Martinez A. — An Introduction to Semiclassical and Microlocal Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Martinez A. — An Introduction to Semiclassical and Microlocal Analysis

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Название: An Introduction to Semiclassical and Microlocal Analysis

Автор: Martinez A.

Аннотация:

"This book presents most of the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics. Both the standard C[superscript [infinite]] pseudodifferential calculus and the analytic microlocal analysis are developed, in a context that remains intentionally global so that only the relevant difficulties of the theory are encountered. The originality lies in the fact that the main features of analytic microlocal analysis are derived from a single and elementary a priori estimate." Various exercises illustrate the chief results of each chapter while introducing the reader to further developments of the theory. Applications to the study of the Schrodinger operator are also discussed, to further the understanding of new notions or general results by placing them in the context of quantum mechanics. This book is aimed at nonspecialists of the subject, and the only required prerequisite is a basic knowledge of the theory of distributions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2002

Количество страниц: 190

Добавлена в каталог: 02.07.2008

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Предметный указатель

Action of the FBI transform on Fourier integral operators      83
Action of the FBI transform on pseudodifferential operators      81
Action of the FBI transform on the Fourier transform      89
Adjoint of a pseudodifferential operator      30 165
Agmon estimates      103 118 167
Agmon inequality      104 167
Airy function      61
Almost analytic extension      68
Analytic representation of a formal analytic symbol      59
Analytic resummation of a formal analytic symbol      59
Analytic singularities      78
Analytic Stationary Phase theorem      60
Analytic symbols      59
Analytic wave front set      78 137
Anti-Wick quantization      67
Asymptotic equivalence of symbols      17
Average impulse      7 8
Average position      7
Bargman transform      74
Bicharacteristic      121
Bochner' tubes theorem      131
Bohr correspondence principle      11
Born      7
Boundary value problems      133
Calderon — Vaillancourt theorem      47
Canonical one-form      144
Canonical symplectic form      84 143
Canonical transformations      84
Canonical transformations of pseudodifferential operators      151
Canonical two-form      84 143
Cauchy problem      133
Change of quantization      39 63 163
Characteristic set      56 116
Classical ellipticity      57 112 113
Classical mechanics      5
Classical observable      6
Classical pseudodifferential operator      46
Classical symbols      46
Classical trajectory      6 58 128
Classically allowed region      118
Classically forbidden region      58 118
Coherent states      74 100 141
Commutator of two pseudodifferential operators      45 166
Compact pseudodifferential operators      65
Composition formulae      45
Composition of pseudodifferential operators      31 34
Continuity of pseudodifferential operators      28 30 47 64 65
Correspondence principle      11
Cotlar-Stein lemma      49 52
de Broglie      7 10
De Broglie relation      8
Degree of a pseudodifferential operator      27
Differential operators      27
Distribution kernel      23
Double wells      136
Egorov theorem      141 152 153 161
Eikonal equation      20 149
Elliptic pseudodifferential operator      57
Elliptic pseudodifferential operators      38
Elliptic symbol      17
Ellipticity in the classical sense      57 112 113
Energy      6 118 128
Energy Observable      6
Energy shell      118
Energy surface      118
Evolution      10 140
Evolution problems      140
Exact Egorov theorem      141
Exponential support      109 110 112 113 115 118 133 136
Exponentially small      109
Fefferman — Phong inequality      98
Forbidden region      58
Formal adjoint of a pseudodifferential operator      30
Formal analytic symbol      59
Formal symbol      18 59
Formulae      163
Fourier integral operators      28 83
Fourier transform      8 22
Fourier-Bros-Iagolnitzer transform      73 81 83 89 101 166
frequency      7
Frequency set      55 81 101
Functional calculus of pseudodifferential operators      69 166
Garding inequality      54 67 98 99
Generating function of a canonical transformation      153 158
Gevrey classes      61 81 102
Gevrey index      61
Gevrey singularities      81
Gevrey symbols      61
Hamilton — Jacobi equations      119
Hamilton's equations      6
Hamiltonian field      45 119 146
Hamiltonian flow      119 147
Hanges theorem      128 138
harmonic oscillator      120 141
Heisenberg      7
Heisenberg uncertainty principle      6 62 73
Helffer — Sjostrand formula      70 166
Hilbert — Schmidt pseudodifferential operators      65
Holmgren theorem      134
Husimi function      105
Image of the FBI transform      101
Impulse      5

Isotropic subspaces      87
JORDAN      7
Kawai — Kashiwara theorem      130
kinetic energy      9
Lagrangian states      106
Lagrangian submanifolds      148
Lagrangian subspaces      87
Laplace operator      9
Left quantization      40
Leibniz formula      14
Levi — Mizohata theorem      102
Linear symbols      64
Linear symplectic transformations      84
Matrix mechanics      7
Matter waves      7 10
Microhyperbolicity      129
Microlocal behavior      55 73 77
Microlocal equivalence of pseudodifferential operators      154
Microlocal exponential estimates      90 167
Microlocal Holmgren theorem      134
Microlocalization      73
Microlocally exponentially small      77
Microlocally infinitely small      55
Microsupport      77
Momentum      5
Newton law      5
Non-characteristic Cauchy problem      133
Null-bicharacteristic      121
Operators of principal type      128
Order function      15
Order of a pseudodifferential operator      27
Oscillatory integrals      21
Parametrix      38 53
Phase function      25
Phase space trajectory      6
Plane wave      7
Plank constant      6 8
Plank-Einstein formula      10
Poisson bracket      45 147
Position-momentum variables      15
Principal symbol      46
Principal type      128
Probability of presence      7
Projection of the microsupport      109 112 115
Propagation of the frequency set      139 162
Propagation of the microsupport      118 122 140
Propagation property of semiclassical measures      106
Pseudodifferential calculus      13
Pseudodifferential operator      26
Pseudodifferential operators with analytic symbols      60
Quantization      13 39 40 67
Quantum evolution      10 140
Quantum flow      141 152
quantum mechanics      7
Quantum observable      9
Range of the FBI transform      101
Regularizing operator      30
Resummation of symbols      18
Right quantization      40
Schrodinger equation      10
Schrodinger operator      9 58 104 113 114 118 121 128 130 136
Schur lemma      50
Schwartz kernel theorem      23
Semiclassical analysis      11
Semiclassical differential operators      27
Semiclassical expansions of symbols      17
Semiclassical measures      105
Sharp Garding inequality      67 98 99
Shrodinger      7
Simple characteristic point      128
Singularities      81
Sobolev singularities      81
Spaces of symbols      16
Standard quantization      40 164 165
State of a particle      7
Stationary Phase theorem      31 60
Stationary points      120
Subprincipal symbol      46
Symbol of a commutator      45 166
Symbol of a pseudodifferential operator      27 41
Symbol of the adjoint operator      165
Symbol of the transposed operator      165
Symbolic calculus      39 43
symbols      14
Symplectic coordinates      147
Symplectic form      84 143
Symplectic transformations      84 144
Trace-Class pseudodifferential operators      66
Transport equations      21
Transposed of a pseudodifferential operator      165
Uncertainty principle      6 62 73
Wave front set      55
Wave function      7
Wave mechanics      7
Wave vector      7
Weighted estimates in the Gevrey case      102
Weighted microlocal estimates      91 95 102 167
Weyl quantization      40 164 165
Weyl symbol      41
Weyl — Hormander calculus      99
WKB asymptotics      104
WKB solutions      20 61 133

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