Characterization of the spectrum of irregular boundary value problem for the Sturm-Liouville operator

We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L2(0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator. In the present paper, we consider the eigenvalue problem for the SturmLiouvulle equation u − q(x)u + λu = 0 (1) on the interval (0, π) with the boundary conditions u(0) + (−1)u(π) + bu(π) = 0, u(0) + (−1)u(π) = 0, (2) where b is a complex number, θ = 0, 1, and the function q(x) is an arbitrary complex-valued function of the class L2(0, π). Denote by c(x, μ), s(x, μ) (λ = μ) the fundamental system of solutions to (1) with the initial conditions c(0, μ) = s(0, μ) = 1, c(0, μ) = s(0, μ) = 0. The following identity is well known c(x, μ)s(x, μ)− c(x, μ)s(x, μ) = 1. (3) Simple calculations show that the characteristic equation of (1), (2) can be reduced to the form ∆(μ) = 0, where ∆(μ) = c(π, μ)− s(π, μ) + (−1)bs(π, μ). (4) The characteristic determinant ∆(μ) of problem (1), (2), given by (4), is referred to as the characteristic determinant corresponding to the triple (b, θ, q(x)). Throughout the following the symbol ||f || stands for 1 ||f ||L2(0,π), < q >= 1 π ∫ π 0 q(x)dx. By Γ(z, r) we denote the disk of radius r centered at a point z. By PWσ we denote the class of entire functions f(z) of exponential type ≤ σ such that ||f(z)||L2(R) <∞, and by PW − σ we denote the set of odd functions in PWσ. The following two assertions provide necessary and sufficient conditions to be satisfied by the characteristic determinant ∆(μ). Theorem 1. If a function ∆(μ) is the characteristic determinant corresponding to the triple (b, θ, q(x)), then ∆(μ) = (−1)b sinπμ μ + f(μ) μ , where f(μ) ∈ PW π . Proof. Let e(x, μ) be a solution to (1) satisfying the initial conditions e(0, μ) = 1, e(0, μ) = iμ, and let K(x, t), K(x, t) = K(x, t) + K(x,−t), and K(x, t) = K(x, t) − K(x,−t) be the transformation kernels [1] that realize the representations e(x, μ) = e + ∫ x