Quasiclassical asymptotics and coherent states for bounded discrete spectra

Abstract

We consider discrete spectra of bound states for nonrelativistic motion in attractive potentials \documentclass[12pt]{minimal}\begin{document}$V_{\sigma }(x) = -|V_{0}|\, |x|^{-\sigma },\, 0<\sigma \le 2$\end{document}Vσ(x)=−|V0||x|−σ,0<σ≤2. For these potentials the quasiclassical approximation for n → ∞ predicts quantized energy levels \documentclass[12pt]{minimal}\begin{document}$e_{\sigma }(n)$\end{document}eσ(n) of a bounded spectrum varying as \documentclass[12pt]{minimal}\begin{document}$e_{\sigma }(n) \sim -n^{-2\sigma /(2-\sigma )}$\end{document}eσ(n)∼−n−2σ/(2−σ). We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behavior. 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 σ in the range 0 < σ ⩽ 2/3 we present exact implementations of such states for the parametrization σ = 2(k − l)/(3k − l) with k and l positive integers satisfying \documentclass[12pt]{minimal}\begin{document}$k>l$\end{document}k>l.