Spectral properties of a nonlocal problem for a second-order differential equation with an involution

For the equation αu″(-x) - u″(x) = λu(x), ™1 < x < 1, where α ∈ (™1, 1), we study the problem with the nonlocal conditions u(™1) = 0, u′(™1) = u′(1). We show that if r = √ ( 1 - α )/( 1 + α ) is irrational, then the system of eigenfunctions is complete and minimal in L 2(™1, 1) but is not a basis. For rational r, we indicate a method for choosing associated functions for which the system of root functions of the problem is an unconditional basis in L 2(™1, 1).