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

TL;DR

This paper investigates bounds on eigenvalues of Schrödinger operators with complex potentials, proving a conjecture for radial potentials and providing partial results for general potentials, including bounds for positive eigenvalues.

Contribution

It proves Laptev and Safronov's conjecture for radial potentials and offers partial results for general potentials, advancing understanding of eigenvalue bounds in complex Schrödinger operators.

Findings

Confirmed the conjecture for radial potentials when 0<γ<ν/2.

Provided near-counterexamples for general potentials when 1/2<γ<ν/2.

Established bounds for positive eigenvalues.

Abstract

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schr\"odinger operator $-\Delta+V$ in $L^2(\mathbb R^\nu)$ with complex potential has absolute value at most a constant times $\|V\|_{\gamma+\nu/2}^{(\gamma+\nu/2)/\gamma}$ for $0<\gamma\leq\nu/2$ in dimension $\nu\geq 2$. We prove this conjecture for radial potentials if $0<\gamma<\nu/2$ and we `almost disprove' it for general potentials if $1/2<\gamma<\nu/2$. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.