On conditions for the finiteness of the spectrum of a second order differential operator with integral boundary conditions

Abstract

In this paper we study the question of the finiteness of the spectrum of a second-order differential operator generated in the space $H=L_2(0,1)$ by integral boundary conditions. We have shown that the spectrum of such an operator is either infinite or empty. Previously, this result was known only in the case of two- or three-point boundary conditions. Next, we obtained a necessary and sufficient condition for the spectrum to be empty in terms of a system of two equations for the potential $q$ and the functions $\sigma_1$ and $\sigma_2$ that define the integral boundary conditions. The left-hand side of the first of these equations for each fixed $q$ is a bilinear form with respect to $\sigma_1$ and $\sigma_2$. This allows us to resolve the indicated equation for $\sigma_1$ within a certain neighborhood of the zero $U$ of the space $H^3$. This scheme is not applicable to the second equation, but it is possible to identify a fairly wide class of functions $(q,\sigma_1\sigma_2)\in U$, on which this equation turns into an identity. Next we explore the question: can the operator in question have an empty spectrum if the functions $\sigma_1, \sigma_2$ are not necessarily close to zero (in the space $H^2$)? We have constructed a class of functions $\sigma_1$ and $\sigma_2$ (in the form of polynomials with arbitrarily large norms) such that the spectrum of the corresponding operator is empty. The operating technique can be extended to the case when $H=L_2(\gamma)$, where $\gamma$ is a curve with a limited slope (that is, the absolute value of the slope of any chord of this curve does not exceed a certain number).