Generalized comparison theorems in quantum mechanics

Abstract

This paper is concerned with the discrete spectra of Schrödinger operators H = −Δ + 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 ℓ: (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 −Δ + V(1)(r) and −Δ + V(2)(r) in ℓ 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.