Remarks on eigenvalue estimates and semigroup domination

TL;DR

This paper reviews recent advances in spectral estimates for magnetic Schrödinger operators, showing how non-magnetic inequalities imply magnetic ones with potential constant adjustments, enhancing understanding of eigenvalue bounds.

Contribution

It establishes a general framework linking magnetic and non-magnetic Lieb-Thirring inequalities, demonstrating that magnetic bounds can be derived from non-magnetic ones in an abstract setting.

Findings

Magnetic and non-magnetic eigenvalue bounds are closely related.

Non-magnetic Lieb-Thirring inequalities imply magnetic counterparts.

Constants in magnetic bounds may be larger than in non-magnetic cases.

Abstract

We present an overview over recent results concerning semi-classical spectral estimates for magnetic Schroedinger operators. We discuss how the constants in magnetic and non-magnetic eigenvalue bounds are related and we prove, in an abstract setting, that any non-magnetic Lieb-Thirring-type inequality implies a magnetic Lieb-Thirring-type inequality with possibly a larger constant.