J.Janas, S.Naboko, E.Sheronova
Jacobi matrices arising in the spectral phase transition
phenomena : asymptotics of generalized eigenvectors in the "double
root" case.
(57K, AMS-Tex)
ABSTRACT. This paper is concerned with asymptotic behavior of generalized
eigenvectors of a class of Hermitian Jacobi matrices $J$ in the
critical case. The last means that the fraction $q_n/\lambda_n $
generated by the diagonal entries $q_n$ of $J$ and its subdiagonal
elements $\lambda_n$ has the limit $\pm2$. In other word, the limit
transfer matrix as $n\to\infty$ contains a Jordan box (=double root
in terms of Birkhoff-Adams theory). This is the situation where the
asymptotic Levinson theorem does not work and one has to elaborate
more special methods for asymptotic analysis. It should be mentioned
that the critical case exactly corresponds to spectral phase
transition phenomena, where the spectral structure changes
dramatically (from discreet spectrum to pure absolutely continuous
one) whenever the parameters in matrix entries cross singular
surfaces \cite{JN02}. Jordan box is the limit transfer matrix for
all values of spectral parameter $\lambda$ simultaneously, it
describes the "moment" of spectral phase transition. Application to
the case of $\lambda_n = n^{\alpha}(1+r_n)$,
$q_n=-2n^{\alpha}(1+p_n)$ with small perturbations $r_n$, $p_n$ and
$\alpha\in(0,1]$ is studied.