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Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction: 1992 2010th Edition
Author(s): Alberto Parmeggiani (Author)
- Publisher: Springer
- Publication Date: 22 April 2010
- Edition: 2010th
- Language: English
- Print length: 272 pages
- ISBN-10: 9783642119217
- ISBN-13: 9783642119217
Book Description
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.
Editorial Reviews
Review
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)
From the Back Cover
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.
