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 .
We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential selfadjointness under the null non-trapping condition.Résumé. -Nous étudions le caractère essentiellement auto-adjoint pour des opérateurs de type principal réel. Contrairement au cas elliptique, nous avons besoin de conditions géométriques même pour des opérateurs sur l'espace euclidien avec coefficients asymptotiquement constants, et nous démontrons le caractère essentiellement auto-adjoint sous la condition de non-capture à énergie zéro.
We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.
Here we discuss a new simplified proof of the essential self-adjointness for formally self-adjoint differential operators of real principal type, previously proved by Vasy (2020) and Nakamura-Taira (2021). For simplicity, here we discuss the second order cases, i.e., Klein-Gordon type operators only.
In this paper, we study the mapping property from Lp to Lq of the resolvent of the Fourier multipliers and scattering theory of generalized Schrödinger operators. Although the first half of the subject was studied by Cuenin [J. Funct. Anal. 272(7), 2987–3018 (2017)], we extend their result away from the duality line, and we also study the Hölder continuity of the resolvent.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.