Quantum transfer operators and quantum scattering

TL;DR

The paper introduces a novel method using quantized transfer operators, akin to open quantum maps, to analyze spectral properties and quantum resonances of scattering Hamiltonians near specific energies.

Contribution

It develops a new semiclassical approach employing finite-dimensional transfer operators to study quantum scattering resonances.

Findings

Quantum resonances are identified as roots of a determinant involving transfer operators.

The method encodes quantum dynamics near a fixed energy using open quantum maps.

Provides a framework for analyzing spectral properties of scattering Hamiltonians.

Abstract

These notes describe a new method to investigate the spectral properties if quantum scattering Hamiltonians, developed in collaboration with J. Sj\"ostrand and M.Zworski. This method consists in constructing a family of "quantized transfer operators" $\{M(z,h)\}$ associated with a classical Poincar\'e section near some fixed classical energy E. These operators are finite dimensional, and have the structure of "open quantum maps". In the semiclassical limit, the family $\{M(z,h)\}$ encode the quantum dynamics near the energy E. In particular, the quantum resonances of the form $E+z$, for $z=O(h)$, are obtained as the roots of $\det(1-M(z,h))=0$.