A New Class of Schr\"odinger Operators without Positive Eigenvalues

TL;DR

This paper proves that certain Schrödinger operators lack positive eigenvalues by demonstrating that eigenfunctions must decay sub-exponentially, and identifying potential classes where this decay condition cannot hold.

Contribution

It introduces a new approach using a different conjugate operator to establish the absence of positive eigenvalues for specific Schrödinger operators.

Findings

Eigenfunctions decay sub-exponentially for a class of potentials

Certain potentials prevent the decay condition, implying no positive eigenvalues

Extends previous methods with a novel conjugate operator approach

Abstract

Following the proof given by Froese and Herbst in [FH82] with another conjugate operator, we show for a class of real potential that possible eigenfunction of the Schr\"odinger operator has to decay sub-exponentially. We also show that, for a certain class of potential, this bound can not be satisfied which implies the absence of strictly positive eigenvalues for the Schr\"odinger operator.