Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

1. Front Matter

   Pages i-xi

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2. Introduction

   • Alberto Parmeggiani

   Pages 1-5

3. The Harmonic Oscillator

   • Alberto Parmeggiani

   Pages 7-13

4. The Weyl–Hörmander Calculus

   • Alberto Parmeggiani

   Pages 15-53

5. The Spectral Counting Function N(λ) and the Behavior of the Eigenvalues: Part 1

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   Pages 55-66

6. The Heat-Semigroup, Functional Calculus and Kernels

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   Pages 67-77

7. The Spectral Counting Function N(λ) and the Behavior of the Eigenvalues: Part 2

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   Pages 79-92

8. The Spectral Zeta Function

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   Pages 93-110

9. Some Properties of the Eigenvalues of \( Q_{\left( {\alpha ,\beta } \right)}^{\rm w} { (x,D)}\)

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   Pages 111-120

10. Some Tools from the Semiclassical Calculus

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    Pages 121-147

11. On Operators Induced by General Finite-Rank Orthogonal Projections

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    Pages 149-159

12. Energy-Levels, Dynamics, and the Maslov Index

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    Pages 161-190

13. Localization and Multiplicity of a Self-Adjoint Elliptic 2×2 Positive NCHO in \(\mathbb{R}^n\)

    • Alberto Parmeggiani

    Pages 191-238

14. Back Matter

    Pages 239-260

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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 (2003–2007) “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.

• calculus
• differential calculus
• dynamics
• eigenvalue
• operator
• quantization
• partial differential equations

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)

• Department of Mathematics, University of Bologna, Bologna, Italy

  Alberto Parmeggiani

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