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.

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