Preservation of logarithmic concavity by the Mellin transform and applications to the Schrödinger equation for certain classes of potentials
type: Item
instance of: scholarly article
title: Preservation of logarithmic concavity by the Mellin transform and applications to the Schrödinger equation for certain classes of potentials
zbMATH Open document ID: 0728.35087
DOI: 10.1007/BF02098444
author: A. K. Common
author: André Martin
author: Joachim Stubbe
published in: Communications in Mathematical Physics
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 4202891
zbMATH Keywords: Mellin transform
zbMATH Keywords: log-concave
zbMATH Keywords: ground state wave function
MaRDI profile type: Publication
cites work: Constraints on the total and kinetic energy of ground states in a class of potential models
full work available at URL: https://doi.org/10.1007/bf02098444
OpenAlex ID: W2022779734
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description: scientific article; zbMATH DE number 4202891