Spectral Portraits in the Semi-Classical Approximation of the Sturm-Liouville Problem with a Complex Potential

Abstract

We study the problem on the limit behavior of the discrete spectrum of the Sturm-Liouville problem − ε y ″ ( x ) + P ( x λ ) y ( x ) = 0 y ( a ) = y ( b ) = 0 , provided that the physical parameter ϵ tends to zero. It is assumed that λ is the spectral parameter, the function P is polynomial on x with analytic coefficients on λ varying on a domain G in the complex λ-plane ℂ. The case P(x, λ) = p(x) - λ with complex valued function p corresponds to the usual linear spectral problem. Boundary conditions are formed by arbitrary complex numbers a, b or ±∞. We shall show that in this case the eigenvalues are concentrated along the so-called limit spectral graph as ϵ → 0. We define three type of curves, forming this graph and find the asymptotic formulae for the eigenvalue distribution along the curves of various types. For the case P(x, λ) = p(x) - λ with real polynomial p(x) these formulae coincide with well known Bohr-Sommerfeld quantization formulae. Non-self-adjoint Sturm-Liouville problems are often found in mathematical physics. We demonstrate this considering the well-known in hydromechanics Orr-Sommerfeld spectral problem.