On one class eigenvalue problem with eigenvalue parameter in the boundary condition at one end-point (original) (raw)
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hrynivÄ math.ucalgary.ca. 3 H.L. acknowledges support of the Fonds zur Fo rderung der wissenschaftlichen Forschung of Austria, Project P 12176 MAT. 4 B.N. initiated this project but unfortunately died before its completion. His research was supported by Ministry of Sciences of Croatia.
Characterization of the spectrum of irregular boundary value problem for the
2012
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