Generalized comparison theorems in quantum mechanics
  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. 
Richard L. Hall and Qutaibeh D. Katatbeh