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The discrete Schrödinger operator. The point spectrum lying on the continuous spectrum. (English. Russian original) Zbl 0828.39005

St. Petersbg. Math. J. 4, No. 3, 559-568 (1993); translation from Algebra Anal. 4, No. 3, 183-195 (1992).
Summary: The point spectrum lying on the continuous spectrum is investigated for the one-dimensional discrete Schrödinger operator with a decaying potential. The continuous spectrum fills the interval \([- 2,2]\). It is well-known that in the case of a potential decaying faster than Coulomb’s, there are no eigenvalues in \((- 2,2)\).
In §2 examples of potentials are constructed which show that, just as in the continuous case, if the potential decays “slightly” more slowly than Coulomb’s, then a dense point spectrum on \([- 2,2]\) may occur.
In §3 the possibility is studied of the existence of an eigenvalue \(\lambda\in (- 2,2)\), in relation to the potential decay and the distance from \(\lambda\) to the boundary of the continuous spectrum. In particular, a very sharp condition is established for the absence of eigenvalues in the open interval \((- 2,2)\).

MSC:

39A70 Difference operators
39A12 Discrete version of topics in analysis
47B39 Linear difference operators
47A10 Spectrum, resolvent
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