Existence, compactness, estimates of eigenvalues and $s$-numbers of a resolvent for a linear singular operator of the Korteweg-de Vries type

Abstract

In this paper, we consider a linear operator of the Korteweg-de Vries type$$ Lu =\frac{\partial u}{\partial y}+R_{2}(y)\frac{\partial^{3}u}{\partial x^{3}}+R_{1}(y)\frac{\partial u}{\partial x}+R_{0}(y)u$$initially defined on $C_{0,\pi}^{\infty} (\overline{\Omega})$, where $\overline{\Omega}=\{(x,y):-\pi\leq x\leq \pi, -\infty <y< \infty\}$. $C_{0,\pi}^{\infty}$ is a set of infinitely differentiable compactly supported function with respect to a variable $y$ and satisfying the conditions:$$u_{x}^{(i)}(-\pi,y)=u_{x}^{(i)}(\pi,y), \;\; i=0,1,2.$$With respect to the coefficients of the operator $L$ , we assume that these are continuous functions in $R(-\infty,+\infty)$ and strongly growing functions at infinity. In the paper, we proved that there exists a bounded inverse operator and found a condition that ensures the compactness of the resolvent under some restrictions on the coefficients in addition to the above conditions. Also, two-sided estimates of singular numbers ($s$-numbers) are obtained and an example is given of how these estimates allow finding estimates of the eigenvalues of the considered operator.