Absence of the point spectrum in a class of tridiagonal operators. (Q1856020)

The paper deals with the tridiagonal operator \[ Te_n= \sqrt{c_n} e_{n+1}+ \sqrt{c_{n-1}} e_{n-1}+ b_n e_n \] with suitable real sequences \((b_n)^\infty_{n=1}\), \((c_n)^\infty_{n=1}\) and the orthonormal basis \((e_n)^\infty_{n=1}\) in a Hilbert space. Sufficient conditions are derived under which the point spectrum of the operator \(T\) outside the interval \([-2\sqrt{c}+ b, 2\sqrt{c}+ b]\) is empty, where \(b= \lim_{n\to\infty} b_n\), \(c= \lim_{n\to\infty} c_n\). The results are illustrated with examples.