Quantum Systems at the Brink: Existence of Bound States, Critical Potentials and Dimensionality

TL;DR

This paper establishes precise conditions for the existence of zero energy bound states in quantum systems, revealing a critical dimension at four and how potential behavior influences spectral properties.

Contribution

It provides necessary and sufficient criteria for zero energy bound states in Schrödinger operators, highlighting the role of dimension and potential asymptotics.

Findings

Zero energy bound states depend on dimension and potential decay.

Dimension four is a critical threshold for spectral phase transition.

Criteria differentiate existence and non-existence of zero energy states.

Abstract

One of the crucial properties of a quantum system is the existence of bound states. While the existence of eigenvalues below zero, i.e., below the essential spectrum, is well understood, the situation of zero energy bound states at the edge of the essential spectrum is far less understood. We present necessary and sufficient conditions for Schr\"odinger operators to have a zero energy bound state. Our sharp criteria show that the existence and non-existence of zero energy ground states depends strongly on the dimension and the asymptotic behavior of the potential. There is a spectral phase transition with dimension four being critical.