Abstract
It is proved that the general formulas, obtained recently for the lower bound of the first eigenvalue, can be further bounded by one or two constants depending on the coefficients of the corresponding operators only. Moreover, the ratio of the upper and lower bounds is no more than four.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chen, M. F., Wang, F. Y., Estimation of spectral gap for elliptic operators, Trans. Amer. Math. Soc., 1997, 349(3): 1239.
Chen, M. F., Wang, F. Y., General formula for lower bound of the first eigenvalue, Sciece in China, Ser. A., 1997, 40 (4):384.
Chen, M. F., Estimation of spectral gap for Markov chains, Acta Math. Sin. New Series, 1996, 12(4):337.
Chen, M. F., Analytic proof of dual variational formula for the first eigenvalue in dimension one, Science in China, Ser. A, 1999, 42(8):805.
Kac, I. S., Krein, M. G., Criteria for the discreteness of the spectrum of a singular string. Izv Vyss Ucebn Zaved Mat (in Russian), 1958, 2:136.
Muckenhoupt, B., Hardy’s inequality with weights, Studia Math., 1972, XLIV:31.
Chen, M. F., Wang, F. Y., Cheegerl’s inequalities for general symmetric forms and existence criteria for spectral gap, Ann. Prob., 2000, 28(1):235; Abstract:Chinese Science Bulletin, 1998, 43(18):1516.
Chen, M. F., From Markov Chains to Nan-Equilibrium Particle Systems, Singapore: World Scientific, 1992.
Chen, M. F., Eigenvalurs, inequalities and ergodic theory (II), Advances in Math (in Chinese), 1999, 28(6):481.
Miclo, L., An example of application of discrete Hardy’s inequalities, Markov Processes Relat. Fields, 1999, 5:319.
Miclo, L., Relations entre isopérimétrie et trou spectral pour les chaînes de Markov finies, Probab. Th. Re1. Fields, 1999, 114: 431.
Robkov, S. G., Götze, F., Discrete isoperimetric and Poincané inequalities, Probab. Th. Rel. Fields, 1999, 114:245.
Bobkov, S. G., Götze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 1999, 163:1.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, M. Explicit bounds of the first eigenvalue. Sci. China Ser. A-Math. 43, 1051–1059 (2000). https://doi.org/10.1007/BF02898239
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02898239