Parmeggiani A. Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

Berlin: Springer, 2010. — 273 p.This volume describes the spectral theory of the Weyl quantization of systems of polynomials in phase-space variables, modelled after the harmonic oscillator. The main technique used is pseudodifferential calculus, including global and semiclassical variants. The main results concern the meromorphic continuation of the spectral zeta function associated with the spectrum, and the localization (and the multiplicity) of the eigenvalues of such systems, described in terms of “classical” invariants (such as the periods of the periodic trajectories of the bicharacteristic flow associated with the eiganvalues of the symbol). The book utilizes techniques that are very powerful and flexible and presents an approach that could also be used for a variety of other problems. It also features expositions on different results throughout the literature.Front Matter
Introduction
The Harmonic Oscillator
The Weyl–Hörmander Calculus
The Spectral Counting Function N (λ) and the Behavior of the Eigenvalues: Part 1
The Heat-Semigroup, Functional Calculus and Kernels
The Spectral Counting Function N(λ) and the Behavior of the Eigenvalues: Part 2
The Spectral Zeta Function
Some Properties of the Eigenvalues of $$ Q_{\left( {\alpha ,\beta } \right)}^{\rm w} { (x,D)}$$
Some Tools from the Semiclassical Calculus
On Operators Induced by General Finite-Rank Orthogonal Projections
Energy-Levels, Dynamics, and the Maslov Index
Localization and Multiplicity of a Self-Adjoint Elliptic 2×2 Positive NCHO in $$\mathbb{R}^n$$
Back Matter