Estimates of 1D resonances in terms of potentials

TL;DR

This paper provides estimates for the sum of negative powers of resonances and eigenvalues of 1D Schrödinger operators with compactly supported potentials, using harmonic analysis techniques.

Contribution

It introduces new bounds relating resonances and eigenvalues to the potential's norm and support size, employing harmonic analysis and Carleson measures.

Findings

Bound on sum of negative powers of resonances and eigenvalues

Estimates depend on potential norm and support diameter

Uses harmonic analysis and Carleson measures techniques

Abstract

We discuss resonances for Schr\"odinger operators with compactly supported potentials on the line and the half-line. We estimate the sum of the negative power of all resonances and eigenvalues in terms of the norm of the potential and the diameter of its support. The proof is based on harmonic analysis and Carleson measures arguments.