Cheeger’s inequalities for general symmetric forms and existence criteria for spectral gap. (English) Zbl 1044.60078

Summary: Some new forms of J. Cheeger’s inequalities [in: Probl. Anal., Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 195–199 (1970; Zbl 0212.44903)] are established for general (maybe unbounded) symmetric forms (Theorems 1.1 and 1.2): the resulting estimates improve and extend the ones obtained by G. F. Lawler and A. D. Sokal [Trans. Am. Math. Soc. 309, No. 2, 557–580 (1988; Zbl 0716.60073)] for bounded jump processes. Furthermore, some existence criteria for spectral gap of general symmetric forms or general reversible Markov processes are presented (Theorems 1.4 and 3.1), based on Cheeger’s inequalities and a relationship between the spectral gap and the first Dirichlet and Neumann eigenvalues on local region.


60J75 Jump processes (MSC2010)
60J25 Continuous-time Markov processes on general state spaces
47A75 Eigenvalue problems for linear operators
Full Text: DOI arXiv Euclid


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