Sturm-Liouville operator with an incomplete system of eigenfunctions and having a finite incompleteness defect

Abstract

In this paper, we study one Sturm-Liouville problem with a nonlinear boundary condition that depends on a parameter. Depending on the value of the boundary parameter, eigenvalue problems can arise with complete, incomplete, and overflowing systems of eigenfunctions in the functional space L2[0, 1]. An example of a Sturm-Liouville operator with an incomplete system of eigenfunctions is given and the finite defect of incompleteness is calculated. Also it is given, the system of eigenfunctions of which has an excess equal to minus one. The interval of variation of the boundary parameter is found for which the system of eigenfunctions of the Sturm-Liouville operator is complete.