Norm estimates of complex symmetric operators applied to quantum systems

TL;DR

This paper explores the application of complex symmetric operator theory to quantum systems, providing new norm estimates and methods for analyzing Schrödinger operators with potential implications for quantum physics research.

Contribution

It introduces a formula for the norm of compact complex symmetric operators and demonstrates its usefulness in quantum mechanical problems.

Findings

Sharp estimates on exponential decay of resolvent and density matrix

New methods for evaluating resolvent norm in complex scaling theory

Potential applications in analyzing Schrödinger operators with spectral gaps

Abstract

This paper communicates recent results in theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schr\"odinger operators. In particular, we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schr\"odinger operators with spectral gaps. Second, we provide new ways of evaluating the resolvent norm for Schr\"odinger operators appearing in the complex scaling theory of resonances.