Eigenvalue bounds for Schr\"odinger operators with complex potentials. III

TL;DR

This paper investigates the spectral properties of Schr"odinger operators with complex potentials, establishing bounds on eigenvalues' real and imaginary parts and providing quantitative relations based on the potential's $L^p$ norm.

Contribution

It introduces new bounds and asymptotic behaviors for eigenvalues of Schr"odinger operators with complex potentials, extending previous spectral analysis results.

Findings

Eigenvalues with real part tending to infinity have imaginary parts tending to zero.

Eigenvalues approaching a non-negative real number have imaginary parts in $ ext{ell}^q$ space.

Quantitative bounds relate eigenvalues' behavior to the $L^p$ norm of the potential.

Abstract

We discuss the eigenvalues $E_j$ of Schr\"odinger operators $-\Delta+V$ in $L^2(\mathbb R^d)$ with complex potentials $V\in L^p$, $p<\infty$. We show that (A) $\mathrm{Re} E_j\to\infty$ implies $\mathrm{Im} E_j\to 0$, and (B) $\mathrm{Re} E_j\to E\in [0,\infty)$ implies $(\mathrm{Im} E_j)\in\ell^q$ for some $q$ depending on $p$. We prove quantitative versions of (A) and (B) in terms of the $L^p$-norm of $V$.