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Localization of eigenfunctions via an effective potential

Abstract : We consider the Neumann boundary value problem for the operator L = − div A grad + V on a Lipschitz domain Ω and, more generally, on manifolds with and without boundary. The eigenfunctions of L are often localized, as a result of disorder of the potential V , the matrix of coefficients A, irregularities of the boundary, or all of the above. In earlier work, two of us introduced the function u solving Lu = 1, and showed numerically that it strongly reflects this localization. In this paper, we deepen the connection between the eigenfunctions and this landscape function u by proving that its reciprocal 1/u acts as an effective potential. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.
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Submitted on : Monday, January 4, 2021 - 11:24:55 AM
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Douglas Arnold, Guy David, Marcel Filoche, David Jerison, Svitlana Mayboroda. Localization of eigenfunctions via an effective potential. Communications in Partial Differential Equations, Taylor & Francis, 2019, 44 (11), pp.1186-1216. ⟨10.1080/03605302.2019.1626420⟩. ⟨hal-02360595⟩

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