On the spectrum of the quadratic pencil of differential operators with periodic coefficients on the semi-axis

Abstract

In this paper, the spectrum and resolvent of the operator L-lambda generated by the differential expression L-lambda(y) = y '' + q(1)(x)y' + [lambda(2) + lambda q(2)(x) + q(3)(x)] y and the boundary condition y'(0) - hy(0) = 0 are investigated in the space L-2(R+). Here the coefficients q(1)(x), q(2)(x), q(3)(x) are periodic functions whose Fourier series are absolutely convergent and Fourier exponents are positive. It is shown that continuous spectrum of the operator L-lambda consists of the interval (-infinity, +infinity). Moreover, at most a countable set of spectral singularities can exists over the continuous spectrum and at most a countable set of eigenvalues can be located outside of the interval (-infinity, +infinity). Eigenvalues and spectral singularities with sufficiently large modulus are simple and lie near the points lambda = +n/2, n is an element of N.