Direct Computation of the Spectral Function

Abstract

We would like to find an explicit formula for the spectral function of the following Sturm-Liouville problem: \[ \left \{ {\begin {array}{*{20}{c}} {Lf \equiv - \frac {{{d^2}}}{{d{x^2}}}f(x) + q(x)f(x),\quad x \geq 0,} \hfill \\ {f’(0) - mf(0) = 0.} \hfill \\ \end {array} } \right .\] A simple operational calculus argument will help us obtain an explicit formula for the transmutation kernel. The expression of the spectral function is then obtained through the nonlinear integral equation found in the Gelfand-Levitan theory.