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

摘要

In this paper, the spectrum and resolvent of the operatorLλ$L_{lambda}$generated by the differential expressionℓλ(y)=y″+q1(x)y′+[λ2+λq2(x)+q3(x)]y$ell_{lambda}(y)=y^{prime prime}+q_{1}(x)y^{prime}+ [ lambda^{2}+lambda q_{2}(x)+q_{3}(x) ] y$and the boundary conditiony′(0)−hy(0)=0$y^{prime}(0)-hy(0)=0$are investigated in the spaceL2(R+)$L_{2}(mathbb{R} ^{+})$. Here the coefficientsq1(x)$q_{1}(x)$,q2(x)$q_{2}(x)$,q3(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 operatorLλ$L_{lambda}$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 pointsλ=±n2$lambda=pmrac{n}{2}$,n∈N$ninmathbb{N}$.