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.
Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices are established. Two cases are considered separately: (i) the case in which the spectral parameter lies in a general gap of the spectru
m of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a many barriers-type criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. As an application, we provide examples of unbounded Jacobi matrices with a spectral mobility edge.
For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end we develop a met
hod for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a wide range of Jacobi matrices.
For the BCS equation with local two-body interaction $lambda V(x)$, we give a rigorous analysis of the asymptotic behavior of the critical temperature as $lambda to 0$. We derive necessary and sufficient conditions on $V(x)$ for the existence of a non-trivial solution for all values of $lambda>0$.
Starting with an adjoint pair of operators, under suitable abstra
