V. Adamjan, H. Langer and M. Langer,
A spectral theory for a λ-rational Sturm–Liouville problem,
J. Differential Equations 171 (2001), 315–345
Abstract:
We consider the regular Sturm–Liouville problem
y'' – py + (λ + q/(u – λ))y = 0,
which contains the eigenvalue parameter rationally.
Under certain assumptions on p, q, and u it is shown that the
spectrum of the problem consists of a continuous component (the range of the function u),
discrete eigenvalues, and possibly a finite number of embedded eigenvalues.
In the considered situation the continuous spectrum
is absolutely continuous, and explicit formulas for the spectral density
and the corresponding Fourier transform are given.