For citation:
Garkavenko G. V., Uskova N. B. Spectral Analysis of a Class of Difference Operators with Growing Potential. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 4, pp. 395-402. DOI: 10.18500/1816-9791-2016-16-4-395-402, EDN: XHPYGT
Spectral Analysis of a Class of Difference Operators with Growing Potential
The similar operator method is used for the spectral analysis of the closed difference operator of the form (A x)(n) = x(n + 1) + x(n − 1) − 2x(n) + a(n)x(n), n ∈ Z under consideration in the Hilbert space l2(Z) of bilateral sequences of complex numbers, with a growing potential a : Z → 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 : Z → C. For the study of the operator, it is represented in the form of A − B, where (Ax)(n) = a(n)x(n), n ∈ Z, x ∈ l2(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 ∈ Z, x ∈ l2(Z), acts as the perturbation.
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