Spectral Analysis of the Laplacian Acting on Discrete Cusps and Funnels
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Complex Analysis and Operator Theory
Spectral Analysis of the Laplacian Acting on Discrete Cusps and Funnels Nassim Athmouni1 · Marwa Ennaceur2 · Sylvain Golénia3 Received: 28 April 2020 / Accepted: 4 November 2020 © Springer Nature Switzerland AG 2020
Abstract 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. Keywords Commutator · Mourre estimate · Limiting absorption principle · Discrete Laplacian Mathematics Subject Classification 81Q10 · 47B25 · 47A10 · 05C63
Contents 1 Introduction . . . . . . . . . 2 The Mourre Theory . . . . . 3 The Free Model . . . . . . . 3.1 Construction of the Graph
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Communicated by Fabrizio Colombo. This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.
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Sylvain Golénia [email protected] Nassim Athmouni [email protected] Marwa Ennaceur [email protected]
1
Université de Gafsa, Campus Universitaire, 2112 Gafsa, Tunisia
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Université de Sfax, Route de la Soukra km 4, B.P. n 802, 3038 Sfax, Tunisia
3
Bordeaux INP, CNRS, IMB, UMR 5251, Univ. Bordeaux, 33400 Talence, France 0123456789().: V,-vol
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3.2 Mourre Estimate on N . . . . . . . . . . . . . . . . . . . 3.3 The Funnel Side . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 A First Step into the Analysis . . . . . . . . . . . . 3.3.2 Construction of the Conjugate Operator . . . . . . . 3.4 The Cusps Side . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Model and the Low/High Energy Decomposition 3.4.2 The Conjugate Operator . . . . . . . . . . . . . . . 3.5 The Compact Part . . . . . . . . . . . . . . . . . . . . . . 3.6 The Whole Graph . . . . . . . . . . . . . . . . . . . . . . 4 The Perturbed Model . . . . . . . . . . . . . . . . . . . . . . . 4.1 Perturbation of the Metric . . . . . . . . . . . . . . . . . . 4.2 The Funnel Side . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Cusp Side: Radial Metric Perturbation . . . . . . . . . 4.4 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
N. Athmouni et al. . . . . . . . . . . . . . . .
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