Schr\"odinger operators with negative potentials and Lane-Emden densities

TL;DR

This paper establishes conditions under which Schr"odinger operators with negative potentials have positive spectra, using Hardy inequalities and Lane-Emden densities, advancing understanding of spectral properties in mathematical physics.

Contribution

It introduces new Hardy-type inequalities linked to Lane-Emden solutions, providing explicit criteria for the positivity of Schr"odinger operator spectra with negative potentials.

Findings

Spectrum of $- abla^2 + V$ is positive under certain conditions.

Ground state energy exceeds the first Dirichlet eigenvalue.

Explicit bounds involving Lane-Emden densities and Hardy inequalities.

Abstract

We consider the Schr\"odinger operator $-\Delta+V$ for negative potentials $V$, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of $-\Delta+V$ is positive, provided that $V$ is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation $-\Delta u=u^{q-1}$ (for some $1\le q< 2$). In this case, the ground state energy of $-\Delta+V$ is greater than the first eigenvalue of the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper.