Critical potentials of the eigenvalues and eigenvalue gaps of Schrodinger operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:314

Article

El Soufi, A ; Moukadem, N

关键词: eigenvalues;    Schrodinger operator;    extremal potential;    extremal gap;   

DOI  :  10.1016/j.jmaa.2005.03.072

来源: Elsevier

【 摘 要 】

Let M be a compact Riemannian manifold with or without boundary, and let -Delta be its LaplaceBeltrami operator. For any bounded scalar potential q, we denote by lambda(i) (q) the ith eigenvalue of the Schrodinger type operator -Delta + q acting on functions with Dirichlet or Neumann boundary conditions in case partial derivative M not equal theta. We investigate critical potentials of the eigenvalues lambda(i) and the eigenvalue gaps G(ij) = lambda(j) - lambda(i) considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q is an element of L-infinity(M) is critical for the functional lambda(2) if and only if q is smooth, lambda(2)(q) = lambda(3)(q) and there exist second eigenfunctions f(1),..., f(k) of -Delta + q such that Sigma(j) f(j)(2) = 1. In particular,),2 (as well as any Xi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional lambda(2) never admits locally minimizing potentials. (c) 2005 Elsevier Inc. All rights reserved.

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