Solution of the Inverse Spectral Problem for an Impedance with IntegrableDerivative. Part 1

摘要

The inverse eigenvalue problem for a Sturm-Liouville equation in impedance formwith Dirichlet boundary conditions on a unit interval is considered. The solution of this nonlinear problem requires the investigation of a combined mapping from the logarithmic derivative of the impedance, assumed to be square integrable, to two sequences of spectral data. The first is the sequence of Dirichlet eigenvalues, shown to be locally bounded with square roots differing from the sequence of integral multiples of pi by a square summable sequence. The second sequence has as first term the mean of the logarithmic derivative of the impedance. Each term in the remainder of the sequence is the logarithm of the product of the impedance and the derivative of an eigenfunction evaluated at an endpoint. It is shown that this is a locally bounded square summable sequence. The combined map is real analytic. The asymptotics and analyticity results follow from a modified Prufer substitution.