Spectral geometry of power-law potentials in quantum mechanics

It is supposed that a single particle moves in ${\mathrm{openR}}^3$ in an attractive central power-law potential $V^{\left(q\right)}$(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=-Δ+$V^{\left(q\right)}$ depend on the power parameter q? Pure power-law potentials have the elementary property that, for p<q, $V^{\left(q\right)}$(r) is a convex transformation of $V^{\left(p\right)}$(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’=-Δ+, $A^{\left(q\right)}$∈openR. This geometrical approach greatly simplifies the description of the spectra and also facilitates the construction of some general eigenvalue bounds and approximation formulas.