The Trace Formula for Schrödinger Operators from Infinite Dimensional Oscillatory Integrals

Abstract

The theory of infinite dimensional oscillatory integrals by finite dimensional approximations is shown to provide new information on the trace formula for Schrodinger operators. In particular, the explicit computation of contributions given by constant and non constant periodic orbits, for potentials which are quadratic plus a bounded nonlinear part, is provided. The heat semigroup as well as the Schrodinger group are discussed and it is shown in particular that their singular supports are contained in an explicit countable set independent of the bounded part of the potential.

Theory and Applications of Infinite Dimensional Oscillatory Integrals

“…The first results were obtained in [16] and further developed in [65] [6] (see also e.g. [3]- [5]). Up to now, only the case where the phase function Φ is of the form (21) has been handled rigorously.…”

“…Other interesting applications are the solution of the Schrödinger equation with a magnetic field [7], the trace formula for the Schrödinger group [4,5] (which includes a rigorous proof of "Gutzwiller's trace formula", of basic importance in the study of quantum chaos, and with interesting connections with number theory), the dynamics of Dirac systems [49] and of quantum open systems [9,39].…”

Feynman path integrals for polynomially growing potentials

“…In fact a heuristic stationary phase argument, in analogy with the behavior of the corresponding finite dimensional oscillatory integrals, shows that the paths contributing to the integral should be those for which the classical action S t is stationary: these are exactly the solutions of the Hamilton equations. Such a consideration and formula (2) are only heuristic: indeed neither the "infinite dimensional Lebesgue measure" D , nor the normalization constant are well defined.…”

“…They can also prove that, for the class of function considered in [6], the infinite dimensional oscillatory integral can be explicitly computed by means of the Parseval type formula proposed by Albeverio and Høegh-Krohn. Such an approach allows a rigorous implementation of an infinite dimensional version of the stationary phase method and was further developed in [4,2] in connection with the study of the asymptotic behaviour of the integral in the limith ↓ 0. Unfortunately, none of the existing approaches can give a rigorous mathematical definition of the Feynman formula (2) for polynomially growing potentials.…”

“…Such an approach allows a rigorous implementation of an infinite dimensional version of the stationary phase method and was further developed in [4,2] in connection with the study of the asymptotic behaviour of the integral in the limith ↓ 0. Unfortunately, none of the existing approaches can give a rigorous mathematical definition of the Feynman formula (2) for polynomially growing potentials. In particular the perturbation V to the harmonic oscillator potential considered by Albeverio, Høegh-Krohn and Ito has to belong to the class of Fourier transforms of measures, so that is bounded.…”