  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.
