Preservation of logarithmic concavity by the Mellin transform and applications to the Schrödinger equation for certain classes of potentials
It is proved that the Mellin transform of a function log-concave (convex) is, after division by \(\Gamma (v+1)\), where v is the argument of the transform, itself log-concave (convex) in v. This theorem is first applied to the moments of the ground state wave function of the Schrödinger equation where the Laplacian of the central potential has a given sign, and generalized to other situations. This is used to derive inequalities linking the \(\ell\)-th derivative of the ground state wave function at the origin for angular momentum \(\ell\) and the expectation value of the kinetic energy, and applied to quarkonium physics. A generalization to higher radial excitations is shown to be plausible by using the WKB approximation. Finally new bounds on ground-state energies in power potentials are obtained. The main mathematical result on properties of the Mellin transform is contained in previous work on the total positivity of moments of polya frequency functions by \textit{S. Karlin}, \textit{F. Proschan} and \textit{R. E. Barlow} [Pac. J. Math. 11, 1023-1033 (1961; Zbl 0141.166)]. The proof given here is more straightforward and differs from that used by the above authors for a wider class of inequalities.
scientific article; zbMATH DE number 4202891
scientific article