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A spectral theory for a λ-rational Sturm-Liouville problem

Adamjan, V. and Langer, Heinz and Langer, M. (2001) A spectral theory for a λ-rational Sturm-Liouville problem. Journal of Differential Equations, 171 (2). pp. 315-345. ISSN 0022-0396

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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.