On the spectrum of one partial integral operator with a degenerate kernel

In this talk we consider a self-adjoint partial integral operator $H=H_{0}-(\gamma T_{1}+\mu T_{2})$, which arises in the theory of discrete Schrödinger operators. We calculate the determinant of the partial integral operator and describe the essential spectrum of the operator $H$ when the kernels of the partial integral operators $T_1$ and $T_2$ are given by special forms. Moreover, we find the lower bound of the essential spectrum for arbitraries $\gamma>0$, $\mu>0$ and prove that the number of negative eigenvalues below this lower boundary is finite.