Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra

TL;DR

This paper analyzes the asymptotic behavior of bound state spectra for specific attractive potentials and constructs coherent states that are complete and normalizable, using advanced mathematical techniques.

Contribution

It introduces a method to construct Klauder-type coherent states for discrete spectra with asymptotic energy levels, employing Hausdorff moment problems and integral transforms.

Findings

Derived asymptotic energy level formulas for potentials with 0<σ≤2.

Constructed explicit coherent states satisfying resolution of unity.

Provided exact implementations for specific σ parametrizations.

Abstract

We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V_{\sigma}(x) = -|V_{0}| |x|^{-\sigma}, 0 < \sigma \leq 2. For these potentials the quasiclassical approximation for n -> \infty predicts quantized energy levels e_{\sigma}(n) of a bounded spectrum varying as e_{\sigma}(n) ~ -n^{-2\sigma/(2-\sigma)}. We construct collective quantum states using the set of wavefunctions of the discrete spectrum taking into account this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For \sigma in the range 0<\sigma\leq 2/3 we present exact implementations of such states for the parametrization \sigma = 2(k-l)/(3k-l), with k and l positive integers satisfying k>l.