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Second eigenvalue of Schrödinger operators and mean curvature. (English) Zbl 0955.58025

Summary: Let \(M\) be a compact immersed submanifold of the Euclidean space, the hyperbolic space or the standard sphere. For any continuous potential \(q\) on \(M\), we give a sharp upper bound for the second eigenvalue of the operator \(-\Delta + q\) in terms of the total mean curvature of \(M\) and the mean value of \(q\). Moreover, we analyze the case where this bound is achieved. As a consequence of this result we obtain an alternative proof for the Alikakos-Fusco conjecture concerning the stability of the interface in the Allen-Cahn reaction diffusion model.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P15 Estimates of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
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