Elementary models of unbounded Jacobi matrices with a few bounded gaps in the essential spectrum (Q2914873)

The main result of the paper displays an explicit example of an unbounded Jacobi operator \(J\) with the following property of its essential spectrum NEWLINE\[NEWLINE \sigma_{ess}(J)=\bigcup_{i=0}^m [\alpha_i,\beta_i], \qquad -\infty=\alpha_0\leq\beta_0\leq\ldots\leq\alpha_m<\beta_m=+\infty. NEWLINE\]NEWLINE Such an operator arises as a compact perturbation of the orthogonal sum \(\bigoplus J_n\) of finite-dimensional Jacobi operators \(J_n\) with the orders \(d(J_n)\to\infty\) as \(n\to\infty\). Actually, \(J_{2n-1}=J_z(k_{2n-1})\) is the truncation of order \(k_{2n-1}\) of an infinite Jacobi matrix \(J_z\) and, similarly, \(J_{2n}=J_c(k_{2n})\) for an appropriate sequence of positive integers \(\{k_j\}\). Here, NEWLINE\[NEWLINE J_z=J(\{a_j^z\},\{b_j^z\}), \qquad J_c=J(\{a_j^c\},\{b_j^c\}) NEWLINE\]NEWLINE are well studied infinite Jacobi matrices with \(b_j^z=b_j^c=0\) for all \(j=1,2,\ldots\), while \(J_z\) is just an \(N\)-periodic Jacobi matrix, and NEWLINE\[NEWLINE a_n^c=n^\alpha+c_n, \qquad 0<\alpha<1, NEWLINE\]NEWLINE \(\{c_n\}\) is a \(2\)-periodic sequence with \(c_1>c_2\), \(2-2^\alpha+c_2>0\).NEWLINENEWLINEThe authors prove that \(\sigma_{ess}(J)=\sigma(J_z)\cup\sigma(J_c)\cup T\), where \(T\) is a finite set.