Recovering of two-point boundary conditions by finite set of eigenvalues of boundary value problems for higher order differential equations

Abstract

The recovering of boundary conditions for higher order differential equations by some set of spectra is difficult because of two facts. First, opposite to second order differential equations, there are not triangle transformation operators for higher order differential equations. Second, non-separable boundary conditions give additional analytic problems while recovering them by the set of spectra. In the present work we provide a new way of normalizing boundary conditions, which is adapted for further recovering by some set of spectra of boundary value problems. In other words, before posing the issue by which data the boundary conditions can be recovered, one should first reduce them to a canonical form. Then, basing on an assumed canonical form, a system of boundary value problems is to be chosen and by the their spectra boundary conditions are to recovered. We propose an algorithm of recovering two-point boundary conditions in a boundary value problem for higher order differential equations. As an additional information, a finite set of eigenvalues of special boundary value problems serve. According the terminology by V.A. Sadovnichii, such problems are called canonical problems.