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A mapping method in inverse Sturm-Liouville problems with singular potentials

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Abstract

In the space L 2[0, π], the Sturm-Liouville operator L D(y) = −y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L 2[0, π], i.e., q ∈ W −12 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L D is solved in the subspace of odd real functions σ(π/2 − x) = −σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    A. M. Savchuk

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  1. A. M. Savchuk
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Correspondence to A. M. Savchuk.

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Original Russian Text © A.M. Savchuk, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 261, pp. 243–248.

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Savchuk, A.M. A mapping method in inverse Sturm-Liouville problems with singular potentials. Proc. Steklov Inst. Math. 261, 237–242 (2008). https://doi.org/10.1134/S0081543808020181

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