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.
Author(s): Alberto Parmeggiani (auth.)
Series: Lecture Notes in Mathematics 1992
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2010
Language: English
Pages: 260
Tags: Partial Differential Equations;Global Analysis and Analysis on Manifolds;Theoretical, Mathematical and Computational Physics
Front Matter....Pages i-xi
Introduction....Pages 1-5
The Harmonic Oscillator....Pages 7-13
The Weyl–Hörmander Calculus....Pages 15-53
The Spectral Counting Function N (λ) and the Behavior of the Eigenvalues: Part 1....Pages 55-66
The Heat-Semigroup, Functional Calculus and Kernels....Pages 67-77
The Spectral Counting Function N(λ) and the Behavior of the Eigenvalues: Part 2....Pages 79-92
The Spectral Zeta Function....Pages 93-110
Some Properties of the Eigenvalues of $$ Q_{\left( {\alpha ,\beta } \right)}^{\rm w} { (x,D)}$$ ....Pages 111-120
Some Tools from the Semiclassical Calculus....Pages 121-147
On Operators Induced by General Finite-Rank Orthogonal Projections....Pages 149-159
Energy-Levels, Dynamics, and the Maslov Index....Pages 161-190
Localization and Multiplicity of a Self-Adjoint Elliptic 2×2 Positive NCHO in $$\mathbb{R}^n$$ ....Pages 191-238
Back Matter....Pages 239-260
