Spectral analysis of a class of hermitian Jacobi matrices in a critical (double root) hyperbolic case

We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic ("double root") situation. For the model with "non-smooth" matrix entries we obtain the asymptotics of generalized eigenvectors and analyze the spectrum. In addition, we reformulate a very helpful theorem from a paper of Janas and Moszynski in its full generality in order to serve the needs of our method

On Cayley Identity for Self-Adjoint Operators in Hilbert Spaces

We prove an analogue to the Cayley identity for an arbitrary self-adjoint operator in a Hilbert space. We also provide two new ways to characterize vectors belonging to the singular spectral subspace in terms of the analytic properties of the resolvent of the operator, computed on these vectors. The latter are analogous to those used routinely in the scattering theory for the absolutely continuous subspace

On a problem in eigenvalue perturbation theory

We consider additive perturbations of the type $K_t=K_0+tW$, $t\in [0,1]$, where $K_0$ and $W$ are self-adjoint operators in a separable Hilbert space $\mathcal{H}$ and $W$ is bounded. In addition, we assume that the range of $W$ is a generating (i.e., cyclic) subspace for $K_0$. If $\lambda_0$ is an eigenvalue of $K_0$, then under the additional assumption that $W$ is nonnegative, the Lebesgue measure of the set of all $t\in [0,1]$ for which $\lambda_0$ is an eigenvalue of $K_t$ is known to be zero. We recall this result with its proof and show by explicit counterexample that the nonnegativity assumption $W\geq 0$ cannot be removed.Comment: 10 pages; added Lemma 2.4, typos removed; to appear in J. Math. Anal. App