Spectrum of One Sturm-Liouville Type Problem on Two Disjoint Intervals (original) (raw)
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We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (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 Sturm-Liouvulle equation u ′′ − q(x)u + λu = 0 (1) on the interval (0, π) with the boundary conditions u ′ (0) + (−1) θ u ′ (π) + bu(π) = 0, u(0) + (−1) θ+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 L 2 (0, π). Denote by c(x, µ), s(x, µ) (λ = µ 2) 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