Opuscula Math. 36, no. 6 (2016), 807-818
http://dx.doi.org/10.7494/OpMath.2016.36.6.807

Opuscula Mathematica

# On the spectrum of periodic perturbations of certain unbounded Jacobi operators

Abstract. It is known that a purely off-diagonal Jacobi operator with coefficients $$a_n=n^{\alpha}$$, $$\alpha\in(0,1]$$, has a purely absolutely continuous spectrum filling the whole real axis. We show that a 2-periodic perturbation of these operators creates a non trivial gap in the spectrum.

Keywords: essential spectrum, spectral gap, periodic perturbation.

Mathematics Subject Classification: 47A10, 47B36, 39A70.

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• Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche - UMR7586, Bâtiment Sophie Germain - case 7012, 5 rue Thomas Mann, 75205 Paris Cedex 13, France
• Communicated by P.A. Cojuhari.
• Accepted: 2016-07-28.
• Published online: 2016-10-29.