zbMATH — the first resource for mathematics

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.

35P15 Estimates of eigenvalues in context of PDEs
60H30 Applications of stochastic analysis (to PDEs, etc.)
Full Text: DOI arXiv
[1] Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, inProblems in Analysis (Gunning, R. C., ed.), pp. 195–199, Princeton Univ. Press, Princeton, N. J., 1970.
[2] Chen, M. F. andWang, F.-Y., Estimation of the first eigenvalue of the second order elliptic operators,J. Funct. Anal. 131 (1995), 345–363. · Zbl 0837.35102
[3] Chen, M. F. andWang, F.-Y., General formula for lower bound of the first eigenvalue on Riemannian manifolds,Sci. China Ser. A 40 (1997), 384–394. · Zbl 0895.58056
[4] Chen, M. F. andWang, F.-Y., Estimation of spectral gap for elliptic operators,Trans. Amer. Math. Soc. 349 (1997), 1239–1267. · Zbl 0872.35072
[5] Donnely, H. andLi, P., Pure point spectrum and negative curvature for noncompact manifolds,Duke Math. J. 46 (1979), 497–503. · Zbl 0416.58025
[6] Kasue, A., On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold,Ann. Sci. École Norm. Sup 17 (1984), 31–44. · Zbl 0553.53026
[7] Kendall, W. S., The radial part of Brownian motion on a manifold: a semimartingale property,Ann. Probab. 15 (1987), 1491–1500. · Zbl 0647.60086
[8] Kumura, H., On the essential spectrum of the Laplacian on complete manifolds,J. Math. Soc. Japan 49 (1997), 1–14. · Zbl 0913.58056
[9] Ledoux, M.; Concentration of Measure and Logarithmic Sobolev Inequalities. to appear inSéminaire de Probabilités XXXIV, Lecture Notes in Math., Springer-Verlag, Berlin-Heidelberg.
[10] Sturm, K. Th., Analysis on local Dirichlet spaces I, reccurrence, conservativeness,L p-Liouville properties,J. Reine Angew. Math. 456 (1994), 173–196. · Zbl 0806.53041
[11] Thomas, L. E., Bound on the mass gap for finite volume stochastic Ising models at low temperature,Comm. Math. Phys. 126 (1989), 1–11. · Zbl 0679.60102
[12] Wang, F.-Y., Spectral gap for diffusion processes on noncompact manifolds,Chinese Sci. Bull. 40 (1995), 1145–1149. · Zbl 0845.58057
[13] Wang, F.-Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds,Probab. Theor. Relat. Fields 109 (1997), 417–424. · Zbl 0887.35012
[14] Wang, F.-Y., Logarithmic Sobolev inequalities: Conditions and counterexamples, Preprint, 1997.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.