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 $\ell_{\lambda}(y)=y^{\prime \prime}+q_{1}(x)y^{\prime}+ [ \lambda^{2}+\lambda q_{2}(x)+q_{3}(x) ] y$ and the boundary condition $y^{\prime}(0)-hy(0)=0$ are investigated in the space $L_{2}(\mathbb{R} ^{+})$ . 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 $(-\infty,+\infty)$ . 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 $(-\infty,+\infty)$ . Eigenvalues and spectral singularities with sufficiently large modulus are simple and lie near the points $\lambda=\pm\frac{n}{2}$ , $n\in\mathbb{N}$ .