Spectral projection, residue of the scattering amplitude, and Schrodinger group expansion for barrier-top resonances

TL;DR

This paper analyzes barrier-top resonances for semiclassical Schrödinger operators, providing resolvent estimates, spectral projection expansions, and residue calculations, which enhance understanding of scattering phenomena and time evolution in quantum systems.

Contribution

It introduces a semiclassical expansion of spectral projections and computes the residue of the scattering amplitude at barrier-top resonances, advancing spectral analysis techniques.

Findings

Spectral projection admits a semiclassical expansion in powers of h.

Explicit leading term of the spectral projection is derived.

Residue of the scattering amplitude at resonances is computed.

Abstract

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrodinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrodinger group in terms of these resonances.