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A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure | | Fritz Gesztesy
; Barry Simon
; | | Date: |
30 Sep 1998 | | Journal: | Ann. of Math. (2) 152 (2000), no. 2, 593-643 | | Subject: | Spectral Theory | math.SP | | Abstract: | We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-kappa^2) =-kappa - int_0^a A(alpha) e^{-2alphakappa} dalpha +O(e^{-(2a -epsilon)kappa}) for all epsilon > 0. We discuss five issues here. First, we extend the theory to general q in L^1 ((0,a)) for all a, including q’s which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure
ho: A(alpha) = -2int_{-infty}^infty lambda^{-frac12} sin (2alpha sqrt{lambda})d
ho(lambda) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b | | Source: | arXiv, math.SP/9809182 | | Services: | Forum | Review | PDF | Favorites | |
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