Uniform bounds of discrete Birman–Schwinger operators

Abstract

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 .

Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schroedinger operators

“…Uniform resolvent bounds for Schrödinger operators in higher dimensions appeared, for instance, in [Fra11], [FS17], [Gut04], [KRS87], [BM18], [Miz19], [RXZ18], and one should also mention the classical Rollnik bound: see, for instance, [RS78, Example 3, p. 150], [Sim71, Sect. I.4], [Yaf10, Proposition 7.1.16] (for a discrete analog in this context see [TT19]). We also mention that the expansions of the integral kernel of the free resolvent (−∆ − zI) −1 are given, for instance, in [JK79] (d = 3), [Jen80] (d ≥ 5), [Jen84] (d = 4), and [JN01] (d ≤ 3).…”

Counterexample to the Laptev--Safronov conjecture

“…where q(ξ) = (ξ 1 − a(ξ ′ ))/p(ξ) > 0, see e.g. [20,Section 14.2], [6,Lemma 3.3] and [37,Section 3.1]. There it is sufficient to work with the limiting distributions corresponding to ε = 0±, which would yield (22) in this case.…”

Limiting absorption principle on $L^p$-spaces and scattering theory

“…where we use the triangle inequality in the first line and Lemma 3.6 in the second line. Thus we may replace χ(D) by (χ j χ)(D) in (30). We may suppose û and f are supported in supp χ and we may suppose ∂ ξ d T = 0 on supp χ by rotating the coordinate and by taking supp χ small enough.…”

“…Moreover, we study the scattering theory of the discrete Schrödinger operator, the fractional Schrödinger operators and the Dirac operators. We note that the limiting absorption principle for free discrete Schrödinger operators is studied in [14], [21] and [30]. In [21], the scattering theory of the discrete Schrödinger operators perturbed by L p -potentials are studied for a range of p. In [30], it is proved that the range of (p, q) which the uniform resolvent estimate holds for the discrete Schrödinger operators differs from the one for the continuous Schrödinger operators when d ≥ 5.…”

“…We define R ± 0 (λ) for λ ∈ I by the Fourier multiplier of the distribution (h 0 (ξ) − (λ ± i0)) −1 , where this distribution is well-defined by virtue of [30,Theoerem 1.8]. Note that we may take λ as a critical value.…”