Spectral geometry of power-law potentials in quantum mechanics.

Abstract

It is supposed that a single particle moves in ${\mathrm{openR}}^{3}$ in an attractive central power-law potential ${V}^{(q)}$(r)=sgn(q)${r}^{q}$, q>-2, and obeys nonrelativistic quantum mechanics. This paper is concerned with the question: How do the discrete eigenvalues ${E}_{\mathrm{nl}}$(q) of the Hamiltonian H=-\ensuremath{\Delta}+${V}^{(q)}$ depend on the power parameter q? Pure power-law potentials have the elementary property that, for p<q, ${V}^{(q)}$(r) is a convex transformation of ${V}^{(p)}$(r). This simple fact makes it possible to use ``kinetic potentials'' to construct a global geometrical theory for the spectrum of H and also for more general operators of the form H'=-\ensuremath{\Delta}+, ${A}^{(q)}$\ensuremath{\in}openR. This geometrical approach greatly simplifies the description of the spectra and also facilitates the construction of some general eigenvalue bounds and approximation formulas.