Localization of eigenfunctions via an effective potential

• Published in: Communications in partial differential equations Vol. 44; no. 11; pp. 1186 - 1216
• Main Authors: Arnold, Douglas N., David, Guy, Filoche, Marcel, Jerison, David, Mayboroda, Svitlana
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.11.2019; Taylor & Francis Ltd
• Subjects: Agmon distance; Schrodinger equation; spectrum; the landscape of localization; Condensed Matter; Disordered Systems and Neural Networks; Eigenvalues; Eigenvectors; Engineering Sciences; Human health and pathology; Landscape; Life Sciences; Localization; Operators (mathematics); Optics; Photonic; Physics; Pulmonology and respiratory tract; Soft Condensed Matter
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2019.1626420

We consider the localization of eigenfunctions for the operator 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.