The Trace Formula And Inverse Nodal Problem For A Class Of Differential Operators With Nonlocal Boundary Conditions

In general, the summation of eigenvalues for unbounded differential operators are diver-gent. To derive the summation, one tries to regularize by substracting some divergent series, and this summation is so-called the trace of Gelfand-Levitan type. Inverse nodal problem consists in constructing operators from the given zeros of their eigenfunctions. The problem of differential operators with nonlocal boundary conditions appear, e.g., in scattering theory, diffusion processes and the other applicable fields, and it is significant to study the spectrum and inverse problems of this kind of differential operators. In this paper, we consider a class of differential operators with nonlocal boundary conditions. First we give trace formula of the operators, and then we show that the potential function can be determined by nodal data, finally reconstruction algorithm is provided.