(untitled) (Journal of Mathematics Mechanics and Computer Science, 2025)
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 σ_1 and σ_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 σ_1 and σ_2. This allows us to resolve the indicated equation for σ_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,σ_1σ_2)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 σ_1, σ_2 are not necessarily close to zero (in the space H^2)? We have constructed a class of functions σ_1 and σ_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(γ), where γ 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).