Singular spectrum near a singular point of Friedrichs model operators of absolute type (Q928427)

Let \(m>0\) and consider the selfadjoint operator \[ (A_mu)(t)=| t|^m u(t)+\int_{\mathbb R}\nu(t,x)u(x)\,dx \] in \(L^2(\mathbb R)\) on its maximal domain. Here it is assumed that the integral operator is nonnegative, belongs to the trace class ideal and that the Hermitian kernel \(\nu(t,x)\) satisfies some additional smoothness conditions. The main objective of the present paper is to study the dependence of the behaviour of the point and singular continuous spectrum on the smoothness of the kernel \(\nu(t,x)\) near the singular point \(t=0\). For the special case \(m=1\), this question has been studied extensively. In the present paper, not only necessary and sufficient conditions for the absence of the singular spectrum near the origin are found, but also counterexamples for the sharpness of these conditions are provided.