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

Boundary value problems

Ashraf D Orujov¹ 

[1] Department of Elementary Education, Cumhuriyet University, Sivas, Turkey

关键词: periodic;    spectrum;    resolvent;    eigenvalue;    spectral singularity;    34L05;    47E05;   

DOI  :  10.1186/s13661-015-0380-y

学科分类：数学（综合）

来源: SpringerOpen

【 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$\lambda=\pm\frac{n}{2}$,n∈N$n\in\mathbb{N}$.

【 授权许可】

CC BY