Microlocal Analysis, Sharp Spectral Asymptotics and Applications V: Applications to Quantum Theory and Miscellaneous Problems

761 Pages·2019·7.963 MB·English

by  Victor Ivrii

Victor Ivrii
Microlocal Analysis,
Sharp Spectral
Asymptotics and
Applications V
Applications to Quantum Theory and
Miscellaneous Problems
Microlocal Analysis, Sharp Spectral Asymptotics
and Applications V
Victor Ivrii
Microlocal Analysis, Sharp
Spectral Asymptotics
and Applications V
Applications to Quantum Theory
and Miscellaneous Problems
123
Victor Ivrii
Department ofMathematics
University of Toronto
Toronto, ON,Canada
ISBN978-3-030-30560-4 ISBN978-3-030-30561-1 (eBook)
https://doi.org/10.1007/978-3-030-30561-1
MathematicsSubjectClassification(2010): 35P20,35S05,35S30,81V70
©SpringerNatureSwitzerlandAG2019
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
The Problem of the Spectral Asymptotics, in particular the problem of the
Asymptotic Distribution of the Eigenvalues, is one of the central problems
in the Spectral Theory of Partial Differential Operators; moreover, it is very
important for the General Theory of Partial Differential Operators.
I started working in this domain in 1979 after R. Seeley [1] justified
a remainder estimate of the same order as the then hypothetical second
term for the Laplacian in domains with boundary, and M. Shubin and
B. M. Levitan suggested me to try to prove Weyl’s conjecture. During the
past almost 40 years I have not left the topic, although I had such intentions
in 1985, when the methods I invented seemed to fail to provide the further
progress and only a couple of not very exciting problems remained to be
solved. However, at that time I made the step toward local semiclassical
spectral asymptotics and rescaling, and new much wider horizons opened.
So I can say that this book is the result of 40 years of work in the Theory
of Spectral Asymptotics and related domains of Microlocal Analysis and
Mathematical Physics (I started analysis of Propagation of singularities
(which plays the crucial role in my approach to the spectral asymptotics) in
1975).
This monograph consists of five volumes. This Volume II concludes the
general theory. It consists of two parts. In the first one we develop methods
of combining local asymptotics derived in Volume I, with estimates of the
eigenvalue counting functions in the small domains (singular zone) derived
by methods of functional analysis. In the second part we derive eigenvalue
asymptotics which either follow directly from the general theory, or require
applications of the developed methods (if the operator has singularities or
degenerations, strong enough to affect the asymptotics).
Victor Ivrii,
Toronto, June 10, 2019.
V
Contents
Preface V
Introduction XX
Part XI. Asymptotics of the Ground State Energy
of Heavy Atoms and Molecules . . . . . XX
Part XII. Articles . . . . . . . . . . . . . . . . . . XXIV
XI Application to Multiparticle Quantum The-
ory 1
25 No Magnetic Field Case 2
25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2
25.1.1 Framework . . . . . . . . . . . . . . . . . . . . . 2
25.1.2 Problems to Consider . . . . . . . . . . . . . . . . 4
25.1.3 Thomas-Fermi Theory . . . . . . . . . . . . . . . 5
25.1.4 Main Results Sketched and Plan of the Chapter . 7
25.2 Reduction to Semiclassical Theory. . . . . . . . . . . . . 8
25.2.1 Lower Estimate . . . . . . . . . . . . . . . . . . . 9
25.2.2 Upper Estimate . . . . . . . . . . . . . . . . . . . 12
25.2.3 Remarks and Dirac Correction . . . . . . . . . . . 14
25.3 Thomas-Fermi Theory . . . . . . . . . . . . . . . . . . . 15
25.3.1 Existence . . . . . . . . . . . . . . . . . . . . . . 16
25.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . 18
25.4 Application of Semiclassical Methods . . . . . . . . . . . 23
25.4.1 Asymptotics of the Trace . . . . . . . . . . . . . . 23
25.4.2 Upper Estimate for E . . . . . . . . . . . . . . . . 26
Estimating |𝜆 −𝜈| . . . . . . . . . . . . . . . . 26
N
VI
CONTENTS VII
Estimating D-Term . . . . . . . . . . . . . . . . . 29
Finally, a tTheorem . . . . . . . . . . . . . . . . . 31
25.4.3 Improved Asymptotics . . . . . . . . . . . . . . . 32
25.4.4 Corollaries and Discussion . . . . . . . . . . . . . 35
Estimates for D(𝜌 −𝜌𝖳𝖥,𝜌 −𝜌𝖳𝖥) . . . . . . . 35
𝝭 𝝭
Estimates for Distance between Nuclei in the Free
Nuclei Model . . . . . . . . . . . . . . . 36
25.5 Negatively Charged sSystems . . . . . . . . . . . . . . . 38
25.5.1 Estimates of the Correlation Function . . . . . . . 38
25.5.2 Excessive Negative Charge . . . . . . . . . . . . . 43
25.5.3 Estimate for Ionization Energy . . . . . . . . . . 47
25.6 Positively Charged sSystems . . . . . . . . . . . . . . . . 48
25.6.1 Estimate from above for Ionization Energy . . . . 48
25.6.2 Estimate from below for Ionization Energy . . . . 51
25.6.3 Estimate for Excessive Positive Charge . . . . . . 60
25.AAppendices . . . . . . . . . . . . . . . . . . . . . . . . . 64
25.A.1 Electrostatic Inequalities . . . . . . . . . . . . . . 64
25.A.2 Hamiltonian Trajectories . . . . . . . . . . . . . . 65
25.A.3 Some Spectral Function Estimates. . . . . . . . . 66
26 The Case of External Magnetic Field 68
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 68
26.1.1 Framework . . . . . . . . . . . . . . . . . . . . . 68
26.1.2 Problems to Consider . . . . . . . . . . . . . . . . 69
26.1.3 Magnetic Thomas-Fermi Theory . . . . . . . . . . 71
26.1.4 Main Results Sketched and Plan of the Chapter . 72
26.2 Magnetic Thomas-Fermi Theory . . . . . . . . . . . . . . 74
26.2.1 Framework and Existence . . . . . . . . . . . . . 74
26.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . 76
26.2.3 Positive Ions . . . . . . . . . . . . . . . . . . . . . 82
26.3 Applying Semiclassical Methods: M =1 . . . . . . . . . 85
26.3.1 Heuristics . . . . . . . . . . . . . . . . . . . . . . 85
Total Charge . . . . . . . . . . . . . . . . . . . . 86
Semiclassical D-Term . . . . . . . . . . . . . . . . 88
|𝜆 −𝜈| and Another D-Term . . . . . . . . . . . 90
N
Trace . . . . . . . . . . . . . . . . . . . . . . . . . 91
Discussion . . . . . . . . . . . . . . . . . . . . . . 92
26.3.2 Smooth Approximation . . . . . . . . . . . . . . . 93
VIII CONTENTS
Trivial Arguments . . . . . . . . . . . . . . . . . 93
Formal Expansion . . . . . . . . . . . . . . . . . . 96
Expansion: Justification . . . . . . . . . . . . . . 98
26.3.3 Rough Approximation . . . . . . . . . . . . . . . 100
Properties of Mollification . . . . . . . . . . . . . 100
Charge Term . . . . . . . . . . . . . . . . . . . . 102
Trace Term . . . . . . . . . . . . . . . . . . . . . 105
Semiclassical D-Term: Local Theory . . . . . . . 108
Semiclassical D-Term: Global Theory . . . . . . . 111
26.4 Applying Semiclassical Methods: M ≥2 . . . . . . . . . 113
26.4.1 Scaling Functions in Zone 𝒳 . . . . . . . . . . . 113
𝟤
26.4.2 Zone 𝒳 : Semiclassical N-Term . . . . . . . . . . 118
𝟤
26.4.3 Zone 𝒳 : Semiclassical D-Term . . . . . . . . . . 122
𝟤
26.4.4 Semiclassical T-Term . . . . . . . . . . . . . . . . 125
Semiclassical T-Term: Zone 𝒳 Extended . . . . . 125
𝟣
Semiclassical T-Term: Zone 𝒳 . . . . . . . . . . 127
𝟤
26.4.5 Zone 𝒳 . . . . . . . . . . . . . . . . . . . . . . . 131
𝟥
26.5 Semiclassical Analysis in the Boundary Strip for M ≥2 . 131
26.5.1 Properties of W𝖳𝖥 if N =Z . . . . . . . . . . . . 132
B
26.5.2 Analysis in the Boundary Strip 𝒴 for N ≥Z . . . 137
26.5.3 Analysis in the Boundary Strip 𝒴 for N <Z . . . 140
𝟦
Case B ≥(Z −N)𝟥 . . . . . . . . . . . . . . . . . 140
+
𝟦
Case B ≤(Z −N)𝟥 . . . . . . . . . . . . . . . . . 145
+
26.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . 146
26.6 Ground State Energy . . . . . . . . . . . . . . . . . . . . 147
26.6.1 Lower Estimates . . . . . . . . . . . . . . . . . . 147
26.6.2 Upper Estimate: General Scheme . . . . . . . . . 149
26.6.3 Upper Estimate as M =1 . . . . . . . . . . . . . 149
Estimate for |𝜆 −𝜈| . . . . . . . . . . . . . . . . 149
N
Estimate for D-Terms . . . . . . . . . . . . . . . . 151
Summary . . . . . . . . . . . . . . . . . . . . . . 152
26.6.4 Upper Estimate as M ≥2 . . . . . . . . . . . . . 153
Estimate for |𝜆 −𝜈| . . . . . . . . . . . . . . . . 153
N
Estimate for D-Terms for Almost Neutral Systems 156
EstimateforD-TermsforPositivelyChargedSystems 158
Summary . . . . . . . . . . . . . . . . . . . . . . 163
26.7 Negatively Charged Systems . . . . . . . . . . . . . . . . 164
CONTENTS IX
26.7.1 Estimates of the Correlation Function . . . . . . . 165
26.7.2 Excessive Negative Charge . . . . . . . . . . . . . 166
26.7.3 Estimate for Ionization Energy . . . . . . . . . . 171
26.8 Positively Charged Systems . . . . . . . . . . . . . . . . 177
26.8.1 Upper Estimate for Ionization Energy: M =1 . . 177
26.8.2 Lower Estimate for Ionization Energy: M =1 . . 182
26.8.3 Estimates for Ionization Energy: M ≥2 . . . . . 191
26.8.4 Free Nuclei Model . . . . . . . . . . . . . . . . . 192
Preliminary Arguments . . . . . . . . . . . . . . . 192
Minimal Distance . . . . . . . . . . . . . . . . . . 193
Estimate of Excessive Positive Charge . . . . . . 196
Estimate for Excessive Negative Charge and Ioniza-
tion Energy . . . . . . . . . . . . . . . . 198
26.AAppendices . . . . . . . . . . . . . . . . . . . . . . . . . 198
26.A.1 Electrostatic Inequalities . . . . . . . . . . . . . . 198
26.A.2 Very Strong Magnetic Field Case . . . . . . . . . 203
26.A.3 Riemann Sums and Integrals . . . . . . . . . . . . 204
26.A.4 Some Spectral Function Estimates. . . . . . . . . 205
26.A.5 Zhislin’s Theorem for Constant Magnetic Field. . 206
27 The Case of Self-Generated Magnetic Field 208
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 208
27.2 Local Semiclassical Trace Asymptotics . . . . . . . . . . 210
27.2.1 Toy-Model . . . . . . . . . . . . . . . . . . . . . . 210
Statement of the Problem . . . . . . . . . . . . . 210
Preliminary Analysis . . . . . . . . . . . . . . . . 211
Estimates . . . . . . . . . . . . . . . . . . . . . . 214
27.2.2 Microlocal Analysis Unleashed . . . . . . . . . . . 217
Sharp Estimates . . . . . . . . . . . . . . . . . . 217
Application . . . . . . . . . . . . . . . . . . . . . 222
Classical Dynamics and Sharper Estimates . . . . 225
27.2.3 Local Theory . . . . . . . . . . . . . . . . . . . . 228
Localization and Estimate from above . . . . . . 228
Estimate from below . . . . . . . . . . . . . . . . 231
27.2.4 Rescaling . . . . . . . . . . . . . . . . . . . . . . 233
Case 𝜅≤1 . . . . . . . . . . . . . . . . . . . . . 234
Case 1≤𝜅≤h−𝟣 . . . . . . . . . . . . . . . . . . 236
X CONTENTS
27.3 Global Trace Asymptotics in the Case of Coulomb-Like
Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 238
27.3.1 Problem . . . . . . . . . . . . . . . . . . . . . . . 238
27.3.2 Estimates to a Minimizer . . . . . . . . . . . . . . 239
Preliminary Analysis . . . . . . . . . . . . . . . . 239
Estimates to a Minimizer. I . . . . . . . . . . . . 241
Estimates to a Minimizer. II . . . . . . . . . . . . 244
Estimates to a Minimizer. III . . . . . . . . . . . 246
27.3.3 Basic Trace Estimates . . . . . . . . . . . . . . . 248
27.3.4 Improved Trace Estimates . . . . . . . . . . . . . 251
Improved Tauberian Estimates . . . . . . . . . . 251
Improved Weyl eEstimates . . . . . . . . . . . . . 253
27.3.5 Single Singularity . . . . . . . . . . . . . . . . . . 255
Coulomb Potential . . . . . . . . . . . . . . . . . 255
Main Theorem . . . . . . . . . . . . . . . . . . . 260
27.3.6 Several Singularities . . . . . . . . . . . . . . . . 261
Decoupling of Singularities . . . . . . . . . . . . . 261
Main Results . . . . . . . . . . . . . . . . . . . . 264
Problems and Remarks . . . . . . . . . . . . . . . 265
27.4 Asymptotics of the Ground State Energy . . . . . . . . . 265
27.4.1 Problem . . . . . . . . . . . . . . . . . . . . . . . 265
27.4.2 Lower Estimate . . . . . . . . . . . . . . . . . . . 267
27.4.3 Upper Estimate . . . . . . . . . . . . . . . . . . . 270
27.4.4 Main Theorems . . . . . . . . . . . . . . . . . . . 272
27.4.5 Free Nuclei Model . . . . . . . . . . . . . . . . . 274
27.5 Miscellaneous Problems. . . . . . . . . . . . . . . . . . . 276
27.5.1 Excessive Negative Charge . . . . . . . . . . . . . 276
27.5.2 Estimates for iIonization Energy . . . . . . . . . . 277
27.5.3 Free Nuclei Model: Excessive Positive Charge . . 278
27.AAppendices . . . . . . . . . . . . . . . . . . . . . . . . . 279
27.A.1 Minimizers and Ground States . . . . . . . . . . . 279
27.A.2 Zhislin’s Theorem . . . . . . . . . . . . . . . . . . 280
27.A.3 L. Erdo¨s–J. P. Solovej’s Lemma . . . . . . . . . . 280
28 The Case of Combined Magnetic Field 284
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 284
28.1.1 Plan of the Chapter. . . . . . . . . . . . . . . . . 285
28.1.2 Unfinished Business . . . . . . . . . . . . . . . . . 285