On singular spectrum of finite dimensional perturbations (to the Aronszajn-Donoghue-Kac theory)

The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations à of an operator A_{0 =} A_0^* defined on a Hilbert space H. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions A_B of a symmetric operator A acting in H in terms of the Weyl function M(·) of the pair {A, A₀} and boundary n-dimensional operator B = B^*. Assuming that the multiplicity of singular spectrum of A₀ is maximal it is established orthogonality of singular parts E^s_Aв and E^s_Ao of the spectral measures E_Aв and E_Ao of the operators A_B and A₀, respectively. It is shown that the multiplicity of singular spectrum of special extensions of direct sums A = A⁽¹⁾ ⊕ A⁽²⁾ cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. In particular, it is obtained a generalization of the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line as well as its clarification. The multiplicity of singular spectrum of special extensions of direct sums A = A⁽¹⁾ ⊕ A⁽²⁾ are investigated. In particular, it is shown that it cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. This result generalizes the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line and clarifies it.