A new approach to inverse spectral theory. I. Fundamental formalism.
成果类型：
Article
署名作者：
Simon, B
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.2307/121061
发表日期：
1999
页码：
1029-1057
关键词：
摘要：
We present a new approach (distinct from Gel'fand-Levitan) to the theorem of Borg-Marchenko that the m-function (equivalently, spectral measure) for a finite interval or half-line Schrodinger operator determines the potential. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(-kappa(2)) = -kappa - integral(0)(b) A(alpha)e(-2 alpha kappa) d alpha + O(e(-(2b-epsilon)kappa)). A on [0,a] is a function of q on [0, a] and vice-versa. A key role is played by a differential equation that A Obeys after allowing x-dependence: delta A/delta x = delta A/delta alpha + integral(0)(alpha) A(beta, x)A(alpha - beta, x) d beta. Among our new results are necessary and sufficient conditions on the m-functions for potentials q(1) and q(2) for q(1) to equal q(2) on [0, a].