Spectral Enclosures for Non-self-adjoint Discrete Schrödinger Operators

Abstract

We study location of eigenvalues of one-dimensional discrete Schrödinger operators with complex p -potentials for 1 ≤ p ≤ ∞. In the case of 1 -potentials, the derived bound is shown to be optimal. For p > 1, two different spectral bounds are obtained. The method relies on the Birman-Schwinger principle and various techniques for estimations of the norm of the Birman-Schwinger operator. V e n := υ n e n , ∀n ∈ Z.

The abstract Birman-Schwinger principle and spectral stability

“…The same machinery has been recently applied to possibly non-self-adjoint biharmonic Schrödinger operators in [33] and the wave operator with complex-valued dampings in [40]. The Birman-Schwinger principle is not limited to continuous spaces; see [34,8] for Schrödinger operators on lattices.…”

“…It is customarily used to transfer a differential equation to an integral equation and has been employed in many circumstances over the last half century since the pioneering works of Birman [7] and Schwinger [48]. In recent years, the method has been revived in the context of spectral theory of non-self-adjoint Schrödinger and Dirac operators with complex potentials as a replacement of unavailable variational techniques (see, e.g., [27,15,22,29,14,26,23,16,24,33,34,10] to quote just a couple of most recent works). While its usefulness is very robust, the method is usually applied to concrete problems ad hoc and not always rigorously.…”

Eigenvalue bounds and spectral stability of Lamé operators with complex potentials

“…Finally, using the weak lower semi-continuity of the norm and the preliminary estimate (51), one has…”

“…We consider the perturbation V : R d → C d×d to be a multiplication operator by a (possibly) nonhermitian matrix: this frames our study into a non-self-adjoint setting. Spectral analysis of non-self-adjoint models has seen a huge development in the last decades and nowadays the literature in this direction is very extensive, see [1,9,17,24,25,[27][28][29][30][31][32]34,35,[37][38][39][42][43][44][45][47][48][49][50][51]59,64,71,72] which is just a selection of the existing material in the subject.…”

“…The usefulness of the Birman-Schwinger principle to localize eigenvalues of self-adjoint and non-self-adjoint Hamiltonians is by no means questionable, as a matter of fact an extensive bibliography on the subject has been produced adopting this methodology. Without any hope of completeness we refer to [38,42,48] for results on Schrödinger operators and [51] for an adaptation to the discrete setting, see also [59] where matrix-valued damped wave operators are concerned. Lower order operators, such as Dirac or fractional Schrödinger models, are investigated in [16,25,27,32,37] (see also [26,41]) and in [17] respectively in the continuous and discrete scenario; as for higher order operators refer to [50].…”