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

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 2 −1 [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.