Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction (Lecture Notes in Mathematics, Vol. 1992) (Lecture Notes in Mathematics, 1992)

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. 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. 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 Alberto Parmeggiani Includes Bibliographical References (p. 249-251) And Index.