Identification of the Domain of the Sturm–Liouville Operator on a Star Graph

Abstract

This article is devoted to the unique recovering of the domain of the Sturm–Liouville operator on a star graph. The domain of the Sturm–Liouville operator is uniquely identified from the set of spectra of a finite number of specially selected canonical problems. In the general case, the domain of the definition of the original operator can be specified by integro-differential linear forms. In the case when the domain of the Sturm–Liouville operator on a star graph corresponds to the boundary value problem, it is sufficient to choose only finite parts of the spectra of canonical problems for a unique identification of the boundary form. Moreover, the above statement is valid only for a symmetric star graph.

Variational Estimation Methods for Sturm–Liouville Problems

“…Kernel of the operator, properties of the first eigenvalue and oscillation properties of eigenfunctions to the nonlocal problem were expressed. Additionally, the solution of the Cauchy problem for the SL equation on a star graph was constructed in [24].…”

“…which is satisfied by the solutions of (22). Let us consider the set V a,b = u|u ∈ V a,b verifies (24) . For I given, let µ(a, b) = inf u∈V J(u).…”