Nonlocal spectral problem for a second-order di˙erential equation with an involution

Abstract

For the spectral problem -u"(x) + au"(—x) = \u(x), —1 < x < 1, with nonlocal boundary conditions u(—1) = /3u( 1), u'(—1) = u'(1), where a € (—1,1), в2 = 1, we study the spectral properties. We show that If r = ^/(1 — a)/(1 + a) is irrational, then the system of eigenfunctions is complete and minimal іп L2(—1,1) but is not a basis. In the case of a rational number r, the root subspace of the problem consists of eigenvectors and an infinite number of associated vectors. In this case, we indicated a method for choosing associated functions that provides the system of root functions of the problem is an unconditional basis in L2 ( 1, 1).