The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems

Abstract

We study the direct and inverse problems for the family of Sturm-Liouville operators, generated by fixed potential q and the family of separated boundary conditions. We prove that the union of the spectra of all these operators can be represented as a smooth surface (as the values of a real analytic function of two variables), which has specific properties. We call this function ”the eigenvalues function of the family of Sturm-Liouville operators (EVF)”. From the properties of this function we select those, which are sufficient for a function of two variables be the EVF a family of Sturm-Liouville operators.

Isospectral Dirac operators

“…The proof of this theorem for the case p, q ∈ AC[0, π] there is in the paper [18]. The detailed proof for the case p, q ∈ L 2 R [0, π] there is in [7] (see also [4-6, 8, 10, 19]).…”

Ways of Radical Change of the Image of Tax and Customs Officers

“…Harutyunyan in 2000 in a paper [HN00]. The properties of this function were studied in detail in paper [Har19a] and were applied for solving the inverse problem.…”