Lecture Notes in Mathematics
cover

Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

Authors: Parmeggiani, Alberto

  • Unites the pseudodifferential calculus, the global and the semiclassical ones
  • Utilizes powerful, flexible techniques
  • Gathers results formerly scattered throughout the literature
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About this book

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.

Reviews

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)

Table of contents (12 chapters)

Table of contents (12 chapters)
  • Introduction

    Pages 1-5

    Parmeggiani, Alberto

  • The Harmonic Oscillator

    Pages 7-13

    Parmeggiani, Alberto

  • The Weyl–Hörmander Calculus

    Pages 15-53

    Parmeggiani, Alberto

  • The Spectral Counting Function

    Pages 55-66

    Parmeggiani, Alberto

  • The Heat-Semigroup, Functional Calculus and Kernels

    Pages 67-77

    Parmeggiani, Alberto

Buy this book

eBook $44.99
price for USA in USD
  • ISBN 978-3-642-11922-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.95
price for USA in USD
  • ISBN 978-3-642-11921-7
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
1992
Copyright
2010
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-11922-4
DOI
10.1007/978-3-642-11922-4
Softcover ISBN
978-3-642-11921-7
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XII, 260
Topics