Semiclassical energy formulas for power-law and log potentials in quantum mechanics

TL;DR

This paper derives semiclassical energy formulas for quantum particles in power-law and logarithmic potentials, proving monotonicity properties of certain functions to improve eigenvalue estimates in various dimensions.

Contribution

It introduces new monotonicity results for semiclassical parameters, refining eigenvalue bounds for quantum systems with power-law and log potentials.

Findings

Proves monotonicity of P_{n ext{ell}}(q) functions.

Establishes that Q(q)=Z(q)P(q) is monotone increasing for n=1.

Provides sharper eigenvalue estimates at the bottom of angular-momentum subspaces.

Abstract

We study a single particle which obeys non-relativistic quantum mechanics in R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2, then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may be represented exactly by the semiclassical expression E_{n\ell}(q) = min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) = ln(r). By writing one power as a smooth transformation of another, and using envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are monotone increasing. Recent refinements to the comparison theorem of QM in which comparison potentials can cross over, allow us to prove for n = 1 that Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q} is monotone decreasing. Thus P(q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvlaues at the bottom of each angular-momentum subspace.