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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Methods of Mathematical Physics

J. Math. Phys. 53, 023501 (2012); http://dx.doi.org/10.1063/1.3679660 (22 pages)

Local trace formulae and scaling asymptotics for general quantized Hamiltonian flows

Roberto Paoletti

Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi 53, 20125 Milano, Italy

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(Received 23 March 2011; accepted 4 January 2012; published online 1 February 2012)

A class of trace formulae in the Toeplitz quantization of Hamiltonian flows on compact symplectic manifolds is given a local intepretation in terms of certain scaling asymptotics near a symplectic fixed locus; these are reminiscent of the near diagonal asymptotics of equivariant Szegö kernels. In spectral theory, the distributional trace of a positive elliptic operator encapsulates information on the asymptotic distribution of its eigenvalues; a trace formula relates the singularities of the trace to Poincaré data of an appropriate Hamiltonian flow. This leads to the asymptotics of certain smoothing kernels, related to the Fourier transform of the trace of a wave operator. In Toeplitz quantization, we consider analogues of these, with the wave operator replaced by a quantized Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz operators. We study the concentration of these smoothing kernels near the fixed loci of the linearized dynamics, and specifically how the Poincaré map controls their scaling asymptotics. This generalizes previous results on holomorphic flows.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. PRELIMINARIES AND NOTATION
  3. PROOF OF THEOREM 1.1

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KEYWORDS and PACS

Keywords

eigenvalues and eigenfunctions, elliptic equations, Fourier transforms, Poincare mapping

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3679660

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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