Bands of pure a.c. spectrum for lattice Schr{ö}dinger operators with a more general long range condition. Part I

Abstract

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schrödinger operators $H_{\mathrm{std}}=\Delta +V$ and $H_{\mathrm{MV}}=D+V$ on ${\ell }^2({\mathbb{Z}}^d)$, with emphasis on $d=1,2,3$. Considered are electric potentials $V$ satisfying a long range condition of the type: $V-{\tau }_j^{\kappa }V$ decays appropriately for some $\kappa \in \mathbb{N}$ and all $1\le j\le d$, where ${\tau }_j^{\kappa }V$ is the potential shifted by $\kappa$ units on the $j^{\text{th}}$ coordinate. More comprehensive results are obtained for specific small values of $\kappa$, such as $\kappa =1,2,3,4$. In this article, we work in a simplified framework in which the main takeaway appears to be the existence of bands where a limiting absorption principle holds, and hence absolutely continuous (a.c.) spectrum, for $\kappa >1$ and $\Delta$ (resp.\ $\kappa >2$ and $D$). Other decay conditions for $V$ arise from an isomorphism between $\Delta$ and $D$ in dimension 2. Oscillating potentials are natural examples in application.

References in corpus (4)

• Criteria for embedded eigenvalues for discrete Schrödinger operators
• Absence of singular continuous spectrum for perturbed discrete Schrödinger operators
• Sub-exponential decay of eigenfunctions for some discrete Schrödinger operators
• Branching form of the resolvent at threshold for multi-dimensional discrete Laplacians