$L^p$-Spectrum of the Schr\"odinger Operator with Inverted Harmonic Oscillator Potential

TL;DR

This paper characterizes the $L^p$-spectral properties of the Schrödinger operator with an inverted harmonic oscillator potential, providing a comprehensive spectral analysis across all $p$-values.

Contribution

It offers the first complete description of the $L^p$-spectrum for this specific quantum mechanical operator with inverted potential.

Findings

Determined the $L^p$-spectrum for all $p$ between 1 and infinity.

Identified spectral differences compared to the standard harmonic oscillator.

Provided insights into the spectral behavior of non-self-adjoint operators.

Abstract

We determine the $L^p$-spectrum of the Schr\"odinger operator with the inverted harmonic oscillator potential $V(x)=-x^2$ for $1 \leq p \leq \infty$.