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E-grāmata: Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1992
  • Izdošanas datums: 23-Jul-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642119224
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Spectral Theory of Non-Commutative Harmonic Oscillators: An IntroductionPalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1992
  • Izdošanas datums: 23-Jul-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642119224
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This book grew out of a series of lectures given at the Mathematics Department of Kyushu University in the Fall 2006, within the support of the 21st Century COE Program (20032007) Development of Dynamical Mathematics with High Fu- tionality (Program Leader: prof. Mitsuhiro Nakao). It was initially published as the Kyushu University COE Lecture Note n- ber 8 (COE Lecture Note, 8. Kyushu University, The 21st Century COE Program DMHF, Fukuoka, 2008. vi+234 pp.), and in the present form is an extended v- sion of it (in particular, I have added a section dedicated to the Maslov index). The book is intended as a rapid (though not so straightforward) pseudodiff- ential introduction to the spectral theory of certain systems, mainly of the form a +a where the entries of a are homogeneous polynomials of degree 2 in the 2 0 2 n n (x,?)-variables, (x,?)? R×R,and a is a constant matrix, the so-called non- 0 commutative harmonic oscillators, with particular emphasis on a class of systems introduced by M. Wakayama and myself about ten years ago. The class of n- commutative harmonic oscillators is very rich, and many problems are still open, and worth of being pursued.

Recenzijas

From the reviews:

The book under review presents the spectral theory of elliptic non-commutative harmonic oscillators, offering also useful information for more general elliptic differential systems. The book consists of 12 chapters, one appendix and a complete list of references on the subject. The book addresses important and difficult topics in mathematics. The results are presented in a rigorous, illuminating and elegant way. (Dumitru Motreanu, Zentralblatt MATH, Vol. 1200, 2011)

1 Introduction
1(6)
2 The Harmonic Oscillator
7(8)
2.1 From the Hamiltonian to the Operator Acting on L2
7(3)
2.2 The Spectrum of the Harmonic Oscillator
10(5)
3 The Weyl-Hormander Calculus
15(40)
3.1 Review of the Weyl-Hormander Calculus
15(10)
3.2 Global Metrics and Global Pseudodifferential Operators
25(16)
3.3 Spectral Properties of Globally Elliptic Pseudodifferential Operators
41(12)
3.4 Notes
53(2)
4 The Spectral Counting Function N(λ) and the Behavior of the Eigenvalues: Part 1
55(12)
4.1 The Minimax Principle
55(3)
4.2 The Spectral Counting Function
58(3)
4.3 The Spectral Counting Function of the (Scalar) Harmonic Oscillator
61(4)
4.4 Consequences on the Spectral Counting Function of an Elliptic Global ψdo
65(2)
5 The Heat-Semigroup, Functional Calculus and Kernels
67(12)
5.1 Elementary Properties of the Heat-Semigroup
67(2)
5.2 Direct Definition of Tre-tA
69(2)
5.3 Abstract Functional Calculus
71(2)
5.4 Kernels
73(4)
5.5 f(A) as a Pseudodifferential Operator
77(1)
5.6 Notes
77(2)
6 The Spectral Counting Function N(λ) and the Behavior of the Eigenvalues: Part 2
79(14)
6.1 A Parametrix Approximation of e-tA
79(4)
6.2 The Karamata Theorem
83(3)
6.3 Use of the Parametrix Approximation of e-tA for Obtaining the Weyl Asymptotics of N(λ)
86(5)
6.4 Remarks on the Heuristics on N(λ) and ζA (S)
91(1)
6.5 Notes
92(1)
7 The Spectral Zeta Function
93(18)
7.1 Robert's Construction of ζA by Complex Powers
93(3)
7.2 The Meromorphic Continuation of ζA via the Parametrix Approximation of e-tA
96(10)
7.3 The Ichinose-Wakayama Theorem
106(4)
7.4 Notes
110(1)
8 Some Properties of the Eigenvalues of Qw(α,β) (x, D)
111(10)
8.1 The Ichinose and Wakayama Bounds
112(3)
8.2 A Better Upper-Bound for the Lowest Eigenvalue
115(5)
8.3 Notes
120(1)
9 Some Tools from the Semiclassical Calculus
121(28)
9.1 The Semiclassical Calculus
121(8)
9.2 Decoupling a System
129(11)
9.3 Some Estimates for Semiclassical Operators
140(3)
9.4 Some Spectral Properties of Semiclassical GPDOs
143(6)
10 On Operators Induced by General Finite-Rank Orthogonal Projections
149(12)
10.1 Reductions by a Finite-Rank Orthogonal Projection
149(6)
10.2 Semiclassical Reduction by a Finite-Rank Orthogonal Projection
155(6)
11 Energy-Levels, Dynamics, and the Maslov Index
161(30)
11.1 Introducing the Dynamics
161(12)
11.2 The Maslov Index
173(16)
11.3 Notes
189(2)
12 Localization and Multiplicity of a Self-Adjoint Elliptic 2x2 Positive NCHO in Rn
191(48)
12.1 The Set 22 and its Semiclassical Deformation
192(14)
12.2 Localization and Multiplicity of the Spectrum in the Scalar Case
206(21)
12.3 Localization and Multiplicity of Spec(Aw2(x, D)), with A2e2s2
227(7)
12.4 Localization and Multiplicity of Spec(Qw(α,β) (x, D))
234(3)
12.5 Localization and Multiplicity of Spec(A2w(x, D)), with A2 e 22\2s2
237(1)
12.6 Notes
238(1)
Appendix
239(6)
A.1 Almost-Analytic Extension and the Dyn' kin-Helffer-Sjostrand Formula
239(6)
Main Notation
245(4)
B.1 General Notation
245(1)
B.2 Symbol, Function and Operator Spaces
246(1)
B.3 The Spectral Counting Function and the Spectral ζ-Function
247(1)
B.4 Dynamical Quantities and Assumptions
247(1)
B.5 Classes of Systems
247(2)
References 249(4)
Index 253
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