Weighted Mourre's commutator theory, application to Schroedinger operators with oscillating potential

We study perturbations of the discrete Laplacian associated to discrete analogs of cusps and funnels. We perturb the metric and the potential in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from the possible embedded eigenvalues. The approach is based on a positive commutator technique. Contents 1. Introduction 1 2. The Mourre theory 4 3. The free model 6 3.1. Construction of the graph 6 3.2. Mourre estimate on N 8 3.3. The funnel side 9 3.3.1. A first step into the analysis 9 3.3.2. Construction of the conjugate operator 10 3.4. The cusps side 11 3.4.1. The model and the low/high energy decomposition 12 3.4.2. The conjugate operator 13 3.5. The compact part 14 3.6. The whole graph 15 4. The perturbed model 15 4.1. Perturbation of the metric 15 4.2. Main result 16 4.3. The funnel side 17 4.4. The cusp side: Radial metric perturbation 23 References 29

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Bands of pure absolutely continuous spectrum for lattice Schrödinger operators with a more general long range condition

Golenia

Mandich²

2021

Journal of Mathematical Physics

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Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov–Vainberg Schrödinger operators, Δ + V and D + V on ℓ2(Zd), with emphasis on d = 1, 2, 3. Considered are electric potentials V satisfying a long range condition of the following type: V−τjκV decays appropriately at infinity for some κ∈N and all 1 ≤ j ≤ d, where τjκV is the potential shifted by κ units on the jth coordinate. More comprehensive results are obtained for small values of κ, e.g., κ = 1, 2, 3, 4. 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, pure absolutely continuous spectrum exists. Other decay conditions at infinity for V arise from an isomorphism between Δ and D in dimension 2. Oscillating potentials are examples in application.

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Weighted Mourre's commutator theory, application to Schroedinger operators with oscillating potential

Cited by 5 publications

References 46 publications

Abstract Theory of Pointwise Decay with Applications to Wave and Schrödinger Equations

Abstract Theory of Pointwise Decay with Applications to Wave and Schrödinger Equations

Spectral Analysis of the Laplacian Acting on Discrete Cusps and Funnels

Bands of pure absolutely continuous spectrum for lattice Schrödinger operators with a more general long range condition

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