A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure

• 成果类型: Article
• 署名作者: Gesztesy, F; Simon, B
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2661393
• 发表日期: 2000
• 页码: 593-643
• 关键词: scattering; line

摘要

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 less than or equal to infinity. A is related to the Weyl-Titchmarsh m-function via m(-kappa (2)) = -kappa-integral (a)(0) A(alpha )e(-2 alpha kappa) d alpha +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 rho: A(alpha) = -2 integral (infinity)(-infinity)lambda-(1/2) sin(2 alpha root lambda) d rho(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 < . 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.