We study the ℓp-perturbations of the discrete Laplacian and obtain the spectral enclosures of corresponding non-self-adjoint Jacobi operator. By invoking the Birman-Schwinger principle and establishing norm estimates for the Birman-Schwinger operator, for p = 1, we get the location of the eigenvalues, and the optimal bounds is given under the rank-one and rank-two perturbations. For 1 p ≤ ∞, we get the spectral bounds.
Fu et al. (Thu,) studied this question.