215 research outputs found
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 , ,
where and are self-adjoint operators in a separable Hilbert space
and is bounded. In addition, we assume that the range of
is a generating (i.e., cyclic) subspace for . If is an
eigenvalue of , then under the additional assumption that is
nonnegative, the Lebesgue measure of the set of all for which
is an eigenvalue of is known to be zero. We recall this
result with its proof and show by explicit counterexample that the
nonnegativity assumption cannot be removed.Comment: 10 pages; added Lemma 2.4, typos removed; to appear in J. Math. Anal.
App
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