On the spectrum of the pencil of high order differential operators with almost periodic coefficients

Abstract

In this paper, the spectrum and the resolvent of the operator $L_{\lambda}$ which is generated by the differential expression $\ell_{\lambda }(y)=y^{(m)}+\sum_{\gamma=1}^{m} ( \sum_{k=0}^{\gamma }\lambda^{k}p_{\gamma k}(x) ) y^{(m-\gamma)}$ has been investigated in the space $L_{2}(\mathbb{R})$ . Here the coefficients $p_{\gamma k}(x)=\sum_{n=1}^{\infty }p_{\gamma kn}e^{i\alpha_{n}x}$ , $k=0,1,\ldots,\gamma-1$ ; $p_{\gamma\gamma }(x)=p_{\gamma\gamma}$ , $\gamma=1,2,\ldots,m$ , are constants, $p_{mm}\neq0$ and $p_{\gamma k}^{(\nu)}(x)$ , $\nu=0,1,2,\ldots,m-\gamma$ , are Bohr almost-periodic functions whose Fourier series are absolutely convergent. The sequence of Fourier exponents of coefficients (these are positive) has a unique limit point at +∞. It has been shown that if the polynomial $\phi (z)=z^{m}+p_{11}z^{m-1}+p_{22}z^{m-2}+\cdots+p_{m-1,m-1}z+p_{mm}$ has the simple roots $\omega_{1},\omega_{2},\ldots,\omega_{m}$ (or one multiple root $\omega_{0}$ ), then the spectrum of operator $L_{\lambda}$ is pure continuous and consists of lines $\operatorname{Re}(\lambda\omega _{k})=0$ , $k=1,2,\ldots,m$ (or of line $\operatorname{Re}(\lambda\omega_{0})=0$ ). Moreover, a countable set of spectral singularities on the continuous spectrum can exist which coincides with numbers of the form $\lambda=0$ , $\lambda _{sjn}=i\alpha_{n} ( \omega_{j}-\omega_{s} ) ^{-1}$ , $n\in \mathbb{N}$ , $s,j=1,2,\ldots,m$ , $j\neq s$ . If $\phi(z)=(z-\omega_{0})^{m}$ , then the spectral singularity does not exist. The resolvent $L_{\lambda}^{-1}$ is an integral operator in $L_{2}(\mathbb{R})$ with the kernel of Karleman type for any $\lambda\in\rho (L_{\lambda})$ .