© 2010

Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

  • One of the main features is the use of techniques that are very powerful and flexible.

  • The approach could be used also for other problems.

  • Another feature constists of an exposition of different results that are scattered throughout the literature.


Part of the Lecture Notes in Mathematics book series (LNM, volume 1992)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Alberto Parmeggiani
    Pages 1-5
  3. Alberto Parmeggiani
    Pages 7-13
  4. Alberto Parmeggiani
    Pages 15-53
  5. Alberto Parmeggiani
    Pages 67-77
  6. Alberto Parmeggiani
    Pages 93-110
  7. Alberto Parmeggiani
    Pages 121-147
  8. Alberto Parmeggiani
    Pages 161-190
  9. Back Matter
    Pages 239-260

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.


calculus differential calculus dynamics eigenvalue operator quantization

Authors and affiliations

  1. 1.Department of MathematicsUniversity of BolognaBolognaItaly

Bibliographic information

Industry Sectors
Energy, Utilities & Environment
Finance, Business & Banking


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)