Eigenstates with Infinite Position Moments

TL;DR

This paper establishes precise conditions under which Schrödinger operators possess zero-energy bound states with bounded moments, extending previous mathematical results in quantum spectral theory.

Contribution

It provides necessary and sufficient criteria for zero-energy bound states with bounded moments, advancing the understanding of spectral properties at the spectrum threshold.

Findings

Derived conditions for zero-energy bound states with bounded moments.

Extended prior results to include higher moment bounds.

Clarified the spectral behavior at the essential spectrum threshold.

Abstract

We prove necessary and sufficient conditions for the Schr\"odinger operators to have zero-energy bound states at the threshold of the essential spectrum such that they have bounded $k$-th moment. This result is the extension of the results published in D. Hundertmark, M. Jex, and M. Lange [Forum Mathematics, Sigma 11(2023)].