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Existence of the spectral gap for elliptic operators. (English) Zbl 1075.35540
Let \(M\) be a connected, noncompact, complete Riemannian manifold and let \(L=\Delta + \nabla V \) for some \(V \in C^2(M)\). The spectral gap of the operator \(L\) with Neumann boundary conditions exists if \(\lambda_1 > 0 \) with \[ \lambda_1 = \inf \left\{\frac{\mu (| \nabla f| ^2)}{\mu(f^2)-\mu(f)^2}: f \in C^1(M)\cap L^2(\mu),f \neq \text{constant} \right\}. \] The paper studies the existence of the spectral gap. Using the fact that \(\lambda_1 > 0\) is equivalent to \(\inf \sigma_{ess}(-L) > 0 \) conditions for the existence of the spectral gap in terms of the distance function are given.

MSC:
35P15 Estimates of eigenvalues in context of PDEs
60H30 Applications of stochastic analysis (to PDEs, etc.)
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