On Singular Spectrum of Finite-Dimensional Perturbations (toward the Aronszajn–Donoghue–Kac Theory)

Abstract

The main results of the Aronszajn–Donoghue–Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations $$\tilde {A}$$ of an operator $${{A}_{0}} = A_{0}^{ * }$$ defined on a Hilbert space $$\mathfrak{H}$$ . By applying the technique of boundary triplets, the singular continuous and point spectra of extensions AB of a symmetric operator A are described in terms of the Weyl function $$M( \cdot )$$ of the pair {A, A0} and an n-dimensional boundary operator B = B*. Assuming that the multiplicity of the singular spectrum of A0 is maximal, we establish the orthogonality of the singular parts $$E_{{{{A}_{B}}}}^{s}$$ and $$E_{{{{A}_{0}}}}^{s}$$ of the spectral measure $${{E}_{{{{A}_{B}}}}}$$ and $${{E}_{{{{A}_{0}}}}}$$ of the operators AB and A0, respectively. The multiplicity of the singular spectrum of special extensions of direct sums $$A = {{A}^{{(1)}}} \oplus {{A}^{{(2)}}}$$ is investigated. In particular, it is shown that this multiplicity cannot be maximal, as distinguished from the multiplicity of the absolutely continuous spectrum. This result generalizes and refines the Kac theorem on the multiplicity of the singular spectrum of the Schrodinger operator on the line.