07-41 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) Feb 21, 07
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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.

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