On the basis property of the root function systems of regular boundary value problems for the Sturm-Liouville operator

We consider the nonselfadjoint Sturm-Liouville operator with regular but not strongly regular boundary conditions. We examine the basis property of the root function system of the mentioned operator. In the present paper we study eigenvalue problems for the nonselfadjoint Sturm-Liouville operator Lu = u′′ − q(x)u (1) defined on the interval (0, 1), where q(x) is an arbitrary complex-valued function of the class L1(0, 1). Our main purpose is to investigate the basis property of the root function system of operator (1) with regular but not strongly regular boundary conditions. Author’s interest to this problem was stimulated by the papers of V.A. Il’in [1-3]. By φ(x, μ), ψ(x, μ) we denote the fundamental for μ 6= 0 system of solutions to the equation u′′ − q(x)u + μu = 0 determined by the initial conditions φ(0, μ) = ψ(0, μ) = 1, φx(0, μ) = iμ, ψ′ x(0, μ) = −iμ. It is well known that the functions φ(x, μ) and ψ(x, μ) satisfy the integral equations φ(x, μ) = e + 1 μ ∫ x 0 sinμ(x− t)q(t)φ(t, μ)dt, (2) ψ(x, μ) = e−iμx + 1 μ ∫ x 0 sinμ(x− t)q(t)ψ(t, μ)dt, (2′)