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Quasiclassical Asymptotics and Coherent States for Bounded Discrete Spectra

We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V_σ(x) = -|V_{0}| |x|^{-σ}, 0 < σ\leq 2. For these potentials the quasiclassical approximation for n -> \infty predicts quantized energy levels e_σ(n) of a bounded spectrum varying as e_σ(n) ~ -n^{-2σ/(2-σ)}. 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 σin the range 0<σ\leq 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 k>l.