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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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References:
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