Schroedinger Operators With Few Bound States

Abstract

We show that whole-line Schrödinger 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 Schrödinger 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 $V\equiv 0$.

10 pages

References in corpus (3)

• Sum Rules and the Szego Condition for Orthogonal Polynomials on the Real Line
• Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign
• Schrödinger Operators with Many Bound States

Cited by in corpus (6)

• Bounds on the Discrete Spectrum of Lattice Schrödinger Operators
• Criteria for embedded eigenvalues for discrete Schrödinger operators
• Bound States of Discrete Schroedinger Operators with Super-Critical Inverse Square Potentials
• Discrete and embedded eigenvalues for one-dimensional Schr"odinger operators
• Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality
• Stahl--Totik regularity for continuum Schrödinger operators