Quantitative bounds versus existence of weakly coupled bound states for Schr\"odinger type operators

TL;DR

This paper explores the relationship between the existence of weakly coupled bound states and quantitative bounds on their number for Schrödinger-type operators, revealing a complementary nature in different dimensions.

Contribution

It establishes a connection between the existence of weakly coupled bound states and semi-classical bounds for a broad class of Schrödinger-type operators, highlighting their complementary behavior.

Findings

Weakly coupled bound states exist only in low dimensions.

Semi-classical bounds are valid when weakly coupled bound states do not exist.

The phenomena are shown to be complementary across different dimensions.

Abstract

It is well-known that for usual Schroedinger operators weakly coupled bound states exist in dimensions one and two, whereas in higher dimensions the famous Cwikel-Lieb-Rozenblum bound holds. We show for a large class of Schr\"odinger-type operators with general kinetic energies that these two phenomena are complementary. In particular, we explicitly get a natural semi-classical type bound on the number of bound states precisely in the situation when weakly coupled bound states exist not.