Generalized comparison theorems in quantum mechanics

This paper is concerned with the discrete spectra of Schrödinger operators H = −∆ + 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 Hl : (i) an optimized lower bound when the potential is a sum of terms V (r) = V (r) + V (r) , and the bottoms of the spectra of −∆+V (r) and −∆+V (r) in Hl are known, and (ii) a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials V (r) and V (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.