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A new approach to inverse spectral theory. II: General real potentials and the connection to the spectral measure. (English) Zbl 0983.34013

As in the previous paper by the second author [ibid. 150, No. 3, 1029-1057 (1999; Zbl 0945.34013)] the authors consider the \(A\)-amplitude and the Weyl-Titchmarsh \(m\)-function for the radial Schrödinger equation on a finite interval or on the half-line with a real-valued locally integrable potential. An asymptotic relation between \(A\) and \(m\) is investigated in more detail, a relation between \(A\) and the spectral measure is obtained, a Laplace transform representation for \(m\) is given, \(m\)-functions associated with other boundary conditions are discussed, and some examples are provided where \(A\) can be computed exactly.

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
34L05 General spectral theory of ordinary differential operators
34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47A10 Spectrum, resolvent

Citations:

Zbl 0945.34013
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