Spectral properties of Jacobi matrices whose dioagonal sequences are multiples of the off diagonal sequence (Q2755561)

The main object under consideration is the class of Jacobi matrices with diagonal sequences \(b_n\) and off diagonal sequences \(a_n\) satisfying \(b_n=\alpha a_n\), where \(\alpha\) is assumed (without loss of generality) to be a non-negative number. The authors study the nature of the spectrum for such matrices which turns out to depend badly on the value \(\alpha\). For instance, they prove that if \(a_n\) is non-decreasing and tends to \(1\), then \(\text{supp}(\mu)=[\alpha-2,\alpha+2]\) and \(\mu\) is pure absolutely continuous as long as \(|\alpha|\leq 2\). For \(|\alpha|>2\) the restriction of \(\mu\) to \((\alpha-2,\alpha+2]\) is absolutely continuous. They also show that if \(|\alpha|>2\) is ``large'', then eigenvalues appear and if \(|\alpha|>2\) is ``small'', then eigenvalues do not appear, and examine the critical value of \(\alpha\).