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Small eigenvalues of large Hankel matrices, at β=1/2, a critical point. Comparison of theoretical prediction with numerical computations.

The smallest eigenvalues of large Hankel matrices generated by the weight w(x)= exp (- x^β), for x ∈ [0,∞), and β>0, was studied in early 2000 by Chen and Lawrence, and later 2015 by Emmart, Chen, Weems, and recently 2018, by Chen, Sikorowski, and Emmart. It can be shown that the classical moment problem associated with w(x) is indeterminate if 0<β<1/2. A characterization theorem of Berg-Chen-Ismail, states that the moment problem is indeterminate if and only if the smallest eigenvalue is bounded away from 0, for all n, including infinity. It is shown that, for β>1/2, the smallest eigenvalue tend to 0 exponentially fast, and the moment problem becomes indeterminate for 0<β<1/2. β=1/2, is a kind of critical point (borrowing from the language of phase transition). A heuristic calculation predicts a SLOW decay of the smallest eigenvalue. Intensive numerical computation on the multi-core machines from University of Messachusetts, Univeristy of Macau, and a private smallish cluster, shows that the theoretical prediction to be remarkable accurate.