We consider the localization of eigenfunctions for the operator L = -div A grad + V on a Lipschitz domain Ω and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to Lu = 1, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, 1/u, acts as an effective potential. Hence from the single measurement of u, we obtain, via 1/u, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum.
Bibliographical noteFunding Information:
Arnold is partially supported by the NSF grant DMS-1719694 and Simons Foundation grant 601937, DNA. David is supported in part by the ANR, programme blanc GEOMETRYA ANR-12-BS01-0014, EC Marie Curie grant MANET 607643, H2020 grant GHAIA 777822, and Simons Foundation grant 601941, GD. Filoche was supported in part by Simons Foundation grant 601944, MF. Jerison was partially supported by NSF grants DMS-1069225 and DMS-1500771, a Simons Fellowship, and Simons Foundation grant 601948, DJ. Mayboroda is supported in part by the Alfred P. Sloan Fellowship, the NSF grants DMS 1344235, DMS 1220089, DMS 1839077, a Simons Fellowship, and Simons Foundation grant 563916, SM. Part of this work was completed during Mayboroda?s visit to Universit? Paris-Sud, Laboratoire de Math?matiques, Orsay, and Ecole Polytechnique, PMC. We thank the corresponding Departments and Fondation Jacques Hadamard for support and hospitality.
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- Agmon distance; Schrodinger equation; spectrum; the landscape of localization