Weighted estimates for the Laplacian on the cubic lattice

Møller, Jacob Schach

doi:10.4310/arkiv.2019.v57.n2.a8

Cited by 18 publications

(14 citation statements)

References 27 publications

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“…It is well known that if λ 0 ∈ Λ is an eigenvalue of H, then z 0 = z(λ 0 ) ∈ D is a zero of D with the same multiplicity. We recall needed results from [KM17]:…”

Section: Complex Potentialsmentioning

confidence: 99%

“…Complex potentials. In this paper we combine classical results about Hardy spaces and estimates of the free resolvent from [KM17], this gives us new trace formulae for discrete Scrödinger operators H = ∆ + V on the lattice Z d , where the potential V is complex and satisfies the condition (1.2). We improve results from [KL16], where potentials are considered under the weaker condition |V | 2 3 ∈ ℓ 1 (Z d ).…”

Section: Introductionmentioning

confidence: 99%

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Trace formulae for Schrödinger operators with complex-valued potentials on cubic lattices

Лаптев

2018

Bull. Math. Sci.

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We consider Schrödinger operators with complex decaying potentials (in general, not from trace class) on the lattice. We determine trace formulae and estimate of eigenvalues and singular measure in terms of potentials. The proof is based on estimates of free resolvent and analysis of functions from Hardy space.It is well-known that the spectrum of the Laplacian ∆ is absolutely continuous and equalsNote that if V satisfies (1.2), then V is the Gilbert-Schmidt operator and thus the essential spectrum of the Schrödinger operator H is given by σ ess (H) = [−d, d]. The operator H has N ∞ eigenvalues {λ n , n = 1, ...., N} outside the interval [−d, d].We define the disc D r ⊂ C with the radius r > 0 byThe function λ(z) has the following properties (see more in Section 3):• The function λ(z) is a conformal mapping from D onto the spectral domain Λ.• The function λ(z) maps the point z = 0 to the point λ = ∞.

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“…It is well known that if λ 0 ∈ Λ is an eigenvalue of H, then z 0 = z(λ 0 ) ∈ D is a zero of D with the same multiplicity. We recall needed results from [KM17]:…”

Section: Complex Potentialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Trace formulae for Schrödinger operators with complex-valued potentials on cubic lattices

Лаптев

2018

Bull. Math. Sci.

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“…For example, the case of Z d and graphene have been fully investigated in [6,26] and [25] respectively. As a related work, [14] provides estimates for the unitary group and the resolvent of the discrete Laplace operator on Z d , from which the authors infer some results for the spectral and the scattering theory of perturbed operators by potentials V vanishing at infinity.…”

Section: Introductionmentioning

confidence: 99%

Spectral and scattering theory for topological crystals perturbed by infinitely many new edges

Richard,

Tsuzu

2021

Preprint

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In this paper we investigate the spectral and scattering theory for operators acting on topological crystals and on their perturbations. A special attention is paid to perturbations obtained by the addition of an infinite number of edges, and / or by the removal of a finite number of them, but perturbations of the underlying measures and perturbations by the addition of a multiplication operator are also considered. The description of the nature of the spectrum of the resulting operators, and the existence and completeness of the wave operators are standard outcomes for these investigations.

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“…The authors would like to thank their supervisor Shu Nakamura for encouraging to write this paper and helpful discussions about the Mourre theory for ultrahyperbolic operators. The authors also would like to thank Evgeny Korotyaev for informing the paper [14]. KT would like to thank Haruya Mizutani for answering many questions about uniform resolvent estimates.…”

Section: Introductionmentioning

confidence: 99%

Uniform bounds of discrete Birman–Schwinger operators

Tadano

Taira

2019

Trans. Amer. Math. Soc.

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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 .

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Weighted estimates for the Laplacian on the cubic lattice

Cited by 18 publications

References 27 publications

Trace formulae for Schrödinger operators with complex-valued potentials on cubic lattices

Trace formulae for Schrödinger operators with complex-valued potentials on cubic lattices

Spectral and scattering theory for topological crystals perturbed by infinitely many new edges

Uniform bounds of discrete Birman–Schwinger operators

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