Garkavenko, Galina V.; Uskova, Natal’ya B. Spectral analysis of a class of difference operators with growing potential. (Russian. English summary) Zbl 1375.39039 Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform. 16, No. 4, 395-402 (2016). Summary: 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 \(a: \mathbb{Z} \rightarrow \mathbb{C}\). The asymptotic estimates of eigenvalue, eigenvectors, spectral estimation of equiconvergence applications for the test operator and the operator of multiplication by a sequence \(a: \mathbb{Z} \rightarrow \mathbb{C}\). 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. Cited in 3 Documents MSC: 39A70 Difference operators 47B39 Linear difference operators 47A10 Spectrum, resolvent Keywords:similar operator method; spectrum; difference operator; spectral projections; eigenvalue; eigenvector PDFBibTeX XMLCite \textit{G. V. Garkavenko} and \textit{N. B. Uskova}, Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform. 16, No. 4, 395--402 (2016; Zbl 1375.39039) Full Text: DOI