On a Class of Integral Systems

Abstract

We study spectral problems for two-dimensional integral system with two given non-decreasing functions R, W on an interval [0, b) which is a generalization of the Krein string. Associated to this system are the maximal linear relation T_{max } and the minimal linear relation T_{min } in the space L^2(dW) which are connected by T_{max }=T_{min }^*. It is shown that the limit point condition at b for this system is equivalent to the strong limit point condition for the linear relation T_{max }. In the limit circle case the Evans–Everitt condition is proved to hold on a subspace T_N^* of T_{max } characterized by the Neumann boundary condition at b. The notion of the principal Titchmarsh–Weyl coefficient of this integral system is introduced. Boundary triple for the linear relation T_{max } in the limit point case (and for T_{N}^* in the limit circle case) is constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh–Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of R and W and the formula relating the principal Titchmarsh–Weyl coefficients of the direct and the dual integral systems is proved. For every integral system with the principal Titchmarsh–Weyl coefficients q a canonical system is constructed so that its Titchmarsh–Weyl coefficient Q is the unwrapping transform of q: Q(z)=z q(z^2).