Perturbation determinants and discrete spectra of semi-infinite non-self-adjoint Jacobi operators

Abstract

We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb–Thirring inequalities for such operators. The spectral enclosure for the discrete spectrum and embedded eigenvalues are also discussed. In memory of Sergey Naboko (1950–2020) Introduction In the last two decades there was a splash of activity around the spectral theory of non-self-adjoint perturbations of some classical operators of mathematical physics, such as the Laplace and Dirac operators on the whole space, their fractional powers, and others. Recently, there has been some interest in studying certain discrete models of the above problem. In particular, the structure of the spectrum for compact, non-self-adjoint perturbations of the free Jacobi and the discrete Dirac operators has attracted much attention lately. Actually the problem concerns the discrete component of the spectrum and the rate of its accumulation to the essential spectrum. Such type of results under the various assumptions on the perturbations are united under a common name Lieb–Thirring inequalities. For a nice account of the existing results on the problem for non-self-adjoint, two-sided Jacobi operators, the reader may consult two recent surveys [7] and [10, Section 5.13] and references therein. The spectral theory of semi-infinite, self-adjoint Jacobi matrices is quite popular owing to their tight relation to the theory of orthogonal polynomials on the real line [19]. In contrast, there are only a few papers where semiinfinite, non-self-adjoint Jacobi matrices are examined [18, 1, 2, 8, 13, 14, 4]. The main object under consideration is a semi-infinite Jacobi matrix (0.1) J({aj}, {bj}, {cj})j∈N =   b1 c1 a1 b2 c2 a2 b3 c3 . . . . . . . . .   , Date: August 11, 2021. 2010 Mathematics Subject Classification. 47B36, 47A10, 47A75.