2002/08/30 by Richard L. Hall, Qutaibeh D. Katatbeh
Physics and Astronomy · Mathematics · #math-ph #math.MP #quant-ph
published as J. Phys. A 35, 8727-8742 (2002) · 25 pages, 5 figures
arxiv created 2002/08/30 · arxiv updated 2009/11/30
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 Hell: (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 Hell 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.