Wavefunctions, Green's functions and expectation values in terms of spectral determinants

TL;DR

This paper develops semiclassical approximations for wavefunctions, Green's functions, and expectation values in chaotic quantum systems using spectral determinants and classical trajectories, providing a simple and trajectory-based resummation method.

Contribution

It introduces a novel semiclassical approach expressing key quantum quantities via spectral determinants and classical trajectories, differing from traditional surface of section techniques.

Findings

Spectral determinants can be used to approximate wavefunctions and Green's functions semiclassically.

Resummation methods yield formulas involving a finite number of classical trajectories.

The approach simplifies calculations by focusing on trajectory periods rather than surface sections.

Abstract

We derive semiclassical approximations for wavefunctions, Green's functions and expectation values for classically chaotic quantum systems. Our method consists of applying singular and regular perturbations to quantum Hamiltonians. The wavefunctions, Green's functions and expectation values of the unperturbed Hamiltonian are expressed in terms of the spectral determinant of the perturbed Hamiltonian. Semiclassical resummation methods for spectral determinants are applied and yield approximations in terms of a finite number of classical trajectories. The final formulas have a simple form. In contrast to Poincare surface of section methods, the resummation is done in terms of the periods of the trajectories.