27 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 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
The finite section method for dissipative operators
We show that for self-adjoint Jacobi matrices and Schrödinger operators, perturbed by dissipative potentials in ℓ 1 (N) and L 1 (0,∞) respectively, the finite section method does not omit any points of the spectrum. In the Schrödinger case two different approaches are presented. Many aspects of the proofs can be expected to carry over to higher dimensions, particularly for absolutely continuous spectrum