In this paper, we define time-independent modifiers to construct a long-range scattering theory for a class of difference operators on Z d , including the discrete Schrödinger operators on the square lattice. The modifiers are constructed by observing the corresponding Hamilton flow on T * T d . We prove the existence and completeness of modified wave operators in terms of the above mentioned time-independent modifiers.In Section 2, we construct modified wave operators with time-independent modifiers, which are proposed by Isozaki and Kitada [6], so called Isozaki-Kitada modifiers. Isozaki-Kitada modifiers are formally defined byWe construct J as an operator of the formwhere the phase function ϕ is a solution to the eikonal equationin the "outgoing" and "incoming" regions and considered in Appendix A.The next theorem is our main result.Theorem 1.5. Under Assumptions 1.1 and 1.3, there exists an operator J of the form (1.3) such that, for any Γ ⋐ h 0 (T d )\T, the modified wave operators. Examples 1.6. i) In [11], a long-range scattering theory of the standard difference Laplacian H 0 u[x] = − 1 Hence, with the help of a partition of unity {ψ j } J j=1 on T d , we observe
In this note, uniform bounds of the Birman-Schwinger operators in the discrete setting are studied. For uniformly decaying potentials, we obtain the same bound as in the continuous setting. However, for non-uniformly decaying potential, our results are weaker than in the continuous setting. As an application, we obtain unitary equivalence between the discrete Laplacian and the weakly coupled systems.2010 Mathematics Subject Classification. Primary 47A10, Secndary 47A40. Key words and phrases. discrete Schrödinger operators, resolvents, limiting absorption principle.Corollary 1.2. Under the condition of Theorem 1.1 (i) or (ii), H = H 0 + λV is unitarily equivalent to H 0 for small λ ∈ R. Remark 1.3. For Theorem 1.1 (ii), we show stronger results in Proposition 3.4. For Theorem 1.1 (i), we also obtain stronger results: Uniform resolvent estimates in Lorentz spaces as in Proposition 3.3. Remark 1.4. In [9] and [20], the authors prove the absence of eigenvalues of H 0 +λV for small λ ∈ R if |V (x)| ≤ C(1 + |x|) −2−ε for some C > 0 and ε > 0 with d = 3 and V ∈ l d 3 (Z d ) with d ≥ 4 respectively. In [1], (1) is proved under stronger assumptions: |V (x)| ≤ C x −2(d+3) with d ≥ 3. Moreover, in [14], (1) is established for V ∈ l p (Z d ) with 1 ≤ p < 6/5 if d = 3 and 1 ≤ p < 3d/(2d + 1) if d ≥ 4. The authors in [14] also study the scattering theory of H 0 + V .
The norm resolvent convergence of discrete Schrödinger operators to a continuum Schrödinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in particular, the convergence of the spectrum with respect to the Hausdorff distance.
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