Description of the Structure of Singular Spectrum for Friedrichs Model Operator near Singular Point

The study of the point spectrum and the singular continuous one is reduced to investigating the structure of the real roots set of an analytic function with positive imaginary partM(λ). We prove a uniqueness theorem for such a class of analytic functions. Combining this theorem with a lemma on smoothness of M(λ) near its real roots permits us to describe the density of the singular spectrum. 2000 Mathematics Subject Classification. 47B06, 47B25. 1. Statement of the problem. We consider a selfadjoint operator A2 given by A2 = t ·+(·,φ)φ (1.1) on the domain of functions u(t) ∈ L2(R) such that t2u(t) ∈ L2(R). Here φ ∈ L2(R) and t is the independent variable. The action of the operator can be written as follows: ( A2u ) (t)= t ·u(t)+φ(t) ∫