A new approach to inverse spectral theory II: general real potentials and the connection to the spectral measure

• Autores: Barry Simon, Fritz Gesztesy
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 152, Nº 2, 2000, págs. 593-643
• Idioma: inglés
• DOI: 10.2307/2661393
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• Resumen

  • We continue the study of the A-amplitude associated to a half-line Schr¿odinger operator, - d2 dx2 +q in L2((0, b)), b = 8. A is related to the Weyl- Titchmarsh m-function via m(-.2) = -.-Ra 0 A(a)e-2a. da+O(e-(2a-e).) for all e > 0. We discuss five issues here. First, we extend the theory to general q in L1((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 .: A(a) = -2 R8 -8 .-1 2 sin(2av.) d.(.) (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 < 8. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.