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 $$r = \sqrt {\left( {1 - \alpha } \right)/\left( {1 + \alpha } \right)} $$ 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).

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