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Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

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Abstract

We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential \({V \in C^{\infty} (\mathbb{S}^{d})}\) to the Laplacian on the sphere results in the existence of geodesics \({\gamma}\) such that the uniform measure supported on \({\gamma}\) cannot be obtained as a weak-\({\star}\) accumulation point of the densities \({(|\psi_{n}|^{2} {vol}_{\mathbb{S}^d})}\) for any sequence of eigenfunctions \({(\psi_n)}\) of \({\Delta_{\mathbb{S}^{d}} - V}\). We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.

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Authors and Affiliations

  1. Universidad Politécnica de Madrid. DCAIN, ETSI Navales, Avda. Arco de la Victoria s/n, 28040, Madrid, Spain

    Fabricio Macià

  2. Laboratoire Paul Painlevé (U.M.R. CNRS 8524), U.F.R. de Mathématiques, Université Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    Gabriel Rivière

Authors
  1. Fabricio Macià
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  2. Gabriel Rivière
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Correspondence to Fabricio Macià.

Additional information

Communicated by S. Zelditch

FM takes part in the visiting faculty program of ICMAT and is partially supported by grants ERC Starting Grant 277778 and MTM2013-41780-P (MEC).

GR is partially supported by the Agence Nationale de la Recherche through the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR project GeRaSic (ANR-13-BS01- 0007-01).

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Macià, F., Rivière, G. Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds. Commun. Math. Phys. 345, 1019–1054 (2016). https://doi.org/10.1007/s00220-015-2504-8

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