Generalized comparison theorems in quantum mechanics

TL;DR

This paper develops generalized comparison theorems for the spectra of Schrödinger operators with attractive potentials, providing optimized bounds and spectral ordering criteria, with applications to power-law and Coulomb-plus-linear potentials in multiple dimensions.

Contribution

It introduces new comparison theorems and bounds for Schrödinger spectra, extending previous results to more general potentials and higher dimensions.

Findings

Optimized lower bounds for spectral bottoms in combined potentials

Generalized comparison theorem predicting spectral ordering

Sharpened energy bounds for Coulomb-plus-linear potential in N dimensions

Abstract

This paper is concerned with the discrete spectra of Schroedinger operators H = -Delta + V, where V(r) is an attractive potential in N spatial dimensions. Two principal results are reported for the bottom of the spectrum of H in each angular-momentum subspace H_{ell}: (i) an optimized lower bound when the potential is a sum of terms V(r) = V^{(1)}(r) + V^{(2)}(r), and the bottoms of the spectra of -Delta + V^{(1)}(r) and -Delta + V^{(2)}(r) in H_{ell} are known, and (ii) a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials V^{(1)}(r) and V^{(2)}(r) intersect in a controlled way. Pure power-law potentials are studied and an application of the results to the Coulomb-plus-linear potential V(r) = -a/r + br is presented in detail: for this problem an earlier formula for energy bounds is sharpened and generalized to N dimensions.