Schroedinger Operators With Few Bound States

TL;DR

This paper investigates the spectral properties of Schr"odinger operators with few bound states, showing conditions under which embedded singular spectrum can or cannot occur, and establishing uniqueness results for certain potentials.

Contribution

It proves that Schr"odinger operators with finitely many bound states lack embedded singular spectrum and characterizes potentials that preserve the ground state energy.

Findings

Operators with finitely many bound states have no embedded singular spectrum.

Embedded singular spectrum can occur when bound states approach the spectrum exponentially.

Potential V must be zero if it preserves the ground state energy when added or subtracted from H.

Abstract

We show that whole-line Schr\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schr\"odinger operators, $H$, with bounded positive ground states: Given a potential $V$, if both $H\pm V$ are bounded from below by the ground-state energy of $H$, then $V\equiv 0$.