ , but with only a finite number of non-zero components. Let us define the following family of finite-dimensional pos itive linear functionals {LΛf } on the functional space C( λ∈(−∞,∞) Rλ ; R)

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LΛf g(Pλs1 , . . . , PλsN ) =

Q

Rλ

g(Pλs1 , . . . , PλsN )

λ∈Λf

   1  × exp − (λ2α + m2 )(Pλ )2  2  λ∈Λf

 ×

d(Pλ

 π(λ2α + m2 )) .

(5.8)

λ∈Λf

Here, Λf = {λs1 , . . . , λsN } is an ordered sequence of real number of the real line which is the index set of the Hamel basis of the algebraic dual of the proposed source space. Note that we have the generalized eigenproblem expansion ((−∆)α + m2 )eiλx = (λ2α + m2 )eiλx .

(5.9)

By the Stone–Weierstrass theorem or the Kolmogoroff theorem applied to the family of finite-dimensional measure in Eq. (5.8), there is a unique extension measure dµ({Pλ }λ∈(−∞,∞) ) to the space Πλ∈(−∞,∞) Rλ = E alg and representing the infinite-volume generating functional on our chosen source space (the usual Riesz–Markov theorem applied to the linear functional L = lim{Λf } sup LΛf on C(Πλ∈(−∞,∞) Rλ , R) leads to this extension measure).10 Z[j(x)] = Z[{jλ }λ∈Λf ] =

Q

Rλ

 ! dµ(0)·(α) ({P }  λ λ∈(−∞,∞) ) exp i

λ∈(−∞,∞)

   1  2 (jλ ) = exp − .  2 λ2α + m2 

 jλ Pλ 

λ∈(−∞,∞)

(5.10)

λ∈Λf

At this point it is very important to remark that the generating functional equation (5.10) has continuous natural extension to any test space (S(RN ), D(RN ), etc.) which contains the continuous functions of compact support as a dense subspace. At this point we consider the following quantum field interaction functional which is a measurable functional in relation to the above-constructed Kolmogoroff measure dµ(0)·(α) ({Pλ }λ∈(−∞,∞) ) for α non-integer in the

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original field variable φ(x) 1 (α) 1 (α) V (α) (φ) = λR φ4 + (Zφ (λR , M ) − 1)φ((−∆)α )φ − [(m2 Zφ (λR , m) − 1 2 2 (α)

− (δm2 )(α) (λR )]φ2 − [Zφ (λR , m)(δ (α) λ)(λR , m)]φ4 .

(5.11)

Here, the renormalization constants are given in the usual analytical finitepart regularization form for a λφ4 -field theory. It is still an open problem in the mathematical-physics of quantum fields to prove the integrability in some distributional space of the cut-off removing α → 1 limit of the interaction lagrangean exp(−V (α) (φ)) (see Sec. 5.4 for the analysis of this cut-off removing on space functions). 5.3. The Support of Functional Measures — The Minlos Theorem Let us now analyze the measure support of quantum field theories generating functional equation (5.3). For higher dimensional space-time, the only available result in this direction is the case that we have a Hilbert structure on E.4−6 At this point, we introduce some definitions. Let ϕ : Z+ → R be an increasing fixed function (including the case ϕ(∞) = ∞). Let E be denoted by H and H Z be the subspace of H alg = (Πλ∈A=[0,1] Rλ ) (with A being the index set of a Hamel basis of H), formed by all sequences {xλ }λ∈A ∈ H alg with coordinates different from zero at most a countable number H Z = {(xλ )λ∈A|xλ = 0 for λ ∈ {λµ }µ∈Z }. Consider the following weighted subset of H alg : Z H(e) = {{xλ }λ∈A ∈ H Z }

and lim

N →∞

N 1 (xλσ(µ) )2 ϕ(N ) n=1

µ(HeZ ).

(5.15)

H alg

But

µ(HeZ )

= lim lim

α→0 N →∞

H alg

= lim lim

α→0 N →∞

1 × exp − 2

dµ(h) exp −

1 2πα N/2 ( ϕ(N ))

ϕ(N ) α

=1

N

=1

N

α 2 xλ 2ϕ(N )

RN

dj1 , . . . , djN

j2

˜ 1 , . . . , JN ), Z(j

(5.16)

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where

˜ 1 , . . . , jN ) = Z(j

πRλ λ∈A

dµ({xλ }) exp i

xλ j

=1

dµ(h) exp i

=

N

H alg

N

xλ j

.

(5.17)

=1

Now due to the continuity and positivity of Z[j] in D(T ); we have that for any ε > 0 → ∃δ such that the inequality below is true since we have that: Z(j1 , . . . , jN ) ≥ 1 − ε − δ22 (j, j)(1)

1 2πα ϕ(N )

N/2

RN

N 1 ϕ(N ) 2 ˜ dj1 · · · djN exp − j Z(j1 , . . . , jN ) 2 α =1

 N 2  ≥1−ε− 2 δ 

(m,n)=1

1

2πα N/2 ϕ(N )

RN

dj1 · · · djN

N 1 ϕ(N ) 2 (1) × exp − j jm jn em , en 2 α =1   N   α 2 =1−ε− 2 δmn T en , T em (0)  δ  ϕ(N ) (m,n)=1

≥1−ε−

2 δ2

α ϕ(N )

ϕ(N ) ≥ 1 − ε −

2 α. δ2

(5.18)

By substituting Eq. (5.18) into Eq. (5.15), we get the result 1 ≥ µ(HeZ ) ≥ 1 − ε −

2 lim α = 1 − ε. δ 2 α→0

(5.19)

Since ε was arbitrary we have the validity of our theorem. As a consequence of this theorem in the case of ϕ(N ) being bounded (so T T ∗ is an operator of Trace Class), we have that HeZ = H which is the usual topological dual of H. At this point, a simple proof may be given to the usual Minlos theorem on Schwartz spaces.5,6

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Let us consider S(RD ) represented as the countable normed spaces of sequences8 ∞

S(RD ) =

2m ,

(5.20)

m=0

where 2m =

N (xn )n∈Z , xn ∈ R (xn )2 nm < ∞

(5.21)

n=0

The topological dual is given by the nuclear structure sum8 ∞ !

S (RD ) =

∞ !

2−n =

n=0

(2n )∗

(5.22)

n=0

We, thus, consider E = S(RD ) in Eq. (9.3) and Z[j(x)] = Z[{jn }n∈Z ] is "∞ "∞ continuous on n n . Since Z[{jn }n∈Z ] ∈ C( n=0 2n , R) we have that for any fixed integer p, Z[{jn }n∈Z ] is continuous on the Hilbert space 2p which, by its turn, may be considered as the domain of the following operator: Tp : 2p c0 → 0 {jn } → {np/2 jn }. It is straightforward to have the estimate N (0) Tp em , Tp en ≤ N (Bp ) (m,n)=1

(5.23)

(5.24)

for some positive integer B and {ei } being the canonical orthonormal basis of l02 . By an application of our theorem for each fixed p; we get that the support of measure is given by the union of weighted spaces supp dµ(h) =

∞ !

(2p )∗ =

p=0

∞ ! p=0

2−p = S (RD ).

(5.25)

At this point we can suggest, without a proof of straightforward (nontopological) generalization of the Minlos theorem.

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Theorem 5.2. Let {Tβ }β∈C be a family of operators satisfying the hypothe# sis of Theorem 5.3. Let us consider the Locally Convex space β∈C Dom(Tβ ) (supposed non-empty) with the family of norms ψ β = Tβ ψ, Tβ ψ1/2 . If the Functional Fourier transform is continuous on this locally convex space, the support of the Kolmogoroff measure equation (5.3) is given by the # following subset of [ β∈C Dom(Tβ )]alg , namely supp dµ(h) =

!

Hϕ2 β ,

(5.26)

β∈C

where ϕβ are the functions given by Theorem 5.1. This general theorem will not be applied in what follows. Let us now proceed to apply the above displayed results by considering the Schwinger generating functionals for two-dimensional Euclidean quantum eletrodynamics in Bosonized parametrization9 −1 1 e2 2 2 2 Z[j(x)] = exp − d x d yj(x) (−∆) + (−∆) (x, y)j(y) , 2 R2 π R2 (5.27) where in Eq. (5.27), the electromagnetic field has the decomposition in Landau Gauge Aµ (x) = (εµν ∂ν φ)(x)

(5.28)

and j(x) is, thus, the Schwinger source for the φ(x) field taken as a basic dynamical variable.9 Since Eq. (5.27) is continuous in L2 (R2 ) with the inner product defined 2 by the trace class operator ((−∆)2 + eπ (−∆))−1 , we conclude on the basis of Theorem 5.1 that the associated Kolmogoroff measure in Eq. (5.3) has its support in L2 (R2 ) with the usual inner product. As a consequence, the quantum observable algebra will be given by the functional space L1 (L2 (R2 ), dµ(h)) and usual orthonormal finite-dimensional approximations in Hilbert spaces may be used safely, i.e. if one considers the basis $∞ expansion h(x) = n=1 hn en (x) with en (x) denoting the eigenfunctions of the operator in Eq. (10.27) we get the result ∞ ! n=1

L1 (RN , dµ(h1 , . . . , hN )) = L1 (L2 (R2 ), dµ(h)).

(5.29)

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It is worth mentioning that if one uses the Gauge vectorial field parametrization for the (Q.E.D)2 — Schwinger functional Z[j1 (x), j2 (x)] −1 1 e2 2 2 = exp − d x d yji (x) −∆ + (x, y)δil jl (x) (5.30) 2 R2 π R2 the associated measure support will now be the Schwartz space S (R2 ) 2 since the operator (−∆ + eπ )−1 is an application of S(R2 ) to S (R2 ). As a consequence, it will be very cumbersome to use Hilbert finite-dimensional approximations8 as in Eq. (5.29). An alternative to approximate tempered distributions is the use of its Hermite expansion in S (R) distributional space associated with the eigen# functions of the Harmonic-oscillator V (x) ∈ L∞ (R) L2 (R) potential pertubation (see Ref. 3 for details with V (x) ≡ 0). d2 − 2 + x2 + V (x) Hn (x) = λn Hn (x). dx

(5.31)

Another important class of Bosonic functionals integrals are those associated with an Elliptic positive self-adjoint operator A−1 on L2 (Ω) with suitable boundary conditions. Here Ω denotes a D-dimensional compact manifold of RD with volume element dν(x). 1 +1 Z[j(x)] = exp − dν(x) dν(y)j(x)A (x, y)j(y) . 2 Ω Ω

(5.32)

If A is an operator of trace class on (L2 (Ω), dν) we have, thus, the validity of the usual eigenvalue functional representation Z[{jn }µ∈Z ] ∞

∞ ∞ 1 2 = d(c λ ) exp − λ c X2 ({cn }n∈Z ) exp i c j 2 =1

=1

=1

with the spectral set A−1 σ = λ σ , j = j, σ

(5.33)

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and the characteristic function set X2 ({cn }n∈Z ) =

  1

if

 

otherwise.

∞

c2n < ∞,

n=0

0

(5.34)

It is instructive to point out the usual Hermite functional basis (see 5.4, Ref. 5) as a complete set in L2 (E alg , dµ(h)), only if the Gaussian Kolmogoroff measure dµ(h) is of the class above studied. A criticism to the usual framework to construct Euclidean field theories is that it is very cumbersome to analyze the infinite volume limit from the Schwinger generating functional defined originally on compact space times. In two dimensions the use of the result that the massive scalar field theory generating functional 1 d2 x d2 yj(x)(−∆ + m2 )−1 (x, y)j(y) (5.35) exp − 2 R2 R2 with j(x) ∈ S(R2 ) is given by the limit of finite volume Dirichlet field theories L 1 L 0 T 1 dx dx dy 0 lim exp − L→∞ 2 −L −T −L T →∞ T 1 0 1 2 −1 1 1 0 0 0 1 × dy j(x , x )(−∆D + m ) (x , y , x , y )j(y , y ) −T

(5.36) may be considered, in our opinion, as the similar claim made that is possible from a mathematical point of view to deduce the Fourier transforms from Fourier series, a very, difficult mathematical task (see Appendix B). Let us comment on the functional integral associated with Feynman propagation of fields configurations used in geometrodynamical theories in the scalar case G[β in (x); β out (x), T ](j) +∞ 1 T d2 ν = exp − dt d x φ − 2 + A φ (x, t) φ(x,0)=β in (x) 2 0 dt −∞ out φ(x,T )=β

× exp i 0

(x)

T

+∞ ν

dt

d xj(x)t)φ(x, t) . −∞

(5.37)

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If we define the formal functional integral by means of the eigenfunctions of the self-adjoint Elliptic operator A, namely: φk (t)ψk (x), (5.38) φ(x, t) = {k}

where Aψk (x) = (λk )2 ψk (x)

(5.39)

it is straightforward to see that Eq. (5.36) is formally exactly evaluated in terms of an infinite product of usual Feynman Wiener path measures G[β in (x); β out (x), T ](j)

= DF [ck (t)] {k}

ck (0)=φk (0) ck (T )=φk (T )

T d2 1 T 2 dtjk (t)ck (t) ck − 2 + λk ck (t)dt exp i × exp − 2 0 dt 0 % &

λk λk exp − + = (φ2k (T ) sin(λk T ) 2 sin(λk T ) {k}

+ φ2k (0)) cos(λk T ) − 2φk (0)φk (T ) 2φk (T ) T 2φk (0) T dtjk (t) sin(λk t) − dtjk (t) sin(λk (T − t)) λk λk 0 0 ' T t 2 × − dt dsjk (t)jk (s) sin(λk (T − t)) sin(λk s) . (λk )2 0 0 −

(5.40) Unfortunately, our theorems do not apply in a straightforward way to infinite (continuum) measure product of Wiener measures in Eq. (5.40) to produce a sensible measure theory on the functional space of the infinite product of Wiener trajectories {ck (t)} (note that for each fixed x, a sample field configuration φ(t, 0) in Eq. (5.36) is a H¨ older continuous function, result opposite to the usual functional integral representation for the Schwinger generating functional equations (5.1)–(5.5)), where it does not make mathematical sense to consider a fixed point distribution φ(t, 0) — see Sec. 5.4, Eq. (5.74).

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Let us call to attention that there is still a formal definition of the above Feynman path propagator for fields equation (5.37) which at large time T → +∞ gives formally the quantum field functional integral equation (5.5) associated with the Schwinger generating functional. We thus consider the functional domain for Eq. (5.37) as composed of field configurations which has a classical piece added with another fluctuating component to be functionally integrated out, namely σ(x, t) = σCL (x, t) + σq (x, t).

(5.41)

Here, the classical field configuration problem (added with all zero modes of the free theory) defined by the kinetic term £ d2 (5.42) − 2 + £ σ CL (x, t) = j(x, t), dt with σ CL (x, −T ) = β1 (x);

σ CL (x, T ) = β2 (x),

(5.43)

namely,

−1 d2 σCL (x, t) = − 2 + £ j(x, t) + (all projection on zero modes of £). dt (5.44)

As a consequence of the decomposition equation (5.43), the formal geometrical propagator with an external source is given by D[σ(x, t)] G[β1 (x), β2 (x), T, [j]] = σ(x,−T )=β1 (x) σ(x,+T )=β2 (x)

1 × exp − 2 × exp i

2 d dtdν xσ(x, t) − 2 + £ σ(x, t) dt −T

T

T

dν xj(x, t)σ(x, t)

dt

(5.45)

−T

may be defined by the following mathematically well-defined Gaussian functional measure 1 T dt dν xj(x, t)σ CL (x, t) dσq (x, t) exp − σq (x,−T )=0 2 −T

1 × exp − 2

σq (x,+T )=0

T

dt −T

d2 d xσq (x, t) − 2 + £ σq (x, t) . dt

ν

(5.46)

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The above claim is a consequence of the result below 1 T d2 D D[σq (x, t)] exp − dt d xσq (x, t) − 2 + £ σq (x, t) σq (x,−T )=0 2 −T dt σq (x,T )=0

( ) d2 −1 = detDir2 − 2 + £ , dt

(5.47)

where the subscript Dirichlet on the functional determinant means that one must impose formally the Dirichlet condition on the domain of the operator d2 D ×[−T, T ]) (or L2 (RD ×[−T, T ] if £−1 belongs to trace (− dt 2 +£) on D (R class). Note that the operator £ in Eq. (5.46) does not have zero modes by the construction of Eq. (5.41). At this point, we remark that at the limit T → +∞, Eq. (5.45) is exactly the quantum field functional equation (5.5) if one takes β1 (x) = β2 (x) = 0 (note that the classical vacuum limit T → ∞ of Wiener measures is mathematically ill-defined (see Theorem 5.1 of Ref. 1). It is an important point to remark that σCL (x, t) is a regular ∞ C ([−T, T ] × Ω) solution of the Elliptic problem equation (5.42) and the fluctuating component σq (x, t) is a Schwartz distribution in view of d2 the Minlos-Dao Xing Theorem 1, since the elliptic operator − dt 2 + £ in Eq. (5.47) acts now on D ([−T, T ] × Ω) with the range D([−T, T ] × Ω), which by its turn shows the difference between this framework and the previous one related to the infinite product of Wiener measures since these objects are functional measures in different functional spaces. Finally, we comment that functional Schrodinger equation, may be mathematically defined for the above displayed field propagators equation (5.37) only in the situation of Eq. (5.40). For instance, with £ = −∆ (the Laplacean), we have the validity of the Euclidean field wave equation for the geometrodynamical path-integral equation (5.37) ∂ G[β1 (x), β2 (x), T, [j]] ∂T ( ) δ2 − |∇β2 (x)|2 + j(x, T ) G[β1 (x), β2 (x), T, [j]] dν x + 2 = δ β2 (x) Ω (5.48) with the functional initial condition lim G[β1 (x), β2 (x), T ] = δ (F ) (β1 (x) − β2 (x)).

T →0+

(5.49)

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5.4. Some Rigorous Quantum Field Path-Integral in the Analytical Regularization Scheme In this core section, we address the important problem of producing concrete non-trivial examples of mathematically well-defined (in the ultra-violet region!) path-integrals in the context of the exposed theorems in the previous sections of this paper, especially Sec. 5.2, Eq. (5.11). Let us thus start our analysis by considering the Gaussian measure associated with the (infrared regularized) α-power (α > 1) of the Laplacean acting on L2 (R2 ) as an operational quadratic form (the Stone spectral theorem) α (−∆)ε = (λ)α dE(λ), (5.50a) εIR ≤λ

(0) Zα,ε [j] IR

1 −α = exp − j, (−∆)ε jL2 (R2 ) 2 = d(0) α,ε µ[ϕ] exp(ij, ϕL2 (R2 ) ).

(5.50b)

Here, εIR > 0 denotes the infrared cut-off. It is worth to call the readers’ attention that due to the infrared regularization introduced in Eq. (5.50a), the domain of the Gaussian measure is given by the space of square integrable functions on R2 by the Minlos theorem of Sec. 3, since for α > 1, the operator (−∆)−α εIR defines a class trace operator on L2 (R2 ), namely 1 −α H Tr 1 ((−∆)εIR ) = d2 k < ∞. (5.50c) (|K|2α + εIR ) This is the only point of our analysis where it is needed to consider the infrared cut-off considered on the spectral resolution equation (5.50a). As a consequence of the above remarks, one can analyze the ultraviolet renormalization program in the following interacting model proposed by us and defined by an interaction gbare V (ϕ(x)) with V (x) denoting a function on R such that it posseses an essentially bounded Fourier transform and gbare denoting the positive bare coupling constant. Let us show that by defining a renormalized coupling constant as (with gren < 1) gbare =

gren (1 − α)1/2

(5.51)

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one can show that the interaction function exp −gbare (α) d2 xV (ϕ(x))

(5.52)

(0)

is an integrable function on L1 (L2 (R2 ), dα,εIR µ[ϕ]) and leads to a welldefined ultraviolet path-integral in the limit of α → 1. The proof is based on the following estimates: Since almost everywhere we have the pointwise limit exp −gbare (α) d2 xV (ϕ(x))

N (−1)n (gbare (α))n dk1 · · · dkn V˜ (k1 ) · · · V˜ (kn ) N →∞ n! R n=0 dx1 · · · dxn eik1 ϕ(x1 ) · · · eikn ϕ(xn ) × (5.53)

× lim

R2

we have that the upper-bound estimate below holds true ∞ (−1)n (gbare (α))n α dk1 · · · dkn V˜ (k1 ) · · · V˜ (kn ) |ZεIR [gbare ]| ≤ n! 2 R n=0 P (0) i N k ϕ(x ) × dx1 · · · dxn dα,εIR µ[ϕ](e =1 ). R2 (5.54a) with ZεαIR [gbare ] =

2 d(0) µ[ϕ] exp −g (α) d xV (ϕ(x)) bare α,εIR

(5.54b) (0)

we have, thus, the more suitable form after realizing d2 ki and dα,εIR µ[ϕ] integrals, respectively |ZεαIR =0 [gbare ]| ≤

∞ n (gbare (α))n ˜ ||V ||L∞ (R) n! n=0 1 ) . × dx1 · · · dxn det− 2 [G(N (x , x )] 1≤i≤N i j α 1≤j≤N

(5.55)

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(N )

Here, [Gα (xi , xj )] 1≤i≤N denotes the N × N symmetric matrix with (i, j) 1≤j≤N

entry given by the Green-function of the α-Laplacean (without the infrared cut-off here! and the needed normalization factors!). Gα (xi , xj ) = |xi − xj |2(1−α)

Γ(1 − α) . Γ(α)

(5.56)

At this point, we call the readers’ attention that we have the formulae on the asymptotic behavior for α → 1.

− 12

lim det

α→1 α>1

) [G(N α (xi , xj )]

∼ (1 − α)

N/2

1 (N − 1)(−1)N − 2 × . π N/2 (5.57)

After substituting Eq. (5.57) into Eq. (5.55) and taking into account the hypothesis of the compact support of the nonlinear V (x) (for instance: supp V (x) ⊂ [0, 1]), one obtains the finite bound for any value grem > 0, and producing a proof for the convergence of the perturbative expansion in terms of the renormalized coupling constant. lim |ZεαIR =0 [gbare (α)]| ≤

α→1

n n ∞ (V˜ L∞ (R) )n gren (1 ) √ (1 − α)n/2 1 n! n 2 (1 − α) n=0 ˜

≤ egren V L∞ (R) < ∞.

(5.58)

Another important rigorously defined functional integral is to consider the following α-power Klein Gordon operator on Euclidean space time L = (−∆)α + m2

(5.59)

with m2 a positive “mass” parameters. Let us note that L−1 is an operator of class trace on L2 (Rν ) if and only if the result below holds true *π νπ + ν 1 (α −2) ¯ cosec < ∞, = C(ν)m TrL2 (Rν ) (L−1 ) = dν k 2α k + m2 2α 2α (5.60) namely if α>

ν . 2

(5.61)

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In this case, let us consider the double functional integral with functional domain L2 (Rν ) Z[j, k] =

(0)

dG β[v(x)]

(0)

d(−∆)α +v+m2 µ[ϕ]

ν × exp i d x(j(x)ϕ(x) + k(x)v(x)) ,

(5.62)

where the Gaussian functional integral , on the fields V (x) has a Gaussian generating functional defined by a 1 -integral operator with a positive defined kernel g(|x − y|), namely Z (0) [k] =

(0) dG β[v(x)] exp i dν xk(x)v(x)

1 = exp − dν x dν y(k(x)g(|x − y|)k(x)) . 2

(5.63)

By a simple direct application of the Fubbini–Tonelli theorem on the exchange of the integration order in Eq. (5.62), lead us to the effective λϕ4 like well-defined functional integral representation Zeff [j] =

(0)

d((−∆)α +m2 ) µ[ϕ(x)] 1 ν ν 2 2 × exp − d xd y|ϕ(x)| g(|x − y|)|ϕ(y)| 2 × exp i dν xj(x)ϕ(x) .

(5.64)

Note that if one introduces from the begining a bare mass parameters m2bare depending on the parameters α, but such that it always satisfies Eq. (5.60) one should obtain again Eq. (5.64) as a well-defined measure on L2 (Rν ). Of course, that the usual pure Laplacean limit of α → 1 in Eq. (5.59), will need a renormalization of this mass parameters (limα→1 m2bare (α) = +∞!) as much as its done in the previous example. Let us continue our examples by showing again the usefulness of the precise determination of the functional-distributional structure of the domain of the functional integrals in order to construct rigorously these path-integrals without complicated limit procedures.

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Let us consider a general Rν Gaussian measure defined by the generating functional on S(Rν ) defined by the α-power of the Laplacean operator −∆ acting on S(Rν ) with a of small infrared regularization mass parameter µ2 . 1α 2 −1 Z(0) [j] = exp − j, ((−∆) + µ0 ) j L2 (Rν ) 2 = d(0) (5.65) α µ[ϕ] exp(iϕ(j)). E alg (S(Rν ))

An explicit expression in momentum space for the Green function of the α-power of (−∆)α + µ20 is given by 1 dν k ik(x−y) +α 2 −1 ((−∆) + µ0 ) (x − y) = e . (5.66) (2π)ν k 2α + µ20 ¯ Here, C(ν) is a ν-dependent (finite for ν-values!) normalization factor. Let us suppose that there is a range of α-power values that can be chosen in such a way that one satisfies the constraint below 2j d(0) ν−1 2 , one can see by the Minlos theorem that the measure support of the Gaussian measure equation (5.65) will be given by the intersection Banach space of measurable Lebesgue functions on Rν instead of the previous one E alg (S(Rν )) N

L2N (Rν ) =

(L2j (Rν )).

(5.69)

j=1

In this case, one obtains that the finite-volume p(ϕ)2 interactions   N   exp − λ2j (ϕ2 (x))j dx ≤ 1   Ω j=1

(5.70)

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is mathematically well-defined as the usual pointwise product of measurable functions and for positive coupling constant values λ2j ≥ 0. As a conse(0) quence, we have a measurable functional on L1 (L2N (Rν ); dα µ[ϕ]) (since it is bounded by the function 1). So, it would make sense to consider mathematically the well-defined path-integral on the full space Rν with those values of the power α satisfying the contraint equation (5.67).   N   2j d(0) µ[ϕ] exp − λ ϕ (x)dx Z[j] = 2j α   L2N (Rν ) Ω j=1

× exp i

j(x)ϕ(x) .

(5.71)

Rν

Finally, let us consider an interacting field theory in a compact space-time Ω ⊂ Rν defined by an integer even power 2n of the Laplacean operator with Dirichlet boundary conditions as the free Gaussian kinetic action, namely 1 Z (0) [j] = exp − j, (−∆)−2n jL2 (Ω) 2 (0) = d(2n) µ[ϕ] exp(ij, ϕL2 (Ω) ). (5.72) W2n (Ω)

Here, ϕ ∈ W2n (Ω) is the Sobolev space of order n which is the functional domain of the cylindrical Fourier transform measure of the generating functional Z (0) [j], a continuous bilinear positive form on W2−n (Ω) (the topological dual of W2n (Ω)). By a straightforward application of the well-known Sobolev immersion theorem, we have that for the case of n−k >

ν 2

(5.73)

including k a real number of the functional Sobolev space W2n (Ω) is contained in the continuously fractional differentiable space of functions C k (Ω). As a consequence, the domain of the Bosonic functional integral can be further reduced to C k (Ω) in the situation of Eq. (5.73) (0) d(2n) µ[ϕ] exp(ij, ϕL2 (Ω) ). (5.74) Z (0) [j] = C k (Ω)

That is our new result generalizing the Wiener theorem on Brownian paths in the case of n = 1 , k = 12 , and ν = 1.

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Since the Bosonic functional domain in Eq. (5.74) is formed by real functions and not by distributions, we can see straightforwardly that any interaction of the form F (ϕ(x))dν x (5.75) exp −g Ω

with the non-linearity F (x) denoting a lower bounded real function (γ > 0) F (x) ≥ −γ

(5.76)

is well-defined and is an integrable function on the functional space (0) (C k (Ω), d(2n) µ[ϕ]) by the direct application of the Lebesque theorem ν exp −g F (ϕ(x))d x ≤ exp{+gγ}.

(5.77)

Ω

At this point we make a subtle mathematical remark that the infinite volume limit of Eqs. (5.74) and (5.75) is very difficult, since one loses the Garding–Poincaré inequality at this limit for those elliptic operators and, thus, the very important Sobolev theorem. The probable correct procedure to consider the thermodynamic limit in our Bosonic path-integrals is to consider solely a volume cut-off on the interaction term Gaussian action as in Eq. (5.71) and there search for vol(Ω) → ∞. As a last remark related to Eq. (5.73) one can see that a kind of “fishnet” exponential generating functional 1 (5.78) Z (0) [j] = exp − j, exp{−α∆}jL2 (Ω) 2 has a Fourier transformed functional integral representation defined on the space of the infinitely differentiable functions C ∞ (Ω), which physically means that all field configurations making the domain of such path-integral has a strong behavior-like purely nice smooth classical field configurations. As a general conclusion, we can see that the technical knowledge of the support of measures on infinite-dimensional spaces — specially the powerful Minlos theorem of Sec. 3 is very important for a deep mathematical physical understanding into one of the most important problem is quantum field theory and turbulence which is the problem related to the appearance of ultraviolet (short-distance) divergences on perturbative path-integral calculations.

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5.5. Remarks on the Theory of Integration of Functionals on Distributional Spaces and Hilbert–Banach spaces Let us first consider a given vector space E with a Hilbertian structure , , namely H = (E, ), where means a inner product and H a complete topological space with the metrical structure induced by the given , . We have, thus, the famous Minlos theorem on the support of the cylindrical measure associated with a given , quadratic form defined by a positive definite class trace operator A ∈ 1 (H, H) (see Appendix for a discussion on Fourier transforms in vector spaces of infinite-dimension) 1 1 1 1 dA µ(v) · exp(iv, b) exp − b, Ab = exp − |A| 2 b, |A| 2 b = 2 2 H (5.79) since any given class trace operator can always be considered as the composition of two Hilbert–Schmidt, each one defined by a function on L2 (M × M, dν ⊗ dν). Here, the cylindrical measure dA µ(v), defined already on the vector space of the linear forms of E, with the topology of pontual convergence — the so-called algebraic dual of E — has its support concentrated on the Hilbert spaces H, through the isomorphism of H and its dual H by means of the Riesz theorem. This result can be understood more easily, if one represents the given Hilbert space H as a square-integrable space of measurable functions on a complete measure space (M, dν)L2 (M, dν). In this case, the class trace positive definite operator is represented by an integral operator with a positivedefinite Kenel K(x, y). Note that it is worth to re-write Eq. (5.79) in the Feynman path-integral notation as 1 dν(x)dν(y)f (x)K(x, s)f (s) exp − 2 M×M

1 = dϕ(x) Z(0) L2 (M,dν) x∈M & ' 1 × exp − dν(x)dν(y)ϕ(x)K −1 (x, y)ϕ(y) 2 M×M × exp dν(x)f (x)ϕ(x) . (5.80) M

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Here, the “inverse Kenel” of the operator A is given by the relation dν(y)K(x, y)K −1 (y, x ) = identity operator (5.81) M

and the path-integral normalization factor is given by the functional deter1 1 minant Z(0) = det− 2 (K) = det 2 (K −1 ). A more invariant and rigorous representation for the Gaussian path-integral equations (5.79) and (5.80) can be exposed through an eigenfunction–eigenvalue harmonic expansion associated with our given class trace operator A, namely Aβn = λn βµ , v=

∞

vn βn ;

ϕ=

n=0

∞ 1 λn |vn |2 exp − 2 n=0

= lim sup RN

N

(5.82) ∞

ϕn βn ,

(5.83)

n=0

dϕ|ϕ1 . . . dϕ|ϕN

N 1 |ϕ|ϕn |2 × exp − 2 n=0 λn

N

1 2πλn n=0

12

exp i

N n=0

  ϕn v¯n



.

(5.84) The above cited theorem for the support characterization of Gaussian path-integrals can be generalized to the highly non-trivial case of a nonlinear functional Z(v) on E, satisfying the following conditions: (a) Z(0) = 1, (b)

N

Z(vj − vk )zj z¯k ≥ 0,

for any {zi }1≤i≤N : zi ∈ N

j,k

and {vi }1≤i≤N : vi ∈ E. (c)

(5.85)

there is an H-subspace of E with an inner product , , such that Z(v) is continuous in relation to a given inner product , A coming from a quadratic form defined by a positive definite class trace operator A on H. In others words, we have the sequential continuity criterium (if H is separable):

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lim Z(vn ) = 0,

lim vn , Avn = 0

n→∞

133

(5.86)

n→∞

if

ch05

/ A ∈ (H)

(5.87)

1

We have, thus, the following path-integral representation: dµ(ϕ) exp(iv, ϕ), Z(v) =

(5.88)

H

where

dµ(ϕ) = 1.

(5.89)

H

Another less mathematically rigorous result is that the one related to an inversible self-adjoint positive-definite operator A in a given Hilbert space (H, , ) — not necessarily a bounded operator in the class trace operator as considered previously. In order to write (more or less) formal path-integrals representations for the Gaussian functional 1 (5.90) Z(j) = exp − j, A−1 j 2 with f ∈ Dom(A−1 ) ⊂ H, we start by considering the usual spectral expression for the following quadratic form: λϕ, dE(λ)ϕ (5.91) ϕ, Aϕ = σ(A)

with σ(A) denoting the spectrum of A (a subset of R+ !) and dE(λ) are the spectral projections associated with the spectral representation of A. In this case one has the result for the path-integral weight as 1 1 λϕ, dE(λ)ϕ = exp − ϕ, Aϕ exp − 2 σ(A) 2    

1 = lim sup . (5.92) exp − ϕ, λdE(λ)ϕ   2 λ∈σFin (A)

Here σFin (A) denotes all subsets with a finite number of elements of σ(A).

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As a consequence, one should define formally the generating functional as 1 −1 Z(j) = exp − j, A j 2  0   +∞ 2 1 λ = lim sup (5.93) dxλ · e− 2 λ(xλ ) eijλ xλ  2π  −∞ λ∈σFin (A)

associated with the self-adjoint operator A acting on a Hilbert space H Z(j) = dA µ(ϕ)eij,ϕ . (5.94) H

Otherwise, one should introduce formal redefinitions of parameters entering in the definition of our action operator A, in such a way to render finite the functional determinant in Eq. (5.80). Let us exemplify such calculational point with the operator (−∆ + m2 ), acting an L2 (RN ) (with domain being given precisely by the Sobolev space H 2 (RN )). We note that TrL2 (RN ) (exp(−t(−∆ + m2 ))) ' 2 & +∞ 2 2 1 = √ dn ke−tk e−tm 2π −∞  0 (N − 2)!! π   if N − 1 is even,  N −1  t 2(2t) 2 = C(N )  ((N − 2/2))! 1   if N − 1 is odd.  N 2 (t) 2

(5.95)

with C(N ) denoting an N -dependent constant. It is worth to note that one must introduce in the path-integral equation (5.93), some formal definition for the functional determinant of the selfadjoint operator A, which by its term leads to the formal process of the “infinite renormalization” in quantum field path-integrals     +∞ dx 2 1 λ √ e− 2 λ(xλ ) dA µ(ϕ) = lim sup   (R-valued net) 2π −∞ H  = lim sup 

λ∈σFin (A)

λ∈σFin (A)

 1  −1 √ = detF 2 [A] λ

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  = lim+ ε→0



= lim

ε→0+

1

λ∈σFin (A)

1 exp + 2

1 exp + 2

1/ε ε

1/ε

ε

2 dt −tλ  e  t

dt T r(e−tA ) . t

(5.96)

In the case of the finitude of the right-hand side of Eq. (5.94) (which means that e−tA is a class trace operator and the finitude up the propertime parameter t), one can proceed as in the appendix to define mathematically the Gaussian path-integral. Let us now consider the proper-time (Cauchy principal value sense) integration process as indicated by Eq. (5.95), for the case of N an even space-time dimensionality

1/ε

I(m2 , ε) = ε

2

dt e−tm = t tN/2

1/ε

dt ε

e−tm t

2

N +2 2

(5.97)

The whole idea of the renormalization/regularization program means a (non-unique) choice of the mass parameter as a function of the proper-time cut-off ε in such a way that the infinite limit of ε → 0+ turns out to be finite, namely lim I(m2 (ε), ε) < ∞.

(5.98)

ε→0+

A slight generalization of the above exposed Minlos–Bochner theorem in Hilbert spaces is the following theorem: Theorem 5.3. Let (H, , 1 ) be a separable Hilbert space with the inner product , 1 . Let (H0 , , 0 ) be a subspace of H, so there is a trace class operator T : H → H, such that the inner product , 0 is given explicitly by g, h1 = g, T h0 = T 1/2 g, T 1/2 h0 . Let us, thus, consider a positive 1

definite functional Z(j) ∈ C((H, , 1 ), R) [if jn −→ j, then Z(jn ) → Z(j) on R+ ]. We obtain that the Bochner path-integral representation of Z(j) is given by a measure supported at these linear functionals, such that their restrictions in the subspace H0 are continuous by the norm induced by the “trace-class” inner product , 1 . With this result in our hands, it became more or less straightforward to analyze the cylindrical measure supports in distributional spaces.

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For instance, the basic Euclidean quantum field distributional spaces of tempered distributions in RN : S (RN ), can always be seen as the strong topological dual of the inductive limit of Hilbert spaces below considered sp = (xn ) ∈ C lim np xn n→∞

= 0, with the inner product (xn ), (yn )sp =

∞

2p

n xn y¯n .

(5.99)

n=1

We note now that ∞

S(RN ) =

sp

(5.100)

s−p .

(5.101)

p≥1

and

N

S (R ) =

∞ ! p≥1

An important property is that sp ⊃ sp+1 and they satisfy the hypothesis of Theorem 5.3, since ∞

n2p |xn |2 =

n=1

∞

n(2p+2)

n=1

1 2 |x | n2 n

(5.102)

and ∞ 1 π2 . = n2 6 n=1

(5.103)

As a consequence ||(xn )||sp ≤

π2 ||(xn )||sp+1 6

(5.104)

If one has an arbitrary continuous positive-definite functional in S(RN ), necessarily its measure support will always be on the topological dual of s , for each p. As a consequence, its support will be on the union set #p+1 ∞ p=1 s−p (since s−p ⊂ s−(p+1) ) and thus it will be the whole distribution space S (RN ). Similar results hold true in other distributional spaces. The application of the above-mentioned result in Gaussian pathintegrals is always made with the use of the famous result of the kernel theorem of Schwartz–Gelfand.

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Theorem 5.4 (Gelfand). Any continuous bilinear form B(j, j) defined in the test space of the tempered distribution S (RN ) has the following explicit representation: (5.105) B(j, j) = dN xdN yj(x)(Dxm Dyn F )(x, y)¯j(y) with j ∈ S(RN ), F (x, y) is a continuous function of polynomial growth and Dxm , Dyn are the distributional derivatives of order m and n, respectively. In all cases of application of this result to our study presented in the previous chapters were made in the context that (Dxm Dyn F )(x, y) is a fundamental solution of a given differential operator representing the kinetic term of a given quantum field Lagrangean. As a consequence, we have the basic result in the Gaussian path-integral in Euclidean quantum field theory − 12 B(j,j) = dµ(T ) exp{i(T (j))}, (5.106) e T ∈S (RN )

where T (j) denotes the action of the distribution T on the test function j ∈ S(RN ). At this point of our exposition let us show how to produce a fundamental solution for a given differential operator P (D) with constant coefficients, namely P (D) =

ap D p .

(5.107)

|ρ|≤m

A fundamental solution for Eq. (5.107) is given by a (numeric) distribution E ∈ S (RN ) such that for any ϕ ∈ S(RN ), we have: (P (D)E)(ϕ) = δ(ϕ) = ϕ(0)

(5.108)

E(t P (D)ϕ) = δ(ϕ),

(5.109)

or equivalently

where t P (D) is the transpose operator through the duality of S(RN ) and S (RN ).

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By means of the use of a harmonic–Hermite expansion for the searched fundamental solution E

S (RN )

=

∞

(E, Hp )Hp =

p=1

∞

Ep Hp

(5.110)

p=1

with Hp denoting the appropriate Hermite polynomials in RN , together with the test function harmonic expansion ϕ

S(RN )

=

∞

(ϕ, Hp )Hp =

p=1

∞

ϕp Hp

(5.111)

p=1

and the use of the relationship between Hermite polynomials t P (D)Hp = Mpq Hq

(5.112)

|q|≤(p)

with (p) depending on the order of Hp and the order of the differential operator t P (D). (For instance in S(R): (d/dx)Hn (x) = 2nHn−1 (x); d d2 Hn (x) = 2n Hn−1 (x) = 4n(n − 1)Hn−2 (x), etc., 2 d x dx one obtains the recurrence equations for the searched coefficients Ep in Eq. (5.110)   ϕn  Mnq Eq  = ϕn Hn (0) (5.113) n

|q|≤(n)

for any (ϕn ) ∈ 2 or equivalently

n

Mnq Eq = Hn (0)

(5.114)

|q|≤(n)

the solution of the above written infinite-dimensional system produces a set of coefficients {Ep }p=1,...,∞ satisfying a condition that it belongs to some space s−r , where r is the order of the fundamental solution being searched (the rigorous proof of the above assertions is left as an exercise to our mathematically oriented reader!). Finally, let us sketch the connection between path-integrals and the operator framework in Euclidean quantum field theory, both still mathematically non-rigorous from a strict mathematical point of view. In other

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¯ former approach, one has a self-adjoint operator H(j) indexed by a set of functions (the field classical sources of the quantum field theory under analysis) belonging to the distributional space S(RN ). This self-adjoint operator is formally given by the space-time integrated Lagrangean field theory and the basic object is the generating functional as defined by the vacuum–vacuum transition amplitude Z(j) = eiH(j) ΩVAC , ΩVAC

(5.115)

with ΩVAC denoting the theory vacuum state. It is assumed that Z(j) is a continuous positive definite functional on S(RN ). As a consequence of the above exposed theorems of Minlos and Bochner, there is a cylindrical measure dµ(T ) on S (RN ) such that the generating functional Z(j) is represented by the quantum field path-integral defined by the above-mentioned measure: dµ(T ) exp{i(T (j)} (5.116) Z(j) = S (RN )

in which the Feynman symbolic notation express itself in the following symbolic-operational Feynman notation: 1 DF [T (x, t)] Z[j(x, t)] = Z(0) S (RN ) 1

× e− 2

R +∞ −∞

dn−1 xdtL(T,∂t T,∂x T ) i

e

R +∞ −∞

dn−1 xdtj(x,t)T (x,t)

,

(5.117)

with L(T, ∂t T, ∂x T ) means generically the Lagrangean density of our field theory under quantization. Let us exemplify the Feynman symbolic Euclidean path-integral as given by Eq. (5.117) in the Gaussian case (free Euclidean field massless theory in RN , N ≥ 2) +∞ (−1) 1 +∞ N d x dN yj(x) exp − j(y) 2 −∞ (n − 2)Γ(n)|x − y|n−2 −∞ DF [T (x)] = S (RN )

+∞ 1 +∞ N N (N ) × exp − d x d yT (x)((−∆)x δ (x − y)T (y) 2 −∞ −∞ +∞ + 12 (N ) N × det ((−∆)xδ (x − y)) exp i d xj(x)T (x) . −∞

(5.118)

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Or for the heat differential operator in S (RN × R+ ) ∞ +∞ ∞ 1 +∞ N d x dt dN y dt exp − 2 −∞ 0 −∞ 0 1 |x − y|2 × exp − θ(t − t )j(y, t ) 4(t − t ) ( 2π(t − t )))N DF [T (x, t)]

= S (RN ×R+ )

& ' ∞ 1 +∞ N ∂ × exp − d x dtT (x, t) ∆x − T (x, t) 2 −∞ ∂t 0 & ' +∞ ∞ ∂ dN x dtJ(x, t)T (x, t) . × det1/2 ∆x − × exp i ∂t −∞ 0 (5.119a) It is worth to point out that the fourth-order path-integral in S (R4 ): +∞ 1 1 +∞ 4 4 2 |x − y| n(x − y) j(y) d x d (x)j(x) exp − 2 −∞ 8π −∞ R +∞ 4 2 1 = DF [ϕ(x)]e− 2 −∞ d x(ϕ(x)(∆ )ϕ(x)) S (R4 )

=

R (R4 )

d∆2 µ(ϕ)eiϕ(j) ei

R +∞ −∞

d4 xj(x)ϕ(x)

(5.119b)

1 |x|2 n|x| holds mathematically true since the locally integrable function 8π 2 4 is a fundamental solution of the differential operator ∆ : S (R ) → S(R4 ) when acting on distributional spaces. As a last important point of this section, we present an important result on the geometrical characterization of massive free field on a Euclidean space-time. Firstly, we announce a slightly improved version of the usual Minlos theorem.

Theorem 5.5. Let E be a nuclear space of test functions and dµ a given σ-measure on its topologic dual with the strong topology. Let , 0 be an inner product in E, inducing a Hilbertian structure on H0 = (E, , 0 ), after its topological completation.

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We assume the following: (a) There is a continuous positive definite functional in H0 , Z(j), with an associated cylindrical measure dµ. (b) There is a Hilbert–Schmidt operator T : H0 → H0 ; invertible, such that E ⊂ Range (T ), T −1 (E) is dense in H0 and T −1 : H0 → H0 is continuous. We have, thus, that the support of the measure satisfies the relationship supp dµ ⊆ (T −1 )∗ (H0 ) ⊂ E ∗ .

(5.120)

At this point we give a non-trivial application of Theorem 5.5. Let us consider a differential inversible operator L : S (RN ) → S(R), together with a positive inversible self-adjoint elliptic operator P : D(P ) ⊂ L2 (RN ) → L2 (RN ). Let Hα be the following Hilbert space: * + Hα = S(RN ), P α ϕ, P α ϕL2 (RN ) = , α , for α a real number . (5.121) We can see that for α > 0, the operators below P −α : L2 (RN ) → H+α ϕ → (P −α ϕ) P α : H+α → L2 (RN ) ϕ → (P α ϕ)

(5.122)

(5.123)

are isometries among the following subspaces D(P −α ), , L2 )

and H+α

since P −α ϕ, P −α ϕH+α = P α P −α ϕ, P α P −α ϕL2 (RN ) = ϕ, ϕL2 (RN ) (5.124) and P α f, P α f L2 (RN ) = f, f H+α .

(5.125)

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If one considers T a given Hilbert–Schmidt operator on Hα , the composite operator T0 = P α T P −α is an operator with domain being D(P −α ) and its image being the Range(P α ). T0 is clearly an invertible operator and S(RN ) ⊂ Range(T ) means that the equation (T P −α )(ϕ) = f has always a non-zero solution in D(P −α ) for any given f ∈ S(RN ). Note that the condition T −1 (f ) be a dense subset on Range(P −α ) which means that T −1 f, P −α ϕL2 (RN ) = 0

(5.126)

has a unique solution of the trivial solution f ≡ 0. Let us suppose that T −1 : S(RN ) → Hα be a continuous application and the bilinear term (L−1 (j))(j) be a continuous application in the Hilbert L2

L2

spaces H+α ⊃ S(RN ), namely: if jn −→ j, then L−1 : P −α jn −→ L−1 P −α j, for {jn }n∈Z and jn ∈ S(RN ). By a direct application of Theorem 5.5, we have the result 1 −1 dµ(T ) exp(iT (j)). Z(j) = exp − [L (j)(j)] = 2 (T −1 )∗ Hα

(5.127)

Here, the topological space support is given by (T −1 )∗ Hα = [(P −α T0 P α )−1 ]∗ (P α (S(RN ))) = [(P α )∗ (T0−1 )∗ (P −α )∗ ]P α (S(RN )) = P α T0−1 (L2 (RN )).

(5.128)

2 N N In the important case of L = (−∆ ) , +2 m N) : S (R ) → S(R ∗ 2 −2β ∗ ∈ (L (R )) since Tr(T T ) = and T0 T0 = (−∆ + m ) 0 0 1 2 N Γ( N )Γ(2β− N ) 1 ( m1 ) 2 2 Γ(β) 2 ∞ 2(m2 )β

for β N4 with the choice P = (−∆ + m2 ), we can see that the support of the measure in the path-integral representation of the Euclidean measure field in RN may be taken as the measurable subset below supp {d (−∆ + m2 )u(ϕ)} = (−∆ + m2 )−α · (−∆ + m2 )+β (L2 (RN )) (5.129)

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since L−1 P −α = (−∆ + m2 )−1−α is always a bounded operator in L2 (RN ) for α > −1. As a consequence each field configuration can be considered as a kind of “fractional distributional” derivative of a square integrable function written as ) ( N (5.130) ϕ(x) = (−∆ + m2 ) 4 +ε−1 f (x) with a function f (x) ∈ L2 (RN ) and any given ε > 0, even if originally all fields configurations entering into the path-integral were elements of the Schwartz tempered distribution spaces S (RN ) certainly very “rough” mathematical objects to characterize from a rigorous geometrical point of view. We have, thus, made a further reduction of the functional domain of the free massive Euclidean scalar field of S (RN ) to the measurable subset as given in Eq. (5.130) denoted by W (RN ) 1 2 −1 exp − [(−∆ + m ) j](j) 2 = d(−∆+m2 ) µ(ϕ)eiϕ(j)

S (RN )

= W (RN )⊂S (RN )

d(−∆+m2 ) µ ˜(f )eif,(−∆+m

2 N ) 4 +ε−1 f L2 (RN )

.

(5.131)

References 1. B. Simon, Functional Integration and Quantum Physics (Academic Press, 1979). 2. B. Simon, The P (φ)2 Euclidean (quantum) field theory, in Princeton Series in Physics (1974). 3. J. Glimm and A. Jaffe, Quantum Physics, — A Functional Integral Point of View (Springer Verlag, 1988). 4. Y. Yamasaki, Measure on Infinite Dimensional Spaces, Vol. 5 (World Scientific, 1985). 5. D.-X. Xia, Measure and Integration Theory on Infinite Dimensional Space (Academic Press, 1972). 6. L. Schwartz, Random Measure on Arbitrary Topological Space and Cylindrical Measures (Tata Institute, Oxford University Press, 1973). 7. K. Symanzik, J. Math. Phys. 7, 510 (1966). 8. B. Simon, J. Math. Phys. 12, 140 (1971). 9. L. C. L. Botelho, Phys. Rev. 33D, 1195 (1986).

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10. E. Nelson, Regular probability measures on function space, Ann. Math. 69(2), 630–643 (1959); V. Rivasseau, From Perturbative to Constructive Renormalization (Princeton University, 1991). 11. L. C. L. Botelho, Il Nuovo Cimento 117B, 37 (2002); Il Nuovo Cimento 117B, 331 (2002); J. Phys. A: Math. Gen. 34, 131–137 (2001); Mod. Phys. Lett. 173, Vol. 13/14, 733 (2003); Il Nuovo Cimento 117B, 15 (2002); Ya. G. Sinai, Topics in Ergodic Theory (Princeton University Press, 1994). 12. R. Teman, Infinite-Dimensional Systems in Mechanics and Physics, Vol. 68 (Springer-Verlag, 1988); J. M. Mour˜ ao, T. Thiemann and J. M. Velhinho, J. Math. Phys. 40(5), 2337 (1995). 13. A. M. Polyakov, Phys. Lett. 103B, 207 (1981); L. C. L. Botelho, J. Math. Phys. 30, 2160 (1989). 14. L. C. L. Botelho, Phys. Rev. 49D, 1975 (1994). 15. A. M. Polyakov, Phys. Lett. 103B, 211 (1981). 16. L. Brink and J. Schwarz, Nucl. Phys. B121, 285 (1977). 17. L. C. L Botelho, Phys. Rev. D33, 1195 (1986). 18. B. Durhuus, Quontum Theorem of String, Nordita Lectures (1982). 19. C. G. Callan Jr., S. Coleman, J. Wess and B. Zumine, Structure of Phenomenological Lagrangians — II, Phys. Rev. 177, 2247 (1969). 20. L. C. L. Boteho, Path integral Bosonization for a non-renormalizable axial four-dimensional. Fermion Model, Phys. Rev. D39(10), 3051–3054 (1989).

Appendix A In this appendix, we have given new functional analytical proofs of the Bochner–Martin–Kolmogorov theorem of Sec. 5.2. Bochner–Martin–Kolmogorov Theorem (Version I). Let f : E → R be a given real function with domain being a vector space E and satisfying the following properties: (1) f (0) = 1. (2) The restriction of f to any finite-dimensional vector subspace of E is the Fourier Transform of a real continuous function of compact support. Then there is a measure dµ(h) on σ-algebra containing the Borelians if the space of linear functionals of E with the topology of pontual convergence denoted by E alg such that for any y ∈ E f (g) = exp(ih(g))dµ(h). (A.1) E alg

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Proof. Let {ˆ eλ∈A } be a Hamel (vectorial) basis of E, and E (N ) a given subspace of E of finite-dimensional. By the hypothesis of the theorem, we have that the restriction of the functions to E (N ) (generated by the elements of the Hamel basis {ˆ eλ1 , . . . , eˆλN } = {eλ }λ∈ΛF ) is given by the Fourier transform N f σλ eˆλ = Q (dPλ1 · · · dPλN ) λ∈ΛF

=1

× exp

Rλ

1N

2 aλ Pλ gˆ(Pλ1 , . . . , PλN ),

(A.2)

=1

with gˆ(Pλ1 , . . . , PλN ) ∈ Cc ( λ∈ΛF Rλ ).

As a consequence of the above result, we consider the following welldefined family of linear positive functionals on the space of continuous function on the product space of the Alexandrov compactifications of R denoted by Rw : 1 LλF ∈ C

2Dual w λ

(R ) ; R

,

(A.3)

λ∈ΛF

with g(Pλ1 , . . . , PλN )] = LΛF [ˆ

Q λ∈ΛF

(Rw )λ

gˆ(Pλ1 , . . . , PλN )(dPλ1 · · · dPλN ). (A.4)

Here, gˆ(Pλ1 , . . . , PλN ) still denotes the unique extension of Eq. (A.2) to the Alexandrov compactification Rw . Now, we remark that the above family of linear continuous functionals have the following properties: (1) The norm of LλF is always the unity since ||Lλ1 || = Q gˆ(Pλ1 , . . . , PλN )dPλ1 · · · dPλN = 1. λ∈ΛF

(A.5)

(Rw )λ

(2) If the index set ΛF , contains ΛF the restriction of the associated linear

functional ΛF , to the space C( λ∈Λf (Rw )λ , R) coincides with LΛF .

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Now, a simple application of the Stone–Weierstrass theorem shows us that the topological closure of the union of the subspace of functions of

finite variable is the space C( λ∈A (Rw )λ , R), namely ! ΛF ⊂A

C

(Rw )λ , R

λ∈ΛF

=C

w λ

(R ) , R ,

(A.6)

λ∈A

where the union is taken over all family of subsets of finite elements of the index set A. As a consequence of the Remark 2 and Eq. (A.6) there is a unique extension of the family of linear functionals {LΛF } to the whole space

C( λ∈A (Rw )λ , R) and denoted by L∞ . The Riesz Markov theorem gives

us a unique measure d¯ µ(h) on λ∈A (Rw )λ representing the action of this

functional on C( λ∈A (Rw )λ , R). We have, thus, the following functional integral representation for the function f (g): ¯ ¯ f (g) = Q exp(ih(g))dµ( h). (A.7) (

w λ λ∈A (R ) )

$N ¯ Or equivalently (since h(g) = i=1 pi ai for some {pi }i∈N < ∞), we have the result (exp ih(g))dµ(h) (A.8) f (g) = Q ( λ∈A Rλ )

which is the proposed theorem with h ∈ ( λ∈λF Rλ ) being the element w λ ¯ on the Alexandrov compactification which has the image of h λ∈λF (R ) . The practical use of the Bochner–Martin–Kolmogorov theorem is difficult by the present day’ non-existence of an algoritm generating explicitly a Hamel (vectorial) basis on function of spaces. However, if one is able to apply the theorems of Sec. 5.3, one can construct explicitly the functional measure by only considering the topological basis as in the Gaussian functional integral equation (5.32). Bochner–Martin–Kolmogorov Theorem (Version II). We now have the same hypothesis and results of Theorem Version 1 but with the more general condition. (3) The restriction of f to any finite-dimensional vector subspace of E is the Fourier Transform of a real continuous function vanishing at “infinite.”

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For the proof of the theorem under this more general mathematical condition, we will need two lemmas and some definitions. Definition A.1. Let X be a normal space, locally compact and satisfying the following σ-compacity condition: ∞ !

X=

Kk ,

(A.9)

n=0

with Kn ⊂ int(Kn+1 ) ⊂ Kn+1 .

(A.10)

We define the following space of continuous function “vanishing” at infinite: C˜0 (X, R) = f (x) ∈ C(X, R) lim sup |f (x)| = 0 . (A.11) n→∞ c x∈(Kn )

We have, thus, the following lemma. Lemma A.1. The topological closure of the functions of compact support contains C˜0 (X, R) in the topology of uniform convergence. Proof. Let f (x) ∈ C˜0 (X, R) and gµ ∈ C(X, R), the (Uryhson) functions ¯ n and (K c ). Now it is straightassociated with the closed disjoints sets K n+1 forward to see that (f · gn )(x) ∈ Ci (X, R) and converges uniformly to f (x) by definition (A.11). At this point, we consider a linear positive continuous functional L on C˜0 (X, R). Since the restriction of L to each subspace C(Kn , R) satisfy the conditions of the Riesz–Markov theorem, there is a unique measure µ(n) on Kn containing the Borelians on Kn and representing this linear functional restriction. We now use the hypothesis equation (A.10) to have a well-defined measure on a σ-algebra containing the Borelians of X (A.12) µ ¯ (A) = lim sup µ(n) A Kn . for A in this σ-algebra and representing the functional L on C˜0 (X, R) L(f ) = f (x)d¯ µ(x). (A.13) X

Note that the normal of the topological space X is a fundamental hypothesis used in this proof by means of the Uryhson lemma.

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λ Unfortunately, the non-countable product space is not a λ∈A R normal topological space (the famous Stone counter example) and we cannot, thus, apply the above lemma to our vectorial case equation (A.8). However, we can overcome the use of the Stone–Weierstrass theorem in the proof of the Bochner–Martin–Kolmogorov theorem by considering directly a certain functional space instead of that given by Eq. (A.6). Thus, we define the following space of infinite-dimensional functions vanishing at finite

! def C˜0 Rλ , R = Rλ , R , (A.14) C0 (R∞ , R) ≡ C0 ΛF ⊂A

λ∈A

λ∈λF

where the closure is taken in the topology of uniform convergence.

If we consider a given continuous linear functional L on C0 ( λ∈A Rλ , R)

there is a unique measure µ∞ on the union of the Borelians λ∈ΛF Rλ representing the action of L on C0 (R∞ , R). Conversely, given a family of consistent measures {µΛF } on

the finite-dimensional spaces ( λ∈ΛF Rλ ) satisfying the property of

µΛF ( λ∈ΛF Rλ ) = 1, there is a unique measure on the cylinders λ∈A Rλ

associated with the functional L on C0 ( λ∈A Rλ , R). By collecting the results of the above written lemmas we will get the proof of Eq. (A.8) in this more general case.

Appendix B. On the Support Evaluations of Gaussian Measures Let us show explicitly by one example of ours of the quite complex behavior of cylindrical measures on infinite-dimensional spaces R∞ . Firstly, we consider the family of Gaussian measures on R∞ = {(xn )1≤n≤∞ , xn ∈ R} with σn ∈ 2 . N x2

1 − 2σn2 (∞) d µ({xn }) = lim sup (B.1) dxn √ e n . N σn π n=1 Let us introduce the measurable sets on R∞ ∞ E(αn ) = (xn ) ∈ R∞ ; x2(xn) = α2n x2n < ∞ n=1

and

∞

α2n σn2 < ∞.

=1

(B.2) Here, {αn } is a given sequence suppose to belonging to 2 either.

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Now, it is straightforward to evaluate the “mass” of the infinitedimensional set E(xn ) , namely & ' P −ε( ∞ α2n x2n ) (∞) ∞ n=1 µ(E(αn ) ) = d µ({xn }) lim e R∞

ε→0+

n

1 lim sup

= lim Note that

ε→0+

0≤≤n

n

(1 + 2ε

(1 + 2ε

(B.3)

=1

1 α2n σn2 )− 2

2 1 α2n σn2 )− 2

≤

=1

1+

1 $n =1

α2n σn2

.

(B.4)

As a consequence, one can exchange the order of the limits in Eq. (B.3) and arriving at the result 1 n 2

1 (∞) µ(E(αn ) ) = lim sup (1 + 2ε α2n σn2 )− 2 lim+ 0≤≤n

ε→0

=1

= lim sup {1} = 1.

(B.5)

0≤≤n

So we can conclude, on the basis of Eq. (B.5) that the support of the measure equation (B.1) is the set of E(αn ) for any possible sequence {αn } ∈ 2 . Let us show that (E(αn ) )C ∩ E(βn ) = {φ}, so that these sets are not coincident. Let the sequences be σn = n−σ , αn = nσ−1 ,

(B.6)

βn = nσ−λ , with γ > 1 and σ > 0. We have that

α2n σn2 =

βn2 σn2 =

1 π2 , = n2 6

n−2λ < ∞.

So E{αn } and E{βn } are non-empty sets on R∞ .

(B.7)

(B.8)

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Let us consider the point {¯ xn } ∈ R∞ and defined by the relation x ¯2n = n−2(σ−1)−ε .

(B.9)

We have that

(¯ xn )2 α2n =

2

n−2(σ−1)−ε · nn

(σ−1)

=

n−ε

and

(¯ xn )2 βn2 =

n−2(σ−1)−ε · n2(σ−λ) =

n2−ε−2λ

If we choose ε = 1; γ > 1 (γ = 32 !), we obtain that the point {¯ xn } $ 2 π2 n = 6 ), however it does not belongs belongs to the set E{βn } (since $ −1 to E{xn } (since ∞ = +∞), although the support of the measure n=0 n equation (B.1) is any set of the form E{γn } with {γn } ∈ 2 .

Appendix C. Some Calculations of the Q.C.D. Fermion Functional Determinant in Two-Dimensions and (Q.E.D.)2 Solubility Let us first define the functional determinant of a self-adjoint, positive definite operator A (without zero modes) by the proper-time method log detF (A) = − lim

β→0+

ε

∞

dt Trf (e−tA ) , t

(C.1)

where the subscript f reminds us the functional nature of the objects under study and so its trace. It is thus expected that the definition equation (C.1) has divergents counter terms as ε → 0+ , since exp(−tA) is a class trace operator only for t ≥ ε. Asymptotic expressions at the short-time limit t → 0+ are wellknown in mathematical literature. However, this information is not useful in the first sight of Eq. (C.1) since one should know TrF (e−tA ) for all t-values in [ε, ∞). A useful remark on the exact evaluation is in the case where the operator A is of the form A = B + m2 1 and one is mainly interested in the effective

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asymptotic limit of large mass m2 → ∞. In this particular case, one can use a saddle-point analysis of the expression in Eq. (C.1) ' ∞ & dt −m2 t 2 −tB e ) . lim [log detF (B + m 1)] = − lim lim TrF (e ε→0 m2 →∞ t t∈0+ ε (C.2) Another very important case is covered by the (formal) Schwarz–Romanov theorem announced below. Theorem C.1. Let A(σ) be a one-parameter family of positive-definite selfadjoints operators and satisfying the parameter derivative condition (0 ≤ σ ≤ 1) d A(σ) = fA(σ) + A(σ)g, dσ

(C.3a)

where f and A are σ-independent objects (may be operators). Then we have an explicit result 1 (σ) 2 detF (A(1)) log dσ lim TrF [f e−ε(A ) ] = + detF (A(0)) ε→0 0 1 −ε(A(σ) )2 + dσ lim TrF [ge ] . 0

ε→0+

(C.3b)

The proof of Eqs. (C.3a) and (C.3b) is based on the validity of the differential equation in relation to the σ-parameter * + 2 2 d [log detF (A(σ))2 ] = lim+ 2 TrF (f e−ε(A(σ)) ) + Trε (ge−ε(A(σ)) ) dσ ε→0 (C.4) which can be seen from the obvious calculations written down in the above equation [log detF (A(σ))2 ] ∞ & = − lim+ dt TrF (f A(σ) + A(σ)g)A(σ) + A(σ)(f A(σ) ε→0

ε

−2

+ A(σ)g)(−(A(σ)) = Eq. (C.4).

' d 2 ) (exp(−t(A(σ)) )) dt (C.5)

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Let us apply the above formulae in order to evaluate the functional determinant of the “Chrially transformed” self-adjoint Dirac operator in a two-dimensional space-time D(σ) = exp(σγ 5 ϕ2 (x)λa )|∂|(exp(σγ 5 ϕa (x)λa ).

(C.6)

Here, the Chiral Phase W [ψ] in Eq. (C.6) takes value in SU (N ) for instance. One can see that 7 1 ( D(σ))2 = −(∂µ + iGµ (σ))2 1 − [γµ , γν ] Fµν (−iGµ (σ)) 4

(C.7a)

with the Gauge field γµ Gµ = γµ (W −1 ∂µ W ).

(C.7b)

Hence, the asymptotics of the operator (C.7a) are easily evaluated (see Chap. 1, Appendix E) lim TrF (exp(−ε(D(σ))2 ))

ε→0+

= lim+ Tr ε→0

1 4πε

1+

g(iεµν γ5 ) Fµν (−iGµ (σ)) , 2

(C.8a)

where we have used the Seeley expansions below for the square of the Dirac operator in the presence of a non-Abelian connection A : C0∞ (R2 ) → C0∞ (R2 ), Aϕ = (−∆) − V1 (ξ1 , ξ2 )

lim

t→0+

∂ ∂ − V2 (ξ1 , ξ2 ) − V0 ((ξ1 , ξ2 ), ∂ξ1 ∂ξ2

(C.8b)

(C.8c)

8 9 TrCc∞ (R2 ) (e−tA ) 1 ∂ 1 ∂ 1 1 + − (V12 + V22 ) − V0 = V1 + V0 − 4πt 8π ∂ξ1 ∂ξ2 16π 4π × (ξ1 , ξ2 ) + O(t),

7 7 ( ∂ − ig Gµ )2 = (−∂ 2 )ξ + (2igGµ ∂m u)ξ & ' igσ µν 2 2 Fµν (G) + g Gµ , + ig(∂µ Gµ ) + 2 ξ

(C.8d)

(C.8e)

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which leads to the following exact integral (non-local) representation for the non-Abelian Dirac determinant detF (D(1)) log detF (∂) ' & 1 i 2 a µν d x TrSU(N ) ϕa (x)λ dσEµν F (−iGµ (σ)) . = 2π 0 (C.9) A more invariant expression for Eq. (C.9) can be seen by considering the decomposition of the “SU (N ) gauge field” Gµ (σ) in terms of its vectorial and axial components: (W −1 (σ)∂µ W (σ)) = Vµ (σ) + γ 5 Aµ (σ)

(C.10)

or equivalently (γµ γ5 = itEµν Yν , γs = iγ0 γ1 , [γµ , γν ] = −2iEµν γ5 ): Gµ (σ) = Vµ (σ) + iEµν Aν (σ).

(C.11)

At this point we point out the formula Fµν (−iGµ (σ)) = {(iDαv (σ)Aα (σ))E µν − [Aµ (σ), Aν (σ)] + Fµν (Vµ (σ))}. (C.12) Here DαV Aβ = ∂α Aβ + [Vα , Aβ ].

(C.13)

Note that Aµ (σ) and Vµ (σ) are not independent fields since the chiral phase W (σ) satisfies the integrability condition Fµν (W ∂µ W ) ≡ 0

(C.14)

Fµν (Vβ (σ)) = −[Aµ (σ), Aν (σ)],

(C.15)

DµV Aν (σ) = DνV Aµ (σ).

(C.16)

or equivalently

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After substituting Eqs. (C.12)–(C.16) in Eq. (C.9), one can obtain the results  1  i  2 a = d x Tr φ (α) dσ(2iDµ (σ)Aµ (σ) λ  a SU(N )   0  2π    −[Aµ (σ),Aν (σ)]   : ;< =     + Eµν ( Fµν (Vα (σ)) ) − [Aµ (σ), Aν (σ)] , (C.17)        i   = d2 x TrSU(N )   2π  detF (D(1)) 1 log a  detF (∂)  × λ φa (x) dσ(2iDµ (σ)Aµ (σ))    0     i    + d2 x TrSU(N )   2π    1    a   × λ φ (x) dσ(−2[A (σ), A (σ)]) a µ ν    0     = I1 (φ) + I2 (φ). (C.18) Let us now show that the term I1 (φ) is a mass term for the physical gauge field Aµ (σ = 1) = Aµ . First, we observe the result Lµ (σ) : ;< = d (W ∂µ W )(σ) TrDirac ⊗ TrSU(N ) γ 5 Aµ (σ) dσ

= TrDirac ⊗ TrSU(N ) {γ 5 φa (x)λa (γ 5 (∂µ Aµ (σ)) − [γ 5 Aµ (σ), Lµ (σ)])(x)} = 2 TrSU(N ) {λa φa (x)(∂µ Aµ (σ) + [Vµ (σ), Aµ (σ)](x))} = 2 TrSU(N ) {λa φa (x)Dµ (σ)Aµ (σ)} = I1 (φ).

(C.19)

By the other side TrDirac ⊗ TrSU(N ) {γ 5 Aµ (σ)Lµ (σ)} d d 5 Vµ (σ) + γ5 Aµ (σ) = TrDirac ⊗ TrSU(N ) γ Aµ (σ) dσ dσ d = TrSU(N ) (Aµ (σ)Aµ (σ)) . (C.20) dσ

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At this point it is worth to see the appearance of a dynamical Higgs mechanism for Q.C.D. in two-dimensions. Let us now analyze the second term I2 (φ) in Eq. (C.18) 1 i d2 x TrSU(N ) dσEµν (2φa (x)λa [Aµ (σ), Aν (σ)](x)) exp − 2π 0 1 1 :; [φ](iγ µ ∂µ )W > [φ]. =W

(C.25)

Here * + Rx µ > [φ] = eiγs φea (x) ≡ PDirac PSU(N ) eiγs −∞ dξ (εµν Gν)(x) . W

(C.26)

Since > )∂µ (W > )−1 (x). (γµ Gµ )(x) = +γµ (W

(C.27)

It is worth now to use the formalism of invariant functional integration — Appendix in Chap. 1 — to change the quantization variables of > (φ). This task is easily the gauge field Aµ (x) to the SU (N )-axial phases W accomplished through the use of Riemannian functional metric on the manifold of the gauge connections dS 2 = =

d2 x TrSU(N ) (δGµ δGµ )(x) 1 4

d2 x TrSU(N )⊗Dirac [(γµ δGµ )(γ µ δGµ )](x)

77 = detF [ D D∗ ]adg ×

−1 −1 > > d x TrSU(N )Axial [(δ W W )(δ W W )] , 2

(C.28) since * + > (−W > −1 (δ W > )W > −1 + ∂µ W > )W > −1 ) γµ (δGµ ) = γµ ∂µ (δ W >W > −1 ) = γµ (∂µ − [Gµ , ])(δ W and >W > −1 ] = δ W >{γµ , W > −1 } + {γµ , δ W > }W > −1 ≡ 0 [γµ , δ W

(C.29)

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and leading to new parametrization for the gauge field measure ? 7 7 @1 > (x)]. DF [Gµ (x)] = detF,adj ( D D∗ ) 2 DHaar [W

(C.30)

1 Here, detF,adj φφ∗ 2 is the functional Dirac operator in the presence of the gauge field and in the adjoint SU (N )-representation. Its explicit evaluations are left to our readers. The full gauge-invariant expression for the Fermion determinant is conjectured to be given on explicit integration of the gauge parameters considered now as dynamical variables in the gauge-fixed result. For instance, in the Abelian case and in the gauge fixed Roskies gauge equation (C.6) result, we have the Schwinger result detF [iγ µ (∂µ − ieAµ )] 2 1 e F 2 1 2 = D [W (x)] exp − d x (Aµ − ∂µ W (x)) 2 π 2 2 e F 2 2 2 = D [W (x)] exp − d x[Aµ + (∂µ W ) + 2Aµ ∂µ W ] 2π 2 & ' e ∂µ ∂ν = exp − (x) . (C.31) d2 x Aµ δµν − A ν π (−∂ 2 ) Note that the Haar measure on the Abelian Group U (1) is 2 = δSW

d2 x[δ(eiW ∂µ e−iW )δ(eiW ∂µ e−iW )](x)

=

d2 x(δW (−∂ 2 )δW )(x).

(C.32)

Let us solve exactly the two-dimensional quantum electrodynamics. Firstly, Eq. (C.10) takes the simple form in term of the chiral phase in the Roskies gauge Gµ (x) = (εµν ∂ν φ)(x).

(C.33)

Now, the somewhat cumbersome non-Abelian equations (C.3a) and (C.3b) has a straightforward form in the Abelian case DF [Gµ ] = detF (−∆)DF [φ]

(C.34)

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and thus, we have the exactly soluble expression for the (Q.E.D.)2 generating functional (a non-gauge invariant object!) 1 ¯ Z[Jµ , η, η¯] = DF [φ]DF [χ]DF (χ] 2 & ' 1 e2 × exp − d2 x φ − ∂ 4 + ∂ 2 φ + Eµν (∂ν Jµ )φ (x) 2 π 7' & 1 0 i∂ χ × exp − d2 x(χ, χ) ¯ 7 i∂ 0 χ ¯ 2 & −igγ φ ' s e 0 η 2 ¯ × exp − d x(χ, χ) 0 e−igγs φ η¯ For instance, correlations functions are exactly solved and possessing a “coherent state” factor given by the φ-average below (after “normal ordenation” at the coincident points) and explicitly given a proof of the confinement of the fermionic fields since one cannot assign LSZ-scattering fields configuration for them by the Coleman theorem since then grown as the 1 factor |x − y| 2g2 at large separation distance e−i(γs )x φ(x) e−i(γs )y φ(y) φ

= cos(ϕ(x)) cos(ϕ(y))φ 1x ⊗ 1x + γs ⊗ 1 sin(ϕ(x)) cos(ϕ(y))φ + γs ⊗ 1 cos(ϕ(x)) sin(ϕ(y))φ − γs ⊗ γs sin(ϕ(x)) sin(ϕ(s))φ ? −1 @ g2 2 2 −(−∂ 2 )−1 (x,y) = e−g −∂ + π 1 g 1 π K0 √ |x − y| + g|x − y| (1x ⊗ 1y ). (C.35) = exp + 2 g 2π π 2π The two-point function for the two-dimensional-electromgnetic field shows clearly the presence of a massive excitation (Fotons have acquired a mass term by dynamical means) Gµ (x)Gν (g)φ =

d2 k 1eik(x−s) (Eµα Eνβ )(k α k β ) 2 (2π) k 2 (k 2 + eπ )

1 = 2π

δ µν

: ;< = δ αβ k 2 eik|x−y| . d2 k [Eµα Eνβ ] 2 k 2 (k 2 + eπ )

(C.36)

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As an important point of this supplementary appendix, we wish to point out that the chirally transformed Dirac operator equation (C.6) in fourdimensions, still have formally an exactly integrability as expressed by the integral representation equation (C.3) (see the asymptotic expansion equations (4.38) in Chap. 4). It reads as follows 2 1 7 det( D(σ)2 ) i 4 a x Tr = − d log SU(N ) φa (x)λ det(∂)2 2π 2 & 1 ' c c × dσFαβ (−iGµ (σ))Fµν (−iGµ (σ))εαβµν λc λc , 0

(C.37) where −iγµ Gµ (σ) = exp (σγ 5 φa (x)λa )(i∂) exp (σγ 5 φa (x)λa )

(C.38)

and we have the formulae19 −iγµ Gµ (σ) = Vµ (σ) + γ 5 Aµ (σ),

(C.39a)

a Aµ (σ) = ∆−1 γs φa λa {sin h(∆γs φa λa ) ◦ ∂µ (γ5 φ λa )},

(C.39b)

a Vµ (σ) = ∆−1 γs φa λa {(1 − cos h(∆γs φa λa )) ◦ ∂µ (γ5 φ λa )}

(C.39c)

with the matrix operation ∆X ◦ Y = [X, Y ]

(C.39d)

(n)

and ∆X denoting its n-power. For a complete quantum field theoretic analysis of the above formulae in an Abelian (theoretical) axial model we refer our work to L. C. L. Botelho.20 Finally and just for completeness and pedagogical purposes, let us deduce the formal short-time expansion associated with the second-order positive differential elliptic operator in Rν used in the previous cited reference L = −(∂ 2 )x + aµ (x)(∂µ )x + V (x).

(C.40)

Its evolution kernel k(x, y, t) = x| − exp(−t L) |y satisfies the heat-kernel equation ∂ K(x, y, t) = −Lx K(x, y, t), ∂t

(C.41)

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K(x, y, 0) = δ (ν) (x − y).

(C.42)

After substituting the asymptotic expansion below into Eq. (C.41) [with K0 (x, y, t) denoting the free kernel (aµ ≡ 0 and V ≡ 0)] 1∞ 2 t→0+ n t Hn (x, y) (C.43) K(x, y, t) K0 (x, y, t) n=0

and by taking into account the obvious relationship for t > 0 (∂µ K0 (x, y, t))|x=y = 0.

(C.44)

We obtain the following recurrence relation for the coefficients Hn (x, x): 8 9 (n + 1)Hn+1 (x, x) = − (−∂x2 )Hn (x, x) + aµ (x) · ∂µ Hn (x, x) + V (x) . (C.45) For the axial Abelian case in R4 , we have the result: 1 s µ ν (egγs φ (i∂)egγs φ )2 = (−∂ 2 )14×4 + gγ [γ , γ ] ∂µ φ(x) ∂ν 2 + [−gγs ∂ 2 φ + (g)2 (∂µ φ)2 ].

(C.46)

Note that in R4 H0 (x, x) = 14×4 , H1 (x, x) = −V (x), H2 (x, x) =

1 [−∂ 2 V + aµ ∂µ V + V 2 ](x). 2

(C.47)

Appendix D. Functional Determinants Evaluations on the Seeley Approach In this technical appendix, we intend to highlight the mathematical evaluation of the functional determinants involved in the theory of random surface path-integral.18 Let us start by considering a differential elliptic self-adjoint operator of second-order acting on the space of infinitely differentiable functions of compact support in R2 ,Cc∞ (R2 , Cq ) with values in Cq A= aα (x)Dxα (D.1) |α|≤2

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with α = (α1 , α2 ) multi-indexes and Dxα

=

1 ∂ i ∂x1

α1

1 ∂ i ∂x2

α2 (D.2)

and Aα (x) ∈ Cc∞ (R2 , Cq ). By introducing the usual square-integrable inner product in Cc∞ (R2 , Cq ) and making the hypothesis that A is a positive definite operator, one may consider the (contractive) semi-group generated by A and defined by the spectral calculus e

−tA

1 = 2πi

e−tλ dλ (λ1q×q − A) C

(D.3)

with C being an (arbitrary) path containing the positive semi-weis λ > 0 (the spectrum of the operator A) with a counter-clockwise orientation. According to Seeley, one must consider the symbol associated with the resolvent pseudo-differential operator (λ1q×q − A)−1 and defined by the relation below σ(A − λ1) = eixξ (A − λ1)eixξ =

2

Aj (x, ξ, λ).

(D.4)

|j|=0

with  Aj (x, ξ, λ) = 



aj (x)(ξ1α1 )(ξ2α2 )

0 ≤ j < 2,

(D.5)

|α|=α1 +α2 =j

 A2 (x, ξ, λ) = −λ1q×q + 

 aj (x)(ξ1α1 )(ξ2α2 )

j = 2.

(D.6)

|α|=α1 +α2 =j

This is basic for symbols calculations, the important scaling properties as written below Aj (x, cξ, c2 λ) = (c)j Aj (x, ξ, λ).

(D.7)

It too is a fundamental result of the Seeley’s theory of pseudo-differential operators that the resolvent operator (A − λ1)−1 (the associated Green

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functions of the operator A) has an expansion of the form in a suitable functional space σ((A − λ1)−1 ) =

∞

C−2−j (x, ξ, λ)

(D.8)

j=0

and satisfies the relations below σ(A − λ1) · σ((A − λ1)−1 ) = 1,   ∞  1  [(Dξα (σ(A − λ1))(x, ξ)Dxα [C−2−j (x, ξ, λ)] = 1.  a!  j=0

(D.9)

(D.10)

|α|≤2

Recurrence relationships for the explicitly determination of the Seeley coefficients of the resolvent operator equation (D.8) can be obtained through the use of the scaling properties ξ = pξ ;

1

1

λ 2 = p(λ ) 2 ,

1

C−2−j (x, pξ , (p(λ ) 2 )2 ) = p−(2+j) C−2−j (x, ξ , λ ).

(D.11) (D.12)

After using Eq. (D.11) in Eq. (D.10) and for each integer j, comparing the resultant power series in the variable 1/p(1 = 1 + 0( p1 ) + · · · + 0( p1 )n + · · · ), one can get the result as C−2 (x, ξ) = (A2 (x, ξ)−1 )

0 = a2 (x, ξ)C−2−j (x, ξ) +

8 > >
α! > :

X

(D.13)

Dξα ak (x, ξ) ((iDxα C−2− (x, ξ, λ)))

<j k−|α|−2−=−j

9 > > = > > ;

(D.14)

For the explicit operator in Cc∞ (R2 , Rq ) is given below ∂2 ø2 A = − g11 2 + g22 2 1q×q ∂x1 ∂x2 − (A1 )q×q

∂ ∂ − (A2 )q×q − (A0 )q×q ∂x1 ∂x2

(D.15)

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with all the coefficients in Cc∞ (R2 , Rq ), one obtains the following results after calculations: A2 (x, ξ, λ) = (g11 (x)ξ12 + g22 ξ22 − λ)1q×q , A2 (x, ξ, λ) = −iA1 (x)ξ1 − iA2 (x)ξ2 ,

(D.16)

A0 (x, ξ, λ) = −A0 (x), and C−2 (x, ξ) = (g11 (x)ξ12 + g22 (x)ξ22 − λ)−1 , C−3 (x, ξ) = i(A1 (x)ξ1 + A2 (x)ξ2 )(C−2 (x, ξ))2 & ' ∂ ∂ −2ig11(x)ξ1 g11 (ξ1 )2 + g22 (ξ2 )2 (C−2 (x, ξ))3 ∂x1 ∂x2 & ' ∂ ∂ −2ig22(x)ξ2 g11 (ξ1 )2 + g22 (ξ2 )2 (C−2 (x, ξ))3 ∂x2 ∂x2 (D.17) By keeping in view evaluations of the heat kernel of our given differential operator, let us write its expansion in terms of the Seeley coefficients of (D.8): 2 ∞ 1 −tA 2 −tA )= d xσ(e )(x, ξ) T r(e 2π R2 j=0 =

2 ∞ 1 1 2π 2πi j=0 &

×

−∞

+∞

=

∞ j=0

=

=

1 1 t (2π)3

∞ 1 j=0 ∞ j=0

1 t (2π)3

t

d(−is)eist

(j−2) 2

R2 ×R2

2

' d2 xd2 ξC−2−j (x, ξ, −is) &

+∞

2

d ξ

d x

R2

R2

−∞

d2 ξ R2 −3

d2 xeis t(

2+j 2

R2

2

(2π)

)

+∞

d x R2

'

1 C−2−j (x, t 2 ξ, −is)

2

d ξ R2

−is )ds e C−2−j (x, ξ, t is

−∞

C−2−j (x, ξ, −is) . (D.18)

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By applying Eq. (D.17) to the differential operator as given in Eq. (D.15), we obtain the short-time Seeley expansion as an asymptotic expansion in the variable t q q 1 −tA 2 √ 2 √ )∼ T r(e d x g11 g22 + d x g11 g22 − R 4πt 4π 6    (− 12 divcov A )  : = ; π since its associated Gell–Mann–Low function has a nontrivial zero for g 2 = π (the model contains a tachyonic excitation).12

Appendix B. Path-Integral Bosonization for an Abelian Nonrenormalizable Axial Four-Dimensional Fermion Model B.1. Introduction The study of two-dimensional fermion models in the framework of chiral anomalous path-integral has been shown to be a powerful nonperturbative technique to analyze the two-dimensional bosonization phenomenon. It is the purpose of this appendix to implement this nonperturbative technique to solve exactly a nontrivial and nonrenormalizable four-dimensional axial fermion model which generalizes for four dimensions the two-dimensional model studied in the previous appendix.

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B.2. The Model Let us start our analysis by considering the (Euclidean) Lagrangian of the proposed Abelian axial model in R4 ¯ φ) = ψγ ¯ µ (i∂µ − gγ5 ∂µ φ)ψ + 1 g 2 (∂µ φ)2 + V (φ), L 1 (ψ, ψ, 2

(B.1)

where ψ(x) denotes a massless four-dimensional fermion field, φ(x) a pseudoscalar field interacting with the fermion field through a pseudoscalar derivative interaction, and V (φ) is a φ self-interaction potential given by V (φ) =

g2 −g 4 φ(∂µ φ)2 (−∂ 1 φ) + 2 (−∂ 2 φ)(−∂ 2 φ). 4 12π 4π

(B.2)

The presence of the above φ potential is necessary to afford the exact solubility of the model as we will show later (Eq. (B.18)). The Hermitian γ matrices we are using satisfy the (Euclidean) relations [γµ , γν ] = 2δµν ,

γ5 = γ0 γ1 γ2 γ3 .

(B.3)

¯ φ) is invariant under the global Abelian and The Lagrangian L 1 (ψ, ψ, chiral Abelian groups ψ → eiα ψ,

ψ → eiγ5 β ψ,

(α, β) ∈ R,

(B.4)

with the Noether conserved currents at the classical level: ¯ 4 γ µ ψ) = 0, ∂µ (ψγ

¯ µ ψ) = 0. ∂µ (ψγ

(B.5)

In the framework of path-integrals, the generating functional of the correlation functions of the mathematical model associated with the ¯ φ) is given by Lagrangian L 1 (ψ, ψ, 1 ¯ Z[J, η, η¯] = D[φ]D[ψ]D[ψ] Z[0, 0, 0] 4 ¯ ¯ (B.6) × exp − d x[L1 (ψ, ψ, φ) + Jφ + η¯ψ + ψη](x) . In order to generalize for four-dimensions the chiral anomalous pathintegral bosonization technique, we first rewrite the full Dirac operator in the following suitable form: D[φ] = iγµ (∂µ + igγ5 ∂µ φ) = exp(igγ5 φ)(iγµ ∂µ ) exp(igγ5 φ).

(B.7)

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Now, we proceed as in the two-dimensional case by decoupling the ¯ φ) fermion field from the pseudoscalar field φ(x) in the Lagrangian L 1 (ψ, ψ, by making the chiral change of variables: ψ(x) = exp[igγ5 ψ(x)]χ(x), ¯ ψ(x) = χ(x) ¯ exp[igγ5 φ(x)].

(B.8)

¯ defined by the eigenOn the other hand, the fermion measure D[ψ]D[ψ], ∗ vectors of the Dirac operator D[φ] D[φ] , is not invariant under the chiral change and yields a nontrivial Jacobian, as we can see from the relationship 4 ¯ ¯ D[ψ]D[ψ] exp − d x(ψ D[φ]ψ)(x) = [Det( D[ψ] D[ψ]∗ )] 4 = J[φ] D[χ]D[χ] ¯ exp − d x(χiγ ¯ µ ∂µ (χ)(x) .

(B.9)

4 Here, J[φ] = Det( D[φ] D[φ]∗ ) Det( D[φ = 0] D[φ = 0]∗ ) is the explicit expression for this Jacobian. It is instructive to point out that the model displays the appearance of the axial anomaly as a consequence of the nontriviality of J[φ], i.e. ¯ µ γ5 ψ)(x) = {(δ/δφ)J[ψ]}(χ). ∂µ (ψγ So, to arrive at a complete bosonization of the model (Eq. (B.6)) we face the problem of evaluating J[φ]. Let us, thus, compute the four-dimensional fermion determinant det( D[φ]) exactly. In order to evaluate it, we introduce a one-parameter family of Dirac operators interpolating the free Dirac operator and the interacting one D(ζ) [φ], namely, D(ζ) [φ] = exp(igγ5 ζφ)(iγµ ∂µ ) exp(igγ5 ζφ),

(B.10)

with ζ ∈ [0, 1]. At this point, we introduce the Hermitian continuation of the operators D(ζ) [φ] by making the analytic extension in the coupling constant g¯ = ig. This procedure has to be done in order to define the functional determinant by the proper-time method since only in this way ( Dζ [φ])2 can be considered as a (positive) Hamiltonian ( Dζ [φ])2 ≡ ( Dζ [φ])( Dζ [φ])∗ . The justification of this analytic extension in the model coupling constant is due to the fact that typical interaction energy densities such as ¯ µ ∆µ ψ, which are real in Minkowski space-time, become complex ¯ 4 ψ, ψγ ψγ

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after continuation in Euclidean space-time.21 As a consequence, the above analytic coupling extension must be done in the proper-time regularization for the Dirac functional determinant. By using the property d Dζ [φ] = g¯γ5 ψ Dζ [φ] + Dζ [φ]γ5 g¯φ, dζ

(B.11)

we can write the following differential equation for the functional determinant19 : d n Det Dζ [φ] dζ g γ5 ψ) exp{−ε( Dζ [φ]2 }|x, = 2 lim d3 x Trx|(¯ ε→0+

(B.12)

where Tr denotes the trace over Dirac indices. The diagonal part of exp{−ε( Dζ [φ]2 } has the asymptotic expansion20,22 1 ε2 H x| exp{−ε( Dζ [φ])2 }|x ∼ (φ) + (φ) , (B.13) 1 + εH 1 2 2! ε→0+ 16π 2 ε2 with the Seeley coefficients given by (see the complements) H1 (φ) = ζ g¯γ5 ∂ 2 φ − g¯2 ζ 2 (∂µ φ)2 ,

(B.14)

and H2 (φ) = 2∂ 2 H1 (φ) + 2ζ g¯γ4 (∂µ φ)∂µ H1 (φ) + 2[H1 (φ)]2 .

(B.15)

By substituting Eqs. (B.14) and (B.15) into Eq. (B.12), we obtain finally the result for the above-mentioned Jacobian: J[φ] = J0 [φ, ε]J1 [φ]. Here, H0 [φ, ε] is the ultraviolet cut-off dependent Jacobian term 2 g 4 2 J0 [φ, ε] = exp d x[φ(−∂ )φ](x) 4π 2 ε and J1 (φ) is the associated Jacobian finite part g2 J1 [φ] = exp − 2 d4 x(−∂ 2 φ)(−∂ 2 φ)(x) 4π 2 g 4 2 2 × exp x[φ(∂ φ) (−ø φ)](x) . d µ 12π 2

(B.16)

(B.17)

(B.18)

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From Eqs. (B.17) and (B.1), we can see that the (bare) coupling constant g gets an additive (ultraviolet) renormalization. Besides the Jacobian term cancels with the chosen potential V (φ) in Eq. (B.2). As a consequence of all these results, we have the following expression for Z[J, η, η¯] with the fermions decoupled: 2

1 D(φ]D(χ]D(χ]J[φ](x) ¯ Z[J, η, η¯] = Z[0, 0, 0] 1 1 1 2 ¯ µ ∂µ )χ + x¯(iγµ ∂µ )χ + gR (∂µ φ)2 × exp − d4 x χ(γ 2 2 2 + η¯ exp(igR γ5 φ)χ + χ ¯ exp(igR γ5 φ)η . (B.19) This expression is the main result of this appendix and should be compared with the two-dimensional analogous generating functional analyzed in Appendix A, Eq. (A.9). Now, we can see that the quantum model given by Eq. (6.6), although being nonrenormalizable by usual power counting and Feynman-diagrammatic analysis, it still has nontrivial and exactly soluble Green’s functions. For instance, the two-point fermion correlation function is easily evaluated and produces the result F (x1 −x2 ; m = 0) exp[−∆F (x1 , x2 .m = 0)]. ψα (x1 )ψ¯β (x2 ) = Sαβ

(B.20)

Here, ∆(x1 − x2 ; m = 0) is the Euclidean Green’s function of the massless free scalar propagator. We notice that correlation functions involving ¯ fermions ψ(x), ψ(x) and the pseudoscalar field φ(x) are easily computed too (see Ref. 17). As a conclusion of our appendix let us comment to what extent our proposed mathematical Euclidean axial model describes an operator quantum field theory in Minkowski space-time. In the operator framework the fields ψ(x) and φ(x) satisfy the following wave equations: iγµ ∂µ ψ = igγµ γ5 (∂µ φ)ψ, ¯ µ γ5 ψ) + δ∆ · φ = ∂µ (ψγ δφ

(B.21)

It is very difficult to solve exactly Eq. (B.12) in a pure operator ¯ µ γ5 ψ) = 0]. framework because the model is axial anomalous [∂µ (ψγ

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However the path-integral study (Eq. (B.19)) shows that the operator solution of Eq. (B.21) (in terms of free (normal-ordered) fields) is given by ψ(x) = exp[iγ5 φ(x)]χ(x),

(B.22)

since it is possible to evaluate exactly the anomalous divergence of the above-mentioned axial-vector current and, thus, choose a suitable model potential V (φ) which leads to the above simple solution. It is instructive to point out that the operator solution (Eq. (B.22)) coincides with the operator solution of the U (1) vectorial model analyzed by Schroer in Ref. 1 (the only difference between the model’s solutions being the γ5 factor in the phase of Eq. (B.22)). Consequently, we can follow Schroer’s analysis to conclude that the Euclidean correlation function (Eq. (B.2)) defines Wightman functions which are distributions over certain class of analytic test functions.23−25 But the model suffers the problem of the nonexistence of time-ordered Green’s functions which means that the proposed axial model in Minkowski spacetime does not satisfy the Einstein causality principle.23 Finally, we remark that the proposed model is to a certain extent less trivial than the vectorial model since for nondynamical φ(x) field the associated S matrix is nontrivial and is given in a regularized form by the result +∞ ¯ µ γ5 ∂µ φψ)(x)d4 x (ψγ S = T exp i

= exp −i

−∞

+∞

φ(x) −∞

δ Jε [φ] (x)d4 x , δφ(x)

(B.23)

with Jε [φ] expressed by Eq. (B.16). B.3. Complement We now briefly calculate the asymptotic term of the second-order positive differential elliptic operator: g γ5 ζ∂µ φ)∂µ − g¯ζγ5 ∂ 2 φ + (∂µ φ)2 . ( Dζ [φ])2 = (−∂ 2 ) − (¯

(B.24)

For this study, let us consider the more general second-order elliptic fourdimensional differential operator (non-necessarily) Hermitian in relation to the usual normal in L2 (RD ) (Ref. 22): L x = −(∂ 2 )x + aµ (x)(∂µ )x + V (x).

(B.25)

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∂ K(x, y.ζ) = −Lx K(x, y; ζ), ∂ζ lim K(x, y; ζ) = δ (D) (x − y).

η→0+

(B.26)

The Greeen’s function K(x, y, ζ) has the asymptotic expansion * ∞ + m ζ Hm (x, y) , (B.27) lim K(x, y; ζ) ∼ K0 (x, y; ζ) ζ→0+

m=0

where K0 (x, y, ζ) is the evolution kernel for the n-dimensional Laplacian (−∂ 2 ). By substituting Eq. (B.27) into Eq. (B.26) and taking the coinciding limit x → y, we obtain the following recurrence relation for the coefficients Hm (x, x): *∞ ∞ ∞ ζn 1 n ζn x ζ Hn+1 (x, x) = − (−∂x2 )Hn (x, x) + an (x) ∂ Hn (x, x) n! n! n! µ n=0 n=0 n=0 + ∞ ζn Hn (x, x) . + V (x) (B.28) n! n=0 For D = 4, Eq. (A.5) yields the Seeley coefficients H0 (x, x) = 14×4 , H2 (x) = −V (x), H2 (x, x) = 2[−∂x2 V (x) + aµ (x)∂µ V (x) + V 2 (x)].

(B.29)

Now substituting the value aµ (x) = [−ζ g¯∂µ φ]14×4 and V (x) = [(−¯ gζγ5 ∂x2 φ + ∂µ φ]14×4 into Eq. (A.6), we obtain the result (Eqs. (B.14) and (B.15)) quoted in the main text of this appendix.

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Chapter 7 Nonlinear Diffusion in RD and Hilbert Spaces: A Path-Integral Study∗

7.1. Introduction The deterministic nonlinear diffusion equation is one of the most important topics in mathematical physics of the nonlinear evolution equation theory.1–3 An important class of initial-value problem in turbulence has been modeled by nonlinear diffusion stirred by random sources.4 The purpose of this complementary chapter for Chap. 4, in mathematical methods for physics, is to provide a model of nonlinear diffusion were one can use and understand the compacity functional analytic arguments to produce the theorem of existence and uniqueness on weak solutions for deterministic stirring in L∞ ([0, T ] × L2 (Ω)). We use these results to give a first step “proof” for the famous Rosen path-integral representation for the Hopf characteristic functional associated to the white-noise stirred nonlinear diffusion model. These studies are presented in Sec. 7.2. In Sec. 7.3, we present a study of a linear diffusion equation in a Hilbert space, which is the basis of the famous loop wave equations in string and polymer surface theory.3,4

7.2. The Nonlinear Diffusion Let us start our chapter by considering the following nonlinear diffusion ¯ denoting a C ∞ -compact domain equation in some strip Ω × [0, T ] with Ω D of R . ∂U (x, t) = (+∆U )(x, t) + ∆(∧) F (U (x, t)) + f (x, t), ∂t ∗ Author’s


(7.1)

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with initial and Dirichlet boundary conditions as U (x, 0) = g(x) ∈ L2 (Ω), U (x, t) |∂Ω ≡ 0

(for t > 0).

(7.2) (7.3)

We note that the nonlinearity of the diffusion-spatial term of the parabolic problem (Eq. (7.1)) takes into account the physical properties of nonlinear porous medium’s diffusion saturation physical situation where this model is supposed to be applied1 — by means of the hypothesis that the regularized Laplacean operator ∆(∧) in the nonlinear term of the governing diffusion equation (7.1) has a cut-off in its spectral range. Additionally, we make the hypothesis that the nonlinear function F (x) is a bounded real continuously differentiable function on the extended interval (−∞, ∞) with its derivative F (x) strictly positive there. The external source f (x, t) is supposed to belong to the space L∞ ([0, T ] × L2 (Ω)) or to be a white-noise external stirring of the form (Ref. 2, p. 61) when in the random case d d (7.4) βn (t)ϕn (·) = w(t). f (·, t) = dt dt n∈Z

Here, {ϕn } denotes a complete orthonormal set on L2 (Ω), and βn (t), n ∈ Z are independent Wiener processes. Let us show the existence and uniqueness of weak solutions for the above diffusion problem stated by means of Galerking method for the case of deterministic f (x, t) ∈ L∞ ([0, T ] × L2 (Ω)). Let {ϕn (x)} be spectral eigen-functions associated to the Laplacean ∆. Note that each ϕn (x) ∈ H 2 (Ω) ∩ H01 (Ω).3 We introduce now the (finitedimensional) Galerkin approximants U (n) (x, t) =

n

(n)

Ui (t)ϕi (x),

i=1

f (n) (x, t) =

n

(f (x, t), ϕi (x))L2 (Ω) ϕi (x),

(7.5)

i=1

subject to the initial-conditions U (n) (x, 0) =

n

(g(x), ϕi )L2 (Ω) ϕi (x),

i=1

here (, )L2 (Ω) denotes the usual inner product on L2 (Ω).

(7.6)

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After substituting Eqs. (7.5) and (7.6) in Eq. (7.1), one gets the weak form of the nonlinear diffusion equation in the finite-dimension approximation as a mathematical well-defined systems of ordinary nonlinear differential equations, as a result of an application of the Peano existence-solution theorem. ∂U (n) (x, t) , ϕj (x) + (−∆U (n) (x, t), ϕj (x))L2 (Ω) ∂t 2 L (Ω) = (∇(∧) · [(F (U (n) (x, t))∇(∧) U (n) (x, t)], ϕj (x))L2 (Ω) + (f (n) (x, t), ϕj (x))L2 (Ω) .

(7.7)

By multiplying the associated system (Eq.(7.7)) by U (n) we get the diffusion equation in the finite-dimensional Galerking sub-space in the integral form: 1 d U (n) 2L2 (Ω) + (−∆U (n) , U (n) )L2 (Ω) 2 dt + d3 x(F (U (n) )(∇(∧) U (n) · ∇(∧) U (n) )(x, t) = (f, U (n) )L2 (Ω) . (7.8) Ω

This result in turn yields a prior estimate for any positive integer p: 1 d 1 (U (n) 2L2 (Ω) ) + γ(Ω)U (n) 2L2 (Ω) + (F (U (n) )) 2 (∇U (n) )2L2 (Ω) 2 dt 1 1 (n) 2 2 ≤ (7.9) pf (x, t)L2 (Ω) + U L2 (Ω) . 2 p Here, γ(Ω) is the Garding–Poincaré constant on the inequality of the quadratic form associated to the Laplacean operator defined on the domain H 2 (Ω) ∩ H01 (Ω). U (n) 2H 1 (Ω) = (−∆U (n) , U (n) )L2 (Ω) ≥ γ(Ω)U (n) ||2L2 (Ω) .

(7.10)

By choosing the integer p big enough and applying the Gronwall lemma, we obtain that the set of function {U (n) (x, t)} forms a bounded set in L∞ ([0, T ], L2 (Ω)) ∩ L∞ ([0, T ], H01 (Ω)) and in L2 ([0, T ], L2 (Ω)). As a consequence of this boundedness property of the function set {U (n) }, ¯ (t, x) ∈ there is a sub-sequence weak-star convergent to a function U ∞ 2 L ([0, T ], L (Ω)), which is the candidate for our “weak” solution of Eq. (7.1).

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Another important estimate is to consider again Eq. (7.9), but now considering the Sobolev space H01 (Ω) on this estimate (Eq. (7.9)), namely, T 1 (n) 2 (n) 2 ¯ (U (T )L2 (Ω) − U (0) ) + C0 dt||U (n) ||2H 1 (Ω) 0 2 0

T T 1 1 2 (n) 2 ¯ < ∞, ≤ p ||f ||L2 (Ω) dt + U L2 (Ω) dt < M 2 2p 0 0 (7.11) since we have the coerciveness condition for the Laplacean operator (−∆U (n) , U (n) )L2 (Ω) ≥ C¯0 (U (n) , U (n) )H01 (Ω) .

(7.12)

Note that U (n) (0)2 ≤ 2g(x)2L2 (Ω) (see Eq. (7.8)), and {U (n) (T )2L2 (Ω) } is a bounded set of real positive numbers. As a consequence of a prior estimate of Eq. (7.11), one obtains that the previous sequence of functions {U (n) } ∈ L∞ ([0, T ], H01(Ω) ∩ H 2 (Ω)) forms a bounded set on the vector-valued Hilbert space L2 ([0, T ], H01 (Ω)) either. Finally, one still has another a prior estimate after multiplying the Galerkin system (Eq. (7.7)) by the time-derivatives U˙ (n) , namely T dUn (t) 2 dt ≤ Real(∆Un (T ), Un (T )) dt L2 (Ω) 0 T dUn (∧) dt ∆ F (Un (t)), − (Un (0), Un (0)) + dt L2 (Ω) 0

2 T T dUn 2 1 1 (∧) ≤ p dt . ∆ F (Un (t)) 2 dt + dt 2 2 2p L (Ω) 0 0 L (Ω) (7.13) By noting that 0

T

(∧)

∆

×

0

F (Un (t))2L2 (Ω) dt

T

≤

∆(∧) 2op

dtUn (t)2L2 (Ω) < ∞,

sup

{F (x)}

2

x ∈ [−∞, ∞] (7.14)

n one obtains as a further result that the set of the derivatives { dU dt } is 2 2 2 −1 bounded in L ([0, T ], L (Ω)) (so in L ([0, T ], H (Ω)).

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At this point, we apply the famous Aubin–Lion theorem3 to obtain the strong convergence on L2 (Ω) of the set of the Galerkin approximants ¯ (x, t), since this set is a compact set in {Un (x, t)} to our candidate U 2 2 L ([0, T ], L (Ω)) (see Appendix A). By collecting all the above results we are lead to the strong convergence of the L2 (Ω)-sequence of functions F (Un (x, t)) to the L2 (Ω) function ¯ (x, t)). F (U We now assemble the above-obtained rigorous mathematical results to ¯ (x, t) as a weak solution of Eq. (7.1) for any test function v(x, t) ∈ obtain U ∞ C0 ([0, T ]), H 2 (Ω) ∩ H01 (Ω))

T dv (n) (n) (n) (∧) lim dt U , − + (−∆U , v)L2 (Ω) (F (U ), −∆ v)L2 (Ω) n→∞ 0 dt L2 (Ω) T = lim dt(f (n) , v), (7.15) n→∞

0

or in the weak-generalized sense mentioned above T ¯ (x, t), − dv(x, t) ¯ (x, t), (−∆v)(x, t))L2 (Ω) dt U + (U dt 2 0 L (Ω) ¯ (x, t), −(∆(∧) v(x, t))L2 (Ω) + (F (U T = dt(f (x, t), v(x, t))L2 (Ω) ,

(7.16)

0

since v(0, x) = v(T, x) ≡ 0 by our proposed space of time-dependent test functions as C0∞ ([0, t], H 2 (Ω) ∩ H01 (Ω)), suitable to be used on the Rosens path-integrals representations for stochastic systems (see Eqs. (7.22a) and (7.22b) in what follows). ¯ (x, t), comes from the following The uniqueness of our solution U 4 lemma. ¯(1) and U ¯(2) in L∞ ([0, T ] × L2 (Ω)) are two functions Lemma 7.1. If U satisfying the weak relationship below

T 0

∂v ¯ ¯ ¯(1) − U ¯(2) , +∆v)L2 (Ω) dt U(1) − U(2) , − + (U ∂t L2 (Ω) ¯(1) − F (U ¯(2) ); + ∆v)L2 (Ω) ≡ 0, × (F (U (7.17)

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¯(1) = U ¯(2) a.e in L∞ ([0, T ] × L2(Ω)). The proof of Eq. (7.17) is easily then U obtained by considering the family of test functions on Eq. (7.16) of the following form vn (x, t) = g(ε) (t)e+αn t ϕn (x) with −∆ϕn (x) = αn ϕn (x) and g(t) = 1 for (ε, T − ε) with ε > 0 arbitrary. We can see that it reduces to the obvious identity (αn > 0). T −ε ¯(2) ), ϕn )L2 (Ω) ≡ 0, dt exp(αn t)(F (U¯(1) ) − F (U (7.18) ε

¯(1) ) = F (U ¯(2) ) a.e on (0, T )×Ω since ε is an arbitrary which means that F (U ¯(2) a.e, as F (x) satisfies the lower bound ¯ number. We have thus U(1) = U estimate by our hypothesis on the kind of nonlinearity considered in our nonlinear diffusion equation (7.1).

inf(F (x)) |F (x) − F (y)| ≥ |x − y|. (7.19) −∞ < x < +∞ Let us now consider a path-integral solution of Eq. (7.1) (with g(x) = 0) for f (x, t) denoting the white-noise stirring4 E(f (x, t)f (x , t )) = λδ (D) (x − x )δ(t − t ),

(7.20)

where λ is the noise strength. The first step is to write the generating process stochastic functional through the Rosen–Feynman path-integral identities4 T Z[J(x, t)] = Ef exp i dt dD xU (x, t, [f ])J(x, t) , (7.21a) 0

F

Z[J(x, t)] = Ef

D [U ]δ

× exp i

Ω

(F )

(∂t U − ∆U − ∆

(F (U )) − f )

T

D

dt

d xU (x, t)J(x, t) ,

0

Z[J(x, t)] = Ef

(∧)

(7.21b)

Ω

DF [U ]DF [λ] exp i 0

T

dD xλ(x, t)

dt Ω

(∧) × (∂t U − ∆U − ∆ (F (U ) − f )) × exp i

0

T

D

dt

d xU (x, t)J(x, t) , Ω

(7.21c)

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Z[J(x, t)] =

DF [U ] exp

−

1 2λ

215

T

dD x

dt 0

ch07

Ω

× (∂t U − ∆U − ∆(∧) (F (U ))2 (x, t)) × exp i

0

T

dD xU (x, t)J(x, t) .

dt

(7.21d)

Ω

The important step made rigorous mathematically possible on the above (still formal) Rosen’s path-integral representation by our previous rigorous mathematical analysis is the use of the delta functional identity on Eq. (7.21b) which is true only in the case of the existence and uniqueness of the solution of the diffusion equation in the weak sense at least for multiplier Lagrange fields λ(x, t) ∈ C0∞ ([0, T ], H 2 (Ω) ∩ H01 (Ω)). As an important mathematical result to be pointed out is that in general case of a nonporous medium4 in R3 , where one should model the diffusion nonlinearity by a complete Laplacean ∆F (U (x, t)), one should observe that ¯ (∧) (x, t)} of Eq. (7.1) still remains a bounded the set of (cut-off) solutions {U ∞ 2 set on L ([0, T ], L (Ω)). Since we have the a priori estimate uniform bound for the U (n) -derivatives below in D = 3 (with G (x) = F (x)). Namely

T T dU (n) 2 dU (n) (t) 3 (n) ≤ dt dt d x(∆F (U (t))) · 0 dt L2 (Ω) 0 dt Ω T (n) d (x, t)) d(U 3 3 (n) ≤ dt Real + d xf (x, t) d x∆G(U (t) dt dt 0 Ω Ω 1 1 · (n) + sup pf (x, t)2L2 (Ω) + U 2L2 (Ω) 2 0≤t≤T p 3 (n) (n) ≤ Real d x(∆G(U (T, x)) − ∆G(U (0, x)) Ω · (n) 1 1 sup pf (x, t)2L2 (Ω) + U L2 (Ω) + 2 0≤t≤T p 1 1 · (n) pf 2L∞ ((0,t),L2 (Ω)) + U (t)L∞ ((0,t),L2 (Ω)) < ∞, (7.22) 2 2p where U (n) (T, x) = U (n) (0, x) = 0 (see Eq. (7.3)). The uniform ≤

∂Ω

∂Ω

bound for the derivatives is achieved by choosing

1 2p

< 1.

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As another point worth to call to attention is that the above considered function space is the dual of the Banach space L1 ([0, T ], L2 (Ω)). So, one ¯ (∞) (x, t), can extract from the above set of cut-off solutions a candidate U ∞ 2 in the weak-star topology of L ([0, T ], L (Ω)) for the above cited case of cut-off removing ∧ = +∞.6 However, we will not proceed thoroughly in this straightforward technical question of cut-off removing our model of nonlinear diffusion in this chapter for general spaces RD . Finally, we remark that in the one-dimensional case Ω ∈ R1 , one can further show by using the same compacity methods the existence and uniqueness of the diffusion equation added with the hydrodynamic ∂ (U (x, t))2 , which turns the diffusion equation (7.1) as a advective term 12 dx kind of nonlinear Burger equation on a porous medium. It appears very important to remark that Galerking methods applied directly to the finite-dimensional stochastic equation (7.7) (see Eq. (7.4)) may be saving-time computer simulation candidates for the “turbulent” path-integral (Eqs. (6.22a)–(6.22d)) evaluations by approximate numerical methods (Ref. 2). 7.3. The Linear Diffusion in the Space L2 (Ω) Let us now present some mathematical results for the diffusion problem in Hilbert spaces formed by square-integrable functions L2 (Ω),5 with the domain Ω denoting a compact set of RD . The diffusion equation in the infinite-dimensional space L2 (Ω) is given by the following functional differential equation (see Ref. 5 for mathematical notation). 1 ∂ψ[f (x); t] = TrL2 (Ω) ([QDf2 ψ[f (x, t)]) ∂t 2

(7.23a)

ψ[f (x), t → 0+ ] = Ω[f (x)].

(7.23b)

Here, ψ[f (x), ·] is a time-dependent functional to be determined through the governing equation (7.23) and belonging to the space L2 (L2 (Ω), dQ µ(f )) with dQ µ(f ) denoting the Gaussian measure on L2 (Ω) associated to Q — a fixed positive self-adjoint trace class operator 1 (L2 (Ω)) — and Df2 is the second — Frechet derivative of the functional ψ[f (x), t] which is given by a f (x)-dependent linear operator on L2 (Ω) with associated quadratic form (Df2 ψ[f (x), t] · g(x), h(x))L2 (Ω) .

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By considering explicitly the spectral base of the operator Q on L2 (Ω) Qϕn = λn ϕn ,

(7.24)

The L2 (Ω)-infinite-dimensional diffusion equation takes the usual form: Ψ fn ϕn , t = ψ (∞) [(fn ), t], (7.25a) n

Ω

fn ϕn = Ω(∞) [(fn )],

(7.25b)

n

∂ψ (∞) [(fn ), t] = [(λn ∆fn )ψ (∞) [(fn ), t]], ∂t n ψ (∞) [(fn ), 0] = Ω(∞) [(fn )], or in the Physicist’ functional derivative form (see Ref. 5). ∂ 1 δ2 D ψ[f (x), t] = ψ[f (x), t], d x dD x Q(x, x ) ∂t 2 Ω δf (x )δf (x) Ω ψ[f (x), 0] = Ω[f (x)].

(7.25c) (7.25d)

(7.26a) (7.26b)

Here, the integral operator Kernel of the trace class operator is explicitly given by Q(x, x ) = (λn ϕn (x)ϕn (x )). (7.26c) n

A solution of Eq. (7.26a) is easily written in terms of Gaussian pathintegrals5 which reads in the physicist’s notations

1 1 −1 F 2 Q D [g(x)]Ω[f (x) + g(x)] × det ψ[f (x), t] = t L2 (Ω) 1 D D −1 × exp − d x d x g(x) · Q (x, x )g(x ) . (7.27) 2t Ω Ω Rigorously, the correct functional measure on Eq. (7.27) is the normalized Gaussian measure with the following generating functional dtQ µ[g(x)] exp i j(x)g(x)dD x Z[j(x)] = L2 (Ω)

= exp −

t 2

dD x Ω

Ω

Ω

dD x j(x)Q+1 (x, x )j(x ) .

(7.28)

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At this point, it becomes important to remark that when writing the solution as a Gaussian path-integral average as done in Eq. (7.27), all the L2 (Ω) functions in the functional domain of our diffusion functional field ψ[f (x), t] belongs to the functional domain of the quadratic form associated to the class trace operator Q the so-called reproducing kernel of the operator Q which is not the whole Hilbert space L2 (Ω) as naively indicated on Eq. (7.27), but the following subset of it: 1

⊂

Dom(ψ[·, t]) = {f (x) ∈ L2 (Ω)|Q− 2 f ∈ L2 (Ω)} = L2 (Ω).

(7.29)

The above result gives a new generalization of the famous Cameron– Martin theorem that the usual Wienner measure (defined by the onedimensional Laplacean with Dirichlet conditions on the interval end-points) Wien

+g] , if is translation invariant, i.e. dWien µ[f + g] = dWien µ[f ] × d dWienµ[f µ[f ] and only if the shift function g(x) is absolutely continuous with derivative d2 2 1 on L ([a, b]). In other words, g ∈ H0 ([a, b]) = Dom − d2 x . Another important point to call to the readers’ attention is that one can write Eq. (7.27) in the usual form of diffusion in finite-dimensional case (see Appendix B):

√ 1 1 F 2 Q ψ[f (x), t] = D [g(x)]Ω[f (x) + tg(x)] × det t L2 (Ω) 1 D D −1 × exp − d x d x g(x)Q (x, x )g(x) , (7.30) 2 Ω Ω

At this point it is worth to call to the readers’ attention that dtQ µ and dQ µ Gaussian measures are singular to each other by a direct application of Kakutani theorem for Gaussian infinite-dimensional measures for any time t > 0: dtQ µ[g(x)]/dQ µ[g(x)] = +∞.

(7.31)

Let us apply the above results for the physical diffusion of polymer

rings (closed strings) described by periodic loops X(σ) ∈ RD , 0 ≤ σ ≤

A,AX(σ+A) A = X(σ) with a nonlocal diffusion coefficient Q(σ, σ ) (such that dσ 0 dσ Q(σ, σ ) = Tr[Q] < ∞). The functional governing equation in 0 loop space (formed by polymer rings) is given by A A

A] ∂ψ (ε) [X(σ); δ2 (ε)

= ψ (ε) [X(σ), dσ dσ Qij (σ, σ ) A],

i (σ)δ X

j (σ) ∂A δX 0 0 (7.32a)

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ψ

(ε)

219

A λ A

[X(σ); 0] = exp − dσ dσ Xi (σ)Mij (σ, σ )Xj (σ ) . 2 0 0 (7.32b)

A] Here, the ring polymer surface probability distribution ψ (ε) [X(σ), depends on the area parameter A, the area of the cylindrical polymer surface of our surface-polymer chain. Note that the presence of a parameter ε on the above objects takes into account the local (the integral operator kernel) case Q(σ, σ ) = δ(σ − σ ) as a limiting case of the rigorously mathematical well-defined (class trace) situation on the end of the observable evaluations 1 (σ − σ )2 (ε) . (7.33) Qij (σ, σ ) = exp − πε ε2 The solution of Eq. (7.32a) is straightforwardly written in the case of a self-adjoint kernel M on L2 ([0, A]): A λ A

i (σ)Mij (σ, σ )X

j (σ ) det− 12 [1 + AλM (Q(ε) )] dσ dσ X exp − 2 0 0 −1 A λ2 A (ε) −1 1

× exp + dσ dσ (M X)i (σ) (λM + (Q ) · (M X)j (σ ) . 2 0 A 0 (7.34) The functional determinant can be reduced to the evaluation of an integral equation 1

det− 2 [1 + AλM (Q(ε) )] 1 (ε) 2 = exp − TrL (Ω) lg(1 + λAM (Q ) 2 λ 1 (ε) (ε) −1 = exp − TrL2 (Ω) dλ [(Q )M )(1 + λ A(Q )M ) ] 2 0 λ 1 = exp − TrL2 (Ω) dλ R(λ ) . (7.35) 2 0 Here, the kernel operator R(λ ) satisfies the integral equation (accessible for numerical analysis) R(λ )(1 + λ A(Q(ε) )M = (Q(ε) )M.

(7.36)

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Which in the local case of ε → 0+ , when considered in the final result (Eqs. (7.34) and (7.35)), produces the explicitly candidate solutions for our polymer-surface probability distribution with M a class trace operator on the loop space, L2 ([0, A]): λ −1 A (ε) −1

dλ [M (Q + λ AM ) ] ψ[X(σ), A] = exp − TrL2 (Ω) 2 0 A λ A

j (σ ) × exp − dσ dσ Xi (σ) · Mij (σ, σ )X 2 0 0 A λ2 A × exp + dσ dσ (M X)i (σ) 2 0 0 −1 (ε) −1 1

× λM + (Q ) · (σ, σ )(M X)j (σ ) . A (7.37) It is worth to call to the readers’ attention that if A ∈ 1 and B is a bounded operator — so A · B is a class trace operator. The functional determinant det[1 + AB] is a well-defined object as a direct result of the obvious estimate, the result which was used to arrive at Eq. (7.37). N

N (1 + λn ) ≤ exp λn = exp (Tr AB). lim N →∞

n=0

n=0

As a last comment on the linear infinite-dimensional diffusion problem (Eq. (7.23)), let us sketch a (rigorous) proof that Eq. (7.27) is the unique solution of Eq.(7.23). First, let us consider the initial .. 2 condition on Eq. (7.23) as belonging to the space of all mappings G.L (Ω) → R that are twice Fréchet differentiable on L2 (Ω) with uniformly continuous and bounded ¯ second derivative Df2 G (a bounded operator of L(L2 (Ω)) with norm C). 2 2 This set of mappings will be denoted by uC [L (Ω), R]. It is, thus, straightforward to see through an application of the mean-value theorem that the following estimate holds true |ψ[f (x), t] − G[f (x)]|

sup 1

f (x)∈Q 2 L2 (Ω)

≤

L2 (Ω)

|G(f (x) + g(x)) − G(g(x))|dtQ µ[g(x)]

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≤

L2 (Ω)

DG(f (x), g(x))L2 (Ω) +

0

≤ C¯

dσ(1 − σ)(D2 G[f (x)

+ σg(x)]g(x), g(x)L2 (Ω) dtQ µ[g(x)]

≤ 0 + C¯

1

221

0

1

dσ(1 − σ)

L2 (Ω)

L2 (Ω)

g(x)2L2 (Ω) dtQ µ[g(x)]

g(x)2L2 (Ω) dtQ µ[g(x)]

≤ C¯ Tr(tQ) = (C¯ Tr(Q)t → 0

as t → 0+ .

(7.38)

We have thus defined a strongly continuous semi-group on the Banach space U C 2 [L2 (Ω), R] with infinitesimal generator given by the infinite1 dimensional Laplacean Tr[QD2 ] acting on the space L2 (Q 2 (L2 (Ω)), R). By the general theory of semi-groups on Banach spaces, we obtain that Eq. (7.27) satisfies the infinite-dimensional diffusion initial-value problem Eq. (7.23), at least for initial conditions on the space uC 2 [L2 (Ω), R]. Since purely Gaussian functionals belong to uC 2 [L2 (Ω), R] and they form a dense set on the space L2 (L2 (Ω), dQ µ), we get the proof of our result for general initial condition on L2 (L2 (Ω), dQ µ). Finally, we point out that the general solution of the diffusion problem on Hilbert space with sources and sinks, namely 1 ∂ ψ[f (x), t] = TrL2 (Ω) [QDf2 ψ[f (x), t]] − V [f (x)]ψ[f (x), t] dt 2

(7.39)

with ψ[f (x), t → 0+ ] = Ω[f (x)],

(7.40)

possesses a generalized Feynman–Wiener–Kac Hilbert L2 (Ω) space-valued path-integral representation, which in the Feynman physicist formal notation reads as DF [X(σ)] ψ[h(x), T ] = C([0,T ],L2 (Ω))

1 × exp − 2

T

dσ 0

dX −1 dX ,Q dσ dσ

(σ)

L2 (Ω)

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×Ω

T

X(σ)dσ 0

× exp −

0

+ X(0)

T

dσV

T

X(σ )dσ

+ X(0)

,

(7.41)

0

where the paths satisfy the end-point constraint X(T ) = h(x) ∈ L2 (Ω); X(0) = f (x) ∈ L2 (Ω).

References 1. I. Sneddon, Fourier Transforms (McGraw-Hill Book Company, INC, USA, 1951); P. A. Domenico and F. W. Schwartz, Physical and Chemical Hydrogeology, 2nd edn. (Wiley, 1998); P. L. Schdev, Nonlinear Diffusive Waves (Cambridge University Press, 1987); W. E. Kohler; B. S. White (ed.), Mathematics of Random Media, Lectures in Applied Mathematics, Vol. 27 (American Math. Soc., Providence, Rhode Island, 1980); R. Z. Sagdeev (ed.), Non-Linear and Turbulent Processes in Physics, Vols. 1–3 (Harwood Academic Publishers, 1987); Geoffrey Grimmett, Percolation (Springer-Verlag, 1989). 2. G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional System, London Math. Soc. Lect. Notes, Vol. 229 (Cambridge University Press, 1996); R. Teman, C. Foias, O. Manley and Y. Treve, Physica 6D, 157 (1983). 3. R. Dortray and J.-L. Lion, Analyse Mathématique et Calcul Numérique, Masson, Paris, Vols. 1–9, A. Friedman, Variational Principles and FreeBoundary Problems — Pure and Applied, Math. (Wiley, 1988). 4. L. C. L. Botelho, J. Phys. A: Math. Gen. 34, L31–L37 (2001); L. C. L. Botelho, II Nuovo Cimento 117 (B1), 15 (2002); J. Math. Phys. 42, 1682 (2001); L. C. L. Botelho, Int. J. Mod. Phys. B 13 (13), 1663 (1999); Int. J. Mod. Phys. 12; Mod. Phys. B 14(3), 331 (2002); A. S. Monin and A. M. Yaglon, Statistical Fluid Mechanics, Vol. 2 (Mit Press, Cambridge, 1971); G. Rosen, J. Math. Phys. 12, 812 (1971); L. C. L. Botelho, Mod. Phys. Lett 13B, 317 (1999); V. Gurarie and A. Migdal, Phys. Rev. E 54, 4908 (1996); U. Frisch, Turbulence (Cambridge Press, Cambridge, 1996); W. D. Mc-Comb, The Physics of Fluid Turbulence (Oxford University, Oxford, 1990); L. C. L. Botelho, Mod. Phys. Lett. B 13, 363 (1999); S. I. Denisov and W. Horsthemke, Phys. Lett. A 282 (6), 367 (2001); L. C. L. Botelho, Int. J. Mod. Phys. B 13, 1663 (1999); L. C. L. Botelho, Mod. Phys. Lett. B 12, 301 (1998); L. C. L. Botelho, Mod. Phys. Lett. B 12, 569 (1998); L. C. L. Botelho, Mod. Phys. Lett. B 12, 1191 (1998); L. C. L. Botelho, II Nuovo Cimento 117 B(1), 15 (2002); J. Math. Phys. 42, 1682 (2001). 5. G. da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, Vol. 293 (London Math. Soc. Lect. Notes, Cambridge, UK, 2002).

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6. J. Wloka, Partial Differential Equations (Cambridge University Press, 1987). 7. Soo Bong Chae, Holomorphy and Calculus in Normed Spaces, Pure and Applied Math. Series (Marcel Denner Inc, NY, USA, 1985).

Appendix A. The Aubin–Lion Theorem Just for completeness in this mathematical appendix for our mathematical oriented readers, we intend to give a detailed proof of the basic result on compacity of sets in functional spaces of the form L2 (Ω) and throughout used in Sec. 2. We have, thus, the Aubin–Lion theorem3 in the Gelfand triplet H01 (Ω) → L2 (Ω) → H −1 (Ω) = (H01 (Ω))∗ Aubin–Lion Theorem. If {Un (x, t)} is a sequence of time-differentiable functions in a bounded set of L2 ([0, T ], H01 (Ω)) such that its time derivatives forms a bounded set of L2 ([0, T ], H0−1(Ω)), we have that {Un (x, t)} is a compact set on L2 ([0, T ], L2 (Ω)). Proof. The basic fact we are going to use to give a mathematical proof of this theorem is the following identity (Ehrling’s lemma). For any given ε > 0, there is a constant C(ε) such that Un L2 (Ω) ≤ εUn H01 (Ω) + C(ε)Un 2H −1 (Ω) .

(A.1)

As a consequence, we have the following estimate T Un − Um 2L2 ([0,T ]),L2 (Ω) 0

≤

0

≤ε

2

T

dt(εUn − Um H01 (Ω) + C(ε)Un − Um H −1 (Ω) )2

T

dtUn −

0

≤ε

2

0

dtUn −

0

Um 2H −1 (Ω)

0

dtUn −

0

T

+ (C(ε)

T

+ 2εC(ε)

dt(Un − Um H01 (Ω) × Un − Um H −1 (Ω) )

0

2

T

+ 2εC(ε)

Um 2H 1 (Ω) 0

Um 2H 1 (Ω) 0

2

dtUn −

dtUn −

+ C(ε)

0

12

T

T

Um 2H 1 (Ω) 0

+ 0

Um 2H −1 (Ω)

12

T

dtUn −

Um 2H −1 (Ω)

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≤ 2ε2 M + 2εC(ε)M 2

1 2

0

dtUn − Um 2H −1 (Ω)

0

T

+ (C(ε) )

12

T

dtUn −

Um 2H −1 (Ω)

.

(A.2)

At this point, we use the Arzela–Ascoli theorem to see that {Un (x, t)} is a compact set on the space C([0, T ], H −1 (Ω)) since we have the set equicontinuity: Un (t) − Um (s)H −1 (Ω) ≤

s

t

Un (τ )H −1 (Ω) dτ 0

(1− 12 )

≤ |t − s|

T 0

12 Un (τ )2H −1 (Ω) dτ

1

¯ |t − s| 2 . ≤M

(A.3)

It is a crucial step now by remarking that H01 (Ω) is compactly immersed in L2 (Ω) (Rellich theorem). Let us note that for each t (almost everywhere in [0, T ]), Un (x, t) is a bounded set on H01 (Ω) since Un (x, t) belongs to a bounded set L2 ([0, T ], H01 (Ω)) by hypothesis. As a consequence, {Unk , (x, t)} is a compact set on L2 (Ω) (Rellich theorem) and so in H −1 (Ω) almost everywhere in [0, T ] since L2 (Ω) → H −1 (Ω). By an application of the Arzela–Ascoli theorem, there is a sub-sequence {Unk (x, t)} of {Un (x, t)} (and still denoted by {Un (x, t)} such that it converges uniformly to a given ¯ (x, t) ∈ C([0, T ], H −1(Ω)). As a direct result of this fact we have function U that (for T < ∞!) for (n, m) → ∞. 0

12

T

Un −

Um 2H −1 (Ω)

≤ (sup |Un − Um |C([0,T ],H −1 (Ω) )

0

12

T

1 · dt

→ 0.

(A.4)

Returning to our estimate (Eq. (A.2)), we see that this sub-sequence is a Cauchy sequence in L2 ([0, T ], L2(Ω)). As a consequence, for each fixed ¯ (x, t) in L2 (Ω). t ∈ [0, T ] (almost everywhere), Un (x, t) converges to U

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Appendix B. The Linear Diffusion Equation Let us show mathematically the basic functional integral representation (Eq. (7.30)) for the L2 (Ω)-space diffusion equation (7.23). As a first step for such proof, let us call the readers’ attention that one should consider the second-order (Laplacean) D2 U (x, t) as a bounded operator in L2 (Ω) in order to the operatorial composition with the positive definite class trace operator Q still be a class trace operator as it is explicitly supposed in the right-hand side of Eq. (7.23). We thus impose as the sub-space of initial condition the diffusion equation (7.23) for the (dense) vector sub-space of C(L2 (Ω), R) composed of all functionals of the form dQ µ(p)F (p) exp(i p, xL2 (Ω) ), (B.1) f (x) = L2 (Ω)

with F (p) ∈ L2 (L2 (Ω), dQ µ). By substituting the initial condition (Eq. (B.1)) into the integral representation equation (7.30) and by using the Fubbini–Toneli theorem to exchange the needed integrations order in the estimate below, we get: U (x, t) =

f (x + L2 (Ω)

=

L2

=

L2

√

tξ)dQ µ(ξ)

dQ µ(ξ)

L2

dQ µ(p)F (p)ei p,x+ 1

√ tξL2

dQ µ(p)F (p) · ei p,xL2 e− 2 t p,QpL2 .

(B.2)

Note that we have already proved that U (x, t) is a bounded functional of C(L2 (Ω) × [0, ∞]; R) on the basis of our hypothesis on the initial functional date (Eq. (B.1)). At this point, we observe that the second-order Frechet derivatives of the functional, exp i p, xL2 , are easily (explicitly) evaluated as7 QD

2

e

i p,xL2

=

∞ =1

∂2 λ 2 ∂ x

P∞ ei( n=1 pn xn ) = − ( p, QpL2 ) ei p,xL2 . (B.3)

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We have thus a straightforward proof of our claim cited above on the basis again of the chosen initial date sub-space Tr[QD2 U (x, t)] ≤ dQ µ(p)|F (p)| p, QpL2 L2 (Ω)

≤

2

L2 (Ω)

12

dQ µ(p)|F (p)|

12 2

L2 (Ω)

dQ µ(p)| p, QpL2 (Ω) |

≤ (Tr Q)2 F 2L2 (L2 (Ω),dQ µ) < ∞.

(B.4)

Now, it is a simple application to verify that Eq. (B.2) satisfies the diffusion equation in L2 (Ω) (or in any other separable Hilbert space). Namely ∂U (x, t) 1 = dQ µ(p)F (p)ei p,xL2 − p, QpL2 (Ω) ∂t 2 L2 (Ω) t

× e− 2 , p,QpL2 (Ω) ,

2

TrL2 (Ω) [QD U (x, t)] =

L2 (Ω)

=

L2 (Ω)

(B.5) t

dQ µ(p)F (p){D2 ei p,x }e− 2 p,QpL2 (Ω) t

dQ µ(p)F (p){− p, QpL2 }ei p,x e− 2 p,QpL2 (Ω) , (B.6)

with

U (x, 0) = L2 (Ω)

dQ µ(p)F (p)e

i p,x

lim e

t→0+

− 2t p,QpL2 (Ω)

= f (x). (B.7)

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Chapter 8 On the Ergodic Theorem∗

8.1. Introduction One of the most important phenomenon in numerical studies of the nonlinear wave motion, especially in the two-dimensional case, is the existence of overwhelming majority of wave motions that wandering over all possible system’s phase space and, given enough time, coming as close as desired (but not entirely coinciding) to any given initial condition.1 This phenomenon is signaling certainly that the famous recurrence theorem of Poincaré is true for infinite-dimensional continuum mechanical systems like that one represented by a bounded domain when subject to nonlinear vibrations. A fundamental question appears in the context of this recurrence phenomenon and concerned to the existence of the time average for the associated nonlinear wave motion and naturally leading to the concept of an infinite-dimensional invariant measure for the nonlinear wave equation,1 the mathematical phenomenon subjacente to the Poincaré recurrence theorem. In this chapter, we intend to give applied mathematical arguments for the validity of the famous Ergodic theorem in a class of polynomial nonlinear and Lipschitz wave motions. Our approach is fully based on Hilbert space methods previously used to study dynamical systems in classical mechanics which in turn simplifies enormously the task of constructing explicitly the associated invariant measure on certain Sobolev spaces — the infinite-dimensional space of wave motion initial conditions. This study is presented in the main section of this chapter, namely Sec. 8.4. In Sec. 8.2, we give a very detailed proof of the RAGE’s theorem, a basic functional analysis rigorous method used in our proposed Hilbert ∗ Author’s


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space generalization of the usual finite-dimensional Ergodic theorem and fully presented in Sec. 8.3. In Sec. 8.5, we complement our studies by analyzing the important case of wave-diffusion under random stirring. In Appendices A and B, we additionally present important technical details related to Sec. 8.4, and they contain material as important as those presented in the previous sections of our work. 8.2. On the Detailed Mathematical Proof of the RAGE Theorem In this purely mathematical section, we intend to present a detailed mathematical proof of the RAGE theorem2 used on the analytical proof of the Ergodic theorem on Sec. 8.3 for Hamiltonian systems of N -particles. Let us, thus, start our analysis by considering a self-adjoint operator L on a Hilbert space (H, (, )), where Hc (L) denotes the associated continuity sub-space obtained from the spectral theorem applied to L. We have the following result. Proposition (RAGE theorem). Let ψ ∈ Hc (L) and ψ˜ ∈ H. We have the Ergodic limit 1 lim T →∞ T

T

˜ exp(itL)ψ)|2 dt = 0. |(ψ,

(8.1)

0

Proof. In order to show the validity of the above ergodic limit, let us rewrite Eq. (8.1) in terms of the spectral resolution of L, namely I=

1 T

1 = T

T

˜ exp(itL)ψ)|2 dt |(ψ,

0 T

+∞

e 0

−∞

+itλ

˜ dλ (ψ, Ej (λ)ψ)

+∞

−∞

e

−itµ

˜ dµ (ψ, Ej (µ)ψ) . (8.2)

Here, we have used the usual spectral representation of L +∞ (g, Lh) = λdλ (g, Ej (λ)h), −∞

with g ∈ H and h ∈ Dom (L).

(8.3)

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Let us remark that the function exp(i(λ − µ)t) is majorized by the function 1 which in turn is an integrable function on the domain [0, T ] × R × R with the product measure dt ˜ Ej (λ)ψ) ⊗ dµ (ψ, ˜ Ej (µ)ψ), ⊗ dλ (ψ, T

(8.4)

dt ˜ Ej (λ)ψ) ⊗ dµ (ψ, ˜ Ej (µ)ψ) = (ψ, ˜ ψ)2 < ∞. ⊗ dλ (ψ, T

(8.5)

since 0

T

At this point, we can safely apply the Fubbini theorem to interchange the order of integration in relation to the t-variable which leads to the partial result below +∞ +∞ i(λ−µ)T e −1 ˜ Ej (λ)ψ ⊗ dµ ψ, ˜ Ej (µ)ψ. (8.6) I= dλ ψ, i(λ − µ)T −∞ −∞ Let us consider two cases. First, we take ψ˜ = ψ. In this case, we have the bound 2 sin (λ−µ) T i(λ−µ)T e 2 − 1 ≤ 1. (8.7) i(λ − µ)T ≤ (λ − µ)T If we restrict ourselves to the case of λ = µ, a direct application of the Lebesgue convergence theorem on the limit T → ∞ yields that I = 0. On the other hand, in the case of λ = µ, we intend now to show that the set µ = λ is a set of zero measure with R2 in respect to the measure dλ ψ, Ej (λ)ψ ⊗ dµ ψ, Ej (µ)ψ. This can be seen by considering the real function on R f (λ) = (ψ, Ej (λ)ψ).

(8.8)

We observe that this function is continuous and nondecreasing since ψ ∈ Hc (L) and range Ej (λ1 )C range Ej (λ2 ) for λ1 ≤ λ2 . Now, f (λ) is ¯ a uniform continuous function on the whole real line R, since for a given ˜ such that ε > 0, there is a constant λ ε ˜ < , (ψ, Ej (−∞, −λ)ψ) 2 ε ˜ ψ 2 − (ψ, Ej (−∞, λ)ψ) < . 2

(8.9) (8.10)

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˜ λ], ˜ and Additionally, f (λ) is uniform continuous on the closed interval [−λ, ε ˜ ˜ ∞), f (λ) cannot make variations greater than 2 on (−∞, −λ] and [+λ, besides being a monotonic function on R. These arguments show the uniform continuity of f (λ) on whole line R. Hence, we have that for a given ε > 0, there exists a δ > 0 such that |(ψ, Ej (λ )ψ) − (ψ, Ej (λ )ψ)| ≤

ε , ψ 2

(8.11)

for |λ − λ | ≤ δ.

(8.12)

In particular, for λ = λ + δ e λ = λ − δ (ψ, Ej (λ + δ)ψ) − (ψ, Ej (λ − δ)ψ) ≤

ε . ψ 2

(8.13)

As a consequence, we have the estimate λ+δ +∞ dλ (Ej (λ)ψ, ψ) dµ (ψ, Ej (µ)ψ) −∞

λ−δ

≤ ψ ((ψ, Ej (λ + δ)ψ) − (ψEj (λ − δ)ψ)) 2

≤ ψ 2

ε ≤ ε. ψ 2

(8.14)

Let us note that for each ε > 0, there exists a set Dδ = {(λ, µ) ∈ R2 ; |λ − µ| < δ} which contains the line λ = µ and has measure less than ε in relation to the measure dλ (Ej (λ)ψ, ψ) ⊗ dµ (Ej (µ)ψ, ψ) as a result of Eq. (8.14). This shows our claim that I = 0 in our special case. In the general case of ψ˜ = ψ, we remark that solely the orthogonal component on the continuity sub-space Hc (L) has a nonvanishing inner product with exp(itL)ψ. By using now the polarization formulae, we reduce this case to the first analyzed result of I = 0. At this point, we arrive at the complete RAGE theorem.2 RAGE Theorem. Let K be a compact operator on (H, (, )). We have thus the validity of the Ergodic limit 1 T lim K exp(itL)ψ 2H dt = 0, (8.15) T →∞ T 0 with ψ ∈ Hc (L).

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We leave the details of the proof of this result for the reader, since any compact operator is the norm operator limit of finite-dimension operators and one only needs to show that

N

1 T

cn (eitL ψ, en )gn dt = 0, (8.16) lim

T →∞ T 0

n=0

H

for cn constants and {en }, {gn } a finite set of vector of (H, (, )) .

8.3. On the Boltzmann Ergodic Theorem in Classical Mechanics as a Result of the RAGE Theorem One of the most important statement in physics is the famous zeroth law of thermodynamics: “any system approaches an equilibrium state.” In the classical mechanics frameworks, one begins with the formal elements of the theory. Namely, the phase-space R6N associated to a system of N -classical particles and the set of Hamilton equations p˙ i = −

∂H ; ∂qi

q˙i =

∂H , ∂pi

(8.17)

where H(q, p) is the energy function. The above cited thermodynamical equilibrium principle becomes the mathematical statement that for each compact support continuous functions Cc (R6N ), the famous ergodic limit should hold true.3

1 T f (q(t); p(t))dt = η(f ), (8.18) lim T →∞ T 0 where η(f ) is a linear functional on Cc (R6N ) given exactly by the Boltzmann statistical weight, and {q(t), p(t)} denotes the (global) solutions of the Hamilton equation (8.12). In this section, we point out a simple new mathematical argument of the fundamental equation (8.18) by means of Hilbert space techniques and the RAGEs theorem. Let us begin by introducing for each initial condition (q(0), p(0)), a function ωq0 ,p0 (t) ≡ (q(t), p(t)), here q(t), p(t) is the assumed global unique solution of Eq. (8.17) with prescribed initial conditions. Let Ut : L2 (R6N ) → L2 (R6N ) be the unitary operator defined by (Ut f )(q, p) = f (ωq0 p0 (t)). We have the following theorem (the Liouville’s theorem).5

(8.19)

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Theorem 8.1. Ut is a unitary one-parameter group whose infinitesimal ¯ where −iL is the essential self-adjoint operator acting on generator is −iL, ∞ 6N C0 (R ) defined by the Poisson bracket. 3N ∂f ∂H ∂f ∂H (Lf )(p, q) = {f, H}(q, p) = − (q, p). ∂qi ∂pi ∂pi ∂qi i=1

(8.20)

The basic result we are using to show the validity of the Ergodic limit equation (8.18) is the famous RAGEs theorem exposed in Sec. 8.1. 2 ¯ ¯ is the continuity (R6N ), here Hc (−iL) Theorem 8.2. Let φ ∈ Hc (−iL)(L ¯ sub-space associated to self-adjoint operator −iL. For every vector β ∈ L2 (R6N ), we have the result

1 lim T →∞ T

T

|β, Ut ψ|2 dt = 0,

(8.21)

0

or equivalently for every ψ ∈ L2 (R6N ) 1 lim T →∞ T

0

T

β, Ut ψdt = β, Pker(−iL) ¯ ψ,

(8.22)

is the projection operator on the (closed) sub-space where Pker(−iL) ¯ ¯ ker(−iL). That Eq. (8.22) is equivalent to Eq. (8.21), is a simple consequence of the Schwartz inequality written below

0

T

(β, Ut ψ)dt ≤

T

(β, Ut ψ)2 dt

12

0

12

T

1dt

,

(8.23)

0

or T T 12 1 1 2 β, Ut ψdt ≤ (β, Ut ψ) dt T T 0 0

(8.24)

As a consequence of Eq. (8.23), we can see that the linear functional η(f ) of the Ergodic theorem is exactly given by (just consider β(p, q) ≡ 1 on supp of Pker(−iL) ¯ (ψ)) η(f ) = Pker(−iL) ¯ (f )(q, p).

(8.25)

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By the Riesz’s theorem applied to η(f ), we can rewrite (represent) Eq. (8.25) by means of a (kernel)-function hη(H) (q, p), namely η(f ) = R6N

d3N qd3N p(f (p, q))hη(H) (p, q),

(8.26)

where the function hη(H) (q, p) satisfies the relationship 3N ∂hη ∂H ∂H ∂hη − = 0, {hη(H) , L} = ∂qi ∂pi ∂qi ∂pi i=1

(8.27)

or equivalently hη(H) (q, p) is a “smooth” function of the Hamiltonian function H(q, p), by imposing the additive Boltzmann behavior for hη(H) (q, p) namely, hη(H1 +H2 ) = hη(H1 ) · hη(H2 ) , one obtains the famous Boltzmann weight as the (unique) mathematical output associated to the Ergodic theorem on classical statistical mechanics in the presence of a thermal reservoir.4 hη(H) (q, p) =

exp{−βH(q, p)} d3N qd3N p exp{−βH(q, p)}

,

(8.28)

with β a (positive) constant which is identified with the inverse macroscopic temperature of the combined system after evaluating the system internal energy in the equilibrium state. Note that hη(H) L2 = 1 since η(f ) = 1. A last remark should be made related to Eq. (8.28). In order to obtain this result one should consider the nonzero value in ergodic limit 1 T →∞ T

lim

0

T

dt(Ut ψ)(q, p)hη(H) (q, p) = hη(H) , Pker(−iL) ¯ (ψ),

(8.29)

or by a point-wise argument (for every t) (U−t hη(H) ) ∈ Pker(−iL) ¯ ,

(8.30)

hη(H) ∈ Pker(−iL) ¯ ⇔ Lh = 0.

(8.31)

that is

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8.4. On the Invariant Ergodic Functional Measure for Some Nonlinear Wave Equations Let us start this section by considering the discreticized (N -particle) wave motion Hamiltonian associated to a nonlinear polynomial wave equation related to the vibration of a one-dimensional string of length 2a H(pi , qi ) =

2 N 2a p

1 2 2k + (qi+1 − qi ) + g(qi ) , 2 2 i

N

l=1

(8.32)

with the initial one boundary Dirichlet conditions qi (0) = qi , pi (0) = pi , qi (−a) = qi (a) = 0.

(8.33)

Here k denotes a positive integer associated to the nonlinearity power and g > 0 the nonlinearity positive coupling constant. Since one can argue that the above Hamiltonian dynamical system possess unique global solutions on the time interval t ∈ [0, ∞) one can, thus, straightforwardly write the associated invariant (Ergodic) measure on the basis of Ergodic theorem exposed in Sec. 8.2 restricted to the configuration space after integrating out the term involving the canonical momenta 1 lim T →∞ T

T

dtF (q1 (t), . . . , qN (t)) = 0

RN

F (q1 , . . . , qN )d(inv) µ(q1 , . . . , qN )

(8.34) here the explicit expression for the Hamiltonian system invariant measure in configuration space is given by [with β = 1] (inv)

di

µ(q1 , . . . , qN )

=

1 (0)

Zn

1 exp − 2 i=1 N

2a N

(qi+1 − q1 )2 a2

× (dq1 · · · dqN ),

g (qi )2k + 2k

(8.35a)

Zn(0) =

RN

d(inv) µ(q1 , . . . , qN ).

(8.35b)

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Now it is a nontrivial result on the theory of integration on infinitedimensional spaces that the cylindrical measures (normalized to unity !) converges in a weak (star) topology sense to the well-defined nonlinear N polynomial Wiener measure at the continuum limit N → ∞, or l=1 () → L −L dx() and qi (t) → U (x, t) (see Appendix B for the relevant mathematical arguments supporting the mathematical existence of such limit). As a consequence of Eqs. (8.35) at the continuum limit, we have the infinite-dimensional analogous of Ergodic result for the nonlinear polynomial wave equation (with the Boltzmann constant equal to unity — see Eq. (8.28) — Sec. 8.3 of this study) 1 T →∞ T

T

dWiener µ[U (σ)] × F (U (σ))

F (U (x, t))dt =

lim

0

C α (−a,a)

g × exp − 2k

a 2k

dx(U (σ))

,

(8.36)

−a

where the wave field U (x, t) ∈ L∞ ([0, ∞), L2k−1 ((−a, a))) satisfies in a generalized weak sense the nonlinear wave equation below (see Appendix A): ∂ 2 U (x, t) ∂ 2 U (x, t) = + g(U (x, t))2k−1 , 2 ∂ t ∂2x

(8.37)

U (x, 0) = U (0) (x) ∈ H 1 ((−a, a)),

(8.38)

Ut (x, 0) = 0,

(8.39)

U (−a, t) = U (a, t) = 0,

(8.40)

dWiener µ[U (0) (x)] denotes the Wiener measure over the unidimensional Brownian field end-points trajectories {U (0) (x)| with U (0) (−a) = U (0) (a) = 0}, and F (U (0) (x)) denotes any Wiener-integrable functional. Note that C α (−a, a) denotes the Banach space of the α-Holder continuous function (for α ≤ 12 ) which is obviously the support of the Wiener measure dWienerµ[U (0) (x)]. It is worth to remark the normalization factor Z of the nonlinear Wiener measure implicitly used on Eq. (8.36). a g 1 dWiener µ[U (σ)] exp − dx(U (σ))2k = 1. (8.41) Z C α (−a,a) 2k −a Let us pass to the important case of existence of a dissipation term on the wave equation problem Eqs. (8.37)–(8.40) with ν the positive viscosity

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parameter and leading straightforwardly to solutions on the whole time propagation interval [0, ∞): ∂ 2 U (x, t) ∂U (x, t) ∂ 2 U (x, t) = −ν + g(U (x, t))2k−1 , 2 ∂ t ∂2x ∂t

(8.42)

U (x, 0) = U (0) (x) ∈ H 1 ((−a, a)),

(8.43)

Ut (x, 0) = U (1) (x) ∈ L2 ((−a, a)).

(8.44)

It is a simple observation that there is a bijective correspondence between the solution of Eq. (8.42) with the Klein–Gordon like wave equation ν for the rescaled wave field β(x, t) = e 2 t U (x, t), namely 2 (2k−1)νt ν ∂ 2 β(x, t) ∂ 2 β(x, t) − = β(x, t) + ge− 2 (β(x, t))2k−1 , (8.45) ∂ 2t ∂2x 4 β(x, 0) = U (0) (x) = H 1 ((−a, a)), ν βt (x, 0) = U (1) (x) − U (0) (x) = 0, 2 β(−a, t) = β(a, t) = 0.

(8.46) (8.47) (8.48)

As a result we have the analogous of the Ergodic theorem applied to Eq. (8.45) and a sort of Caldirola–Kanai action is obtained,5 a new result of ours in this nonlinear dissipative case (with the path-integral identification x → σ; U (0) (x) = X(σ)) 1 lim T →∞ T

T

dtF (e

+ ν2 t

U (x, t)) =

dWienerµ[X(σ)]F (X(σ))

1

C 2 (−a,a)

0

a g − (2k−1)νσ 2k 2 × exp − dσe × (X(σ)) 2k −a 2 a ν 2 × exp dσ(X(σ)) . (8.49) 4 −a

Concerning the higher-dimensional case, let us consider A a strongly positive elliptic operator of order 2m associated to the free vibration of a domain Ω in a general space Rν , and satisfying the Garding coerciviness condition (−1)|α| Dxα (aαβ (x))Dxβ , (8.50) A= |α|≤m |β|≤m

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with the Sobolev spaces operator’s domain D(A) = H 2m (Ω) ∩ H0m (Ω).

(8.51)

Note there exists a constant C0 (Ω) > 0(C0 (Ω) → 0 if vol (Ω) → ∞) such that the Garding coerciviness condition holds true on D(A) Real(AU, U )L2 (Ω) ≥ C0 (Ω) f H 2m (Ω) .

(8.52)

We can thus associate a Lipschitz nonlinear external vibration on the domain Ω as governed by the wave equation below ∂ 2 U (x, t) = (AU )(x, t) + G (U (x, t)), ∂2t

(8.53)

U (x, 0) = U (0) (x) ∈ H 2m (Ω) ∩ H0m (Ω),

(8.54)

Ut (x, 0) = g(x) ∈ L2 (Ω).

(8.55)

Here G(z) denotes a differentiable Lipschitizian function, like G(z) = A sin (Bz), e−γz , etc. and thus, leading to the uniqueness and existence of global weak solutions on C([0, ∞), H 2m (Ω) ∩ H0m (Ω)) by the fixed point theorem.6 Analogously to the discretization/N -particle technique of the 1 + 1 case, one obtains as a mathematical candidate for the Ergodic invariant measure, the following rigorously measure on the Hilbert space of functions H 2m (Ω) (with β = 1): 1

d(inv) µ(φ(x)) = dA µ(φ(x)) × e− 2

R Ω

dν x G(φ(x))

.

(8.56)

Here dA µ(φ(x)) denotes the Gaussian measure generated by the quadratic form associated to the operator A on its domain D(A) for real initial conditions functions φ(x): L2 (Ω)

dA µ(φ(x))(φ(x1 )φ(x2 )) = (A−1 )(x1 , x2 ).

(8.57)

Note that the Green function of the operator A belongs to the trace class operators on H 2m (Ω), which results on Eq. (8.57) through the application of the Minlo’s theorem.7

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At this point, we remark that dinv µ(φ(x)) defines a mathematical rigorous Radon measure in H 2m (Ω) since the function (functional): 1 dν xG(φ(x)) , F (φ(x)) = exp − 2 Ω

(8.58)

belongs to L1 (L2 (Ω), dA µ(φ(x))) due to the Lipschitz property of the nonlinearity G(z), i.e. there exist constants c+ and c− , such that c− φ(x) ≤ G(φ(x)) ≤ c+ φ(x).

(8.59)

As a consequence, we have the obvious estimate below e−c

+

R Ω

dn uxφ(x)

1

≤ e− 2

R Ω

dν xG(φ(x))

≤ e−c

−

R Ω

dν xφ(x)

,

(8.60)

so by the Lebesgue comparison theorem, we get the functional integrability of the nonlinearity term of the Ergodic invariant measure Eq. (8.56): R + 1 (c− +c+ )2 TrH 2m (Ω) (A−1 ) − 12 Ω dν xG(φ(x)) dA µ(φ(x))e < ∞, ≤e 2 L2 (Ω) (8.61a) and leading thus to the mathematical validity of Eq. (8.56). At this point, it is worth to remark that one can easily generalize the type of our allowed Lipschitz nonlinearity for those of the following form γ(x) 2 (φ) (x) + dν xG(φ(x)) ≥ + dν x dν xC(x)φ(x) , 2 Ω Ω Ω (8.61b)

with γ(x) and C(x) real positive L∞ (Ω) functions. This claim is a result of the straightforward Gaussian-infinitedimensional integration ν γ(x) ν ¯ ¯ (φ φ)(x) + d µ(φ) exp − d x d C(x) φ A (x) 2m 2 H (Ω) Ω Ω −1 γ(y) ν −1 2 1 + d yA (x − y) · ≤ det 2 1 × exp + dν x dν yC(x) · A−1 (x, y)C(y) < ∞. (8.61c) 2 Ω Ω

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In the usual (Euclidean) Feynman path-integral formal notation, the Ergodic theorem takes the physicist’s form after reintroducing the reservoir temperature parameter β = 1/kT R ν 1 1 T 1 F (U (x, t)) = DF [ϕ(x)]e− 2 β Ω d x(ϕ(x)·(Aϕ(x))) lim T →∞ T 0 Z L2 (Ω) × F (ϕ(x))e−β

R

Ω

dν xG(ϕ(x))

,

with the normalization factor R R ν ν 1 DF [ϕ(x)]e 2 β Ω (ϕ(x)(Aϕ)(x))d x e−β Ω d xG(ϕ(x)). Z=

(8.62)

(8.63)

H 2m (Ω)

The functionals F (ϕ(x)) are objects belonging to the space L (L2 (Ω), dA (ϕ(x))) ⊂ C(L2 (Ω), R). At this point, let us remark that by a direct application of the mean value theorem there is a sequence of growing times {tn }n∈Z (with tn → ∞) such that R ν 1 1 DF [ϕ(x)]e− 2 β Ω d x(ϕ·Aϕ)(x) lim F (φ(x, tn )) = n→∞ 2 L2 (Ω) 1

× F (ϕ(x))e−β

R

Ω

dν xG(ϕ(x))

.

(8.64)

It is worth to recall that the (positive) parameter β is not determined directly from the parameters of the underlying mechanical system and can be thought of as a remnant of the initial conditions “averaged out” by the Ergodic integral (see Sec. 8.3) and sometimes related to the presence of an intrinsic dissipation mechanism involved on the Ergodictime average process responsible for the extensivity of the system’s thermodynamics (see Eq. (8.28) — Sec. 8.3). For instance, at the limit of vanishing temperature (absolute Newtonian–Boltzmann–Maxwell zero temperature β → ∞ and somewhat related to the so called Ergodic mixing microconomical ensemble), one can see the appearance of an “atractor” on the space H 2m (Ω) ∩ H0m (Ω) and exactly given by the stationary strong unique solution of the wave equation (8.53) since, we always have a soluble elliptic problem on H 2m (Ω)∩H0m (Ω) by means of the Lax–Milgram–Gelfand theorem.6 (Aϕ∗ )(x) = G (ϕ∗ (x))

(8.65)

ϕ∗ (x)|∂Ω = 0.

(8.66)

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Note that the stationary strong solution ϕ∗ (x) is not a real atractor in the sense of the theory of dynamical systems, since we can only say that there is a specific sequence of growing {tn }n∈Z , with tn → ∞, such that at the leading asymptotic limit β → ∞ we have the time-infinite limit for every solution of Eqs. (8.65) and (8.66): lim F (ϕ(x, tn )) = F (ϕ∗ (x)).

tn →∞

(8.67)

A (somewhat) formal path-integral calculation of the next term on the result (Eq. (8.41)) for F (z) being smooth real-valued functions can be implemented by means of the usual path-integrals Saddle-point techniques applied to the (normalized) functional integral equation (8.62).8 lim F (ϕ(x, tn )) 2 1δ F ∗ ∗ = F (ϕ (x)) + (ϕ (x)) β δ2ϕ 2 δ G ∗ d × (ϕ ) + α + O(β −2 ) lg det 1 + A−1 dα δ2 ϕ α=0 2 1 δ F ∗ (ϕ (x)) = F (ϕ∗ (x)) + β δ2 ϕ −1 2 δ G ∗ −1 (ϕ ) × TrH 2m (Ω) 1+A + O(β −2 ) δ2 ϕ 1 δ2 F ∗ ∗ (ϕ (x)) = F (ϕ (x)) + β δ2 ϕ n 2 ∞ n −1 δ G ∗ (ϕ ) × + O(β −2 ) (−1) TrH 2m (Ω) A 2ϕ δ n=0

tn →∞

(8.68)

(8.69)

Another important remark to be made is related to the existence of timeindependent simple constraints on the discreticized Hamiltonian equation (8.32), namely H (a) (q1 , . . . , qN ) = 0;

a = 1, . . . , n.

(8.70)

In this case, one should consider that the continuum version of Eq. (8.70) leads to (strongly) continuos functionals on H 2m (Ω) ∩ H0m (Ω): H (a) (φ(x, t)) = 0;

a = 1, . . . , n.

(8.71)

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They have a direct consequence of restricting the wave motion to a closed nonlinear manifold of the original vector Hilbert space H 2m (Ω) ∩ H0m (Ω). Now it is straightforward to apply the Ergodic theorem on the effective mechanical configuration space and obtain, thus, as the Ergodic invariant measure the constraint path-integral. 1 T →∞ T

1 = Z

T

dt F (φ(x, t))

lim

0 1

DF [ϕ(x)]e− 2 β H 2m (Ω)

× F (ϕ(x))

n

R Ω

dν x(ϕ(x)·Aϕ(x)) −β

e

R Ω

dν xG(ϕ(x))

δ (F ) (H (a) (ϕ(x))) ,

(8.72)

a=1

where δ (F ) (·) denotes the usual formal delta-functionals.

8.5. An Ergodic Theorem in Banach Spaces and Applications to Stochastic–Langevin Dynamical Systems In this complementary Sec. 8.5, we intend to present our approach to study long-time/Ergodic behavior of infinite-dimensional dynamical systems by analyzing the somewhat formal diffusion equation with polynomial terms and driven by a white noise stirring.4 In order to implement such studies, let us present the author’s generalization of the RAGE theorem for a contraction self-adjoint semi-group T (t) on a Banach space X. We have, thus, the following theorem. Theorem 8.3. Let f ∈ X. We have the Ergodic generalized theorem 1 lim T →∞ T

T 0

dt(e−tA f ) = Pker(A) (f ),

(8.73)

where A is the infinitesimal generator of T (t), supposed to be self-adjoint. Let us sketch the proof of the above claimed theorem of ours. As a first step, one should consider Eq. (8.73) rewritten in terms of the “resolvent operator of A” by means of a Laplace transform (the

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Hile–Yosida–Dunford spectral calculus) i∞ 1 1 T zt −1 lim dt dze ((z + A) f ) . T →∞ T 0 2πi −i∞

(8.74)

Now it is straightforward to apply the Fubbini theorem to exchange the order of integrations (dt, dz) in Eq. (8.74) and get, thus, the result 1 T dt(e−tA f ) = lim− +((z + A)−1 f ). (8.75) lim T →∞ T 0 z→0 The z → 0− limit of the integral of Eq. (8.75) (since Real (z) ⊂ ρ(A) ⊂ (−∞, 0)) can be evaluated by means of saddle point techniques applied to Laplace’s transforms. We have the following result ∞ e−zt (e−tA f ) dt lim ((z + A)−1 f ) = lim z→0−

z→0−

= lim (e t→∞

0 −tA

f ) = Pker(A) (f ).

(8.76)

Let us apply the above theorem to the Lengevin equation. Let us consider the Fokker–Planck equation associated to the following Langevin equation: ∂V j dq i (t) = − (q (t)) + η i (t), dt ∂qi

(8.77)

where {η i (t)} denotes a white-noise stochastic time process representing the thermal coupling of our mechanical system with a thermal reservoir at temperature T . Its two-point function is given by the “fluctuation–dissipation” theorem η i (t)η j (t ) = kT δ(t − t ).

(8.78)

The Fokker–Planck equation associated to Eq. (8.74) has the following explicit form ∂P i i (q , q¯ ; t) = +(kT )∆qi P (q i , q¯i ; t) + ∇qi (∇qi V · P (q i , q¯i ; t)) ∂t lim P (q i , q¯i ; t) = δ (N ) (q i − q¯i ).

t→0+

(8.79) (8.80)

By noting that we can associate a contractive semi-group to the initial value problem Eq. (8.80) in the Banach space L1 (R3N ), namely: q i )d¯ q i = (e−tAf ). (8.81) P (q i , q¯i , t)f (¯

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Here, the closed positive accretive operator A is given explicitly by −A(·) = kT ∆qi (·) + ∇qi [(∇qi V )(·)],

(8.82)

and acts first on X = C∞ (R3N ). It is instructive to point out that the perturbation accretive operator B = ∇qi · [(∇qi V ) · (·)] on C∞ (R3N ) with V (q i ) ∈ Cc∞ (R3N ) is such that it satisfies the estimate on S(R3N ) : Bf S(R3N ) ≤ a ∆f S(R3N ) + b f S(R3N ) with a > 1 and for some b. As a consequence A(·) = kT ∆qi (·) + (B(·)) generates a contractive semi-group on C∞ (R3N ) or by an extension argument on the whole L1 (R3N ) since the L1 -closure of C∞ (R3N ) is the Banach space L1 (R3N ). At this point, we may apply our Theorem 8.3 to obtain the Langevin– Brownian Ergodic theorem applied to our Fokker–Planck equation 1 lim T →∞ T

T

dt

dq(e

−tA

R3N

0

f )(q) = Pker(A) (f )(q) dN q¯i f (¯ q i )Peq (¯ q i ),

=

(8.83)

RN

where the equilibrium probability distribution is given explicitly by the unique normalizable element of the closed sub-space ker(A). O = +(kT )∆qi Peq (¯ q i ) + ∇q¯i [(∇q¯i V )P(¯ q i )]

(8.84)

or exactly, we have the Boltzmann’s weight for our equilibrium Langevin– Brownian probability distribution 1 eq i i (V (¯ q )) . q ) = exp − (8.85) P (¯ kT For the general Langevin equation in the complete phase space {qi , pi } as in the bulk of this note, one should reobtain the complete Boltzmann statistical weight as the equilibrium ergodic probability distribution N 1 1 i 2 1 (¯ p ) + V (¯ q , p¯ ) = exp − q) , P (¯ Z kT 2 eq

i

i

(8.86)

l=1

with the normalization factor d¯ pi Z= RN

d¯ q i Peq (¯ q i , p¯i ).

RN

(8.87)

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Let us apply the above exposed result for the following nonlinear (polynomial) stochastic diffusion equation on a one-dimensional domain Ω = (−a, a) with all positive coefficients {λj } on the nonlinear polynomial term   2k−1  1 d2 U (x, t)  ∂U (x, t)  =+ − λj U j    (x, t) + η(x, t), ∂t 2 d2 x j=0

(8.88)

j=odd

U (x, t)|∂Ω = 0,

(8.89)

U (x, 0) = f (x) ∈ L2 (Ω).

(8.90)

Here, {η(x, t)} are samplings of a white-noise stirring. In this case, one can show by standard techniques that the global solution U (x, t) ∈ C([0, ∞), L2 (−a, a)) and by using the discreticized approach of Sec. 8.4, the invariant Ergodic measure associated to the nonlinear diffusion equation is given by the mathematically functional equilibrium Langevin–Brownian probability distribution       a  2k   λj j  eq   U (x) × d− 1 d2 µ(U (x)). dx  P (U (x)) = exp − 2 dx2   j −a   j=0   jeven

(8.91) Here, d− 1

d2 2 dx2

µ(U (x)) denotes the Gaussian functional measure (nor-

malized to unity) on the Wiener space C α (−a, a). In the Feynman pathintegral notation (with β = 1) R a dU 2 1 1 T dt F (U (x, t)) = DF [U (x)]e− 2 −a | dx | (x) lim T →∞ T 0 C α ((−a,a)) # $ ×e

−

Ra

−a

dx

P2k

j=0 j=even

λj j

U j (x)

× F [U (x)]. (8.92)

Note that Eq. (8.91) defines a rigorous mathematical object since it does not need to be renormalized from a calculational point of view on the Wiener space C α (−a, a)10 as its nonlinear exponential term is bounded by the unity (Lebesgue theorem). It is worth to point out that for nonlinearities of Lipschitz type on general domains ΩCRν , one can see that the support

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of the Gaussian measure is readily L2 (Ω) and the equilibrium Langevin– Brownian probability distribution eq D (8.93) P (U (x)) = exp − d xG(U (x)) × d(− 12 ∆) µ, (U (x)), Ω

is a perfect well-defined random measure on C α ((−a, a)) by the same Lebesgue convergence theorem. 8.6. The Existence and Uniqueness Results for Some Nonlinear Wave Motions in 2D In this technical section, we give an argument for the global existence and uniqueness solution of the Hamiltonian motion equations associated first to Eq. (8.32) — Sec. 8.3 and by secondly to Eq. (8.53) — Sec. 8.3. Related to the two-dimensional case, let us equivalently show the weak existence and uniqueness of the associated continuum nonlinear polynomial wave equation in the domain (−a, a) × R+ . ∂ 2 U (x, t) ∂ 2 U (x, t) − + g(U (x, t))2k−1 = 0, ∂2t ∂ 2x

(8.94)

U (−a, t) = U (a, t) = 0, U (x, 0) = U0 (x) ∈ H 1 ([−a, a]), Ut (x, 0) = U1 (x) ∈ L2 ([−a, a]).

(8.95) (8.96)

Let us consider the Galerkin approximants functions to Eqs. (8.95) and (8.96) as ¯n (t) ≡ U

n

U (t) sin

=1

π x . a

Since there exists a γ0 (a) positive such that d2 ¯ ¯ ¯n , U ¯n )H 1 ([−a,a]) , , U ≥ γ0 (a)(U − 2 U n n d x L2 ([−a,a])

(8.97)

(8.98)

we have the a priori estimate for any t 0 ≤ ϕ(t) ≤ ϕ(0),

(8.99)

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with ϕ(t) =

1 ¯˙ ¯n 2k2k . ¯n 2 1 + 1 U Un (t) 2L2 + γ0 (a) U H L 2 2k

(8.100)

As a consequence of the bound equation (8.100), we get the bounds for any given T (with Ai constants) ¯ n 2 1 sup ess U H (−a,a) ≤ A1 ,

(8.101)

¯˙ n 2 2 sup ess U L (−a,a) ≤ A2 ,

(8.102)

¯n 2k2k sup ess U L (−a,a) ≤ A3 .

(8.103)

0≤t≤T

0≤t≤T

0≤t≤T

By usual functional-analytical theorems on weak-compactness on ¯n (t) Banach–Hilbert spaces, one obtains that there exists a sub-sequence U such that for any finite T ∗

−−−−−−−−→ ¯ ¯n (t)− U (weak − star) U (t) in L∞ ([0, T ], H 1 (−a, a)), ∗

−−−−−−−−→ ¯˙ n (t)− U (weak − star) v¯(t) ∗

−−−−−−−−→ ¯n (t)− U (weak − star) p¯(t)

(8.104)

in L∞ ([0, T ], L2(−a, a)),

(8.105)

in L∞ ([0, T ], L2k (−a, a)).

(8.106)

At this point, we observe that for any p > 1 (with A˜i constants) and T < ∞, we have the relationship

T

0

0

T

p

¯ n p 1 2 U H (−a,a) dt ≤ T (A1 ) ⇔ p

¯˙ n p 2 2 U L (−a,a) dt ≤ T (A2 ) ⇔

0

0

T

T

¯ n p 2 ˜ U L (−a,a) ≤ A1 ,

(8.107)

¯˙ p 2k U dt ≤ A˜2 , L (−a,a)

(8.108)

since we have the continuous injection below H 1 (−a, a) → L2 (−a, a),

(8.109)

L2k (−a, a) → L2 (−a, a).

(8.110)

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As a consequence of the Aubin–Lion theorem,6 one obtains straightforwardly the strong convergence on LP ((0, T ), L2 (−a, a)) together with the almost everywhere point-wise equality among the solutions candidate

t

¯n → U(t) ¯ U = p¯(t) =

v¯(s)ds.

(8.111)

0

By the Holder inequality applied to the pair (q, k) ¯ Lq (−a,a) ≤ Un − U ¯ 1−θ ¯ θ Un − U L2 (−a,a) Un − U L2k (−a,a) ,

(8.112)

with 0 ≤ θ ≤ 1 1−θ θ 1 = + , q 2 2k

(8.113)

in particular with q = 2k − 1, one obtains the strong convergence of Un (t) k ( 3−2k in the general Banach space L∞ ((0, T ), L2k−1 (−a, a)), with θ¯ = 1−k 2k−1 ). As a consequence of the above results, one can safely pass the weak limit on d2 ¯ (Un , v)L2 (−a,a) C ∞ ((0, T ), L2 (−a, a)) lim n→∞ d2 t

d2 ¯ 2k−1 ¯ + − 2 Un , v + g(Un , v)L2 (−a,a) d x L2 (−a,a) d2 ¯ d2 ¯ ¯ 2k−1 , v)L2 (−a,a) = 0, 2 = 2 (U , v)L (−a,a) + − 2 U , v + g(U d t d x L2 (−a,a) (8.114) for any v ∈ C ∞ ((0, T ), L2 (−a, a)). At this point, we shetch a rigorous argument to prove the problem’s uniqueness. Let us consider the hypothesis that the finite function a(x, t) =

¯ )2k+1 (x, t) − (¯ v )2k+1 (x, t)) ((U , ¯ (x, t) − v¯(x, t)) (U

(8.115)

¯ (x, t) and is essentially bounded on the domain [0, ∞) × (−a, a) where U v¯(x, t) denote two hypothesized different solutions for the 2D-polynomial wave equations (8.94)–(8.96). It is straightforward to see that its difference

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¯ − v¯)(x, t) satisfies the “Linear” wave equation problem W (x, t) = (U ∂2W ∂2W − + (aW )(x, t) = 0, ∂2t ∂x

(8.116)

W (0) = 0,

(8.117)

Wt (0) = 0.

(8.118)

At this point, we observe the estimate (where H 1 → L2 !) 1 d d 2 2 W L2 (−a,a) + W H 1 (−a,a) 2 dt dt

dW

≤ a L∞ ((0,T )(−a,a)) W L2(−a,a) ×

dt L2 (−a,a)

dW 2 ≤M

+ W 2L2 (−a,a)

2 dt L (−a,a)

dW 2 + W 2H 1 (−a,a) , ≤M

(8.119)

2 dt L (−a,a) which after the application of the Gronwall’s inequality give us that

d 2 2 W + W

2 H 1 (−a,a) (t) dt L (−a,a)

dW 2 (0) + W 2H 1 (−a,a) (0) = 0, (8.120) ≤

2 dt L (−a,a) which proves the problem’s uniqueness under the not proved yet hypothesis that in the two-dimensional case (at least for compact support infinite differentiable initial conditions) sup

a(x, t) ≤ M.

(8.121)

x∈(−a,a) t∈[0,∞)

As a last comment on the 1 + 1-polynomial nonlinear wave motion, we call the readers’ attention that one can easily extend the technical results analyzed here to the complete polynomial nonlinearity j

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