Absence of the point spectrum in a class of tridiagonal operators. (English) Zbl 1053.47025
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.
Reviewer: J. Saurer (Chamerau)
MSC:
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 33C70 | Other hypergeometric functions and integrals in several variables |
| 39A70 | Difference operators |
| 47A10 | Spectrum, resolvent |
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