Spectral analysis of a class of difference operators with growing potential

G. V. Garkavenko ; N. B. Uskova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 4, pp. 395-402

Résumé

The similar operator method is used for the spectral analysis of the closed difference operator of the form $ (\mathcal{A} x)(n) = x(n + 1) + x(n-1)-2x(n) + a (n)x(n), n \in \mathbb{Z} $ under consideration in the Hilbert space $ l_ {2} (\mathbb{Z}) $ of bilateral sequences of complex numbers, with a growing potential $ a: \mathbb{Z} \to \mathbb{C} $. The asymptotic estimates of eigenvalue, eigenvectors, spectral estimation of equiconvergence applications for the test operator and the operator of multiplication by a sequence $ a: \mathbb{Z} \to \mathbb{C} $. For the study of the operator, it is represented in the form of $ A-B $, where $ (Ax) (n) = a (n) x (n)$, $n \in \mathbb{Z}$, $x \in l_2 (\mathbb{Z}) $ with the natural domain. This operator is normal with known spectral properties and acts as the unperturbed operator in the method of similar operators. The bounded operator $ (Bx)(n)=-x(n+1)-x(n-1)+2x(n)$, $n \in \mathbb{Z}$, $x \in l_2(\mathbb{Z})$, acts as the perturbation.