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On Persson’s theorem in local Dirichlet spaces. (English) Zbl 0908.31004
The author considers a strongly, local, regular and irreducible Dirichlet form $$\epsilon.$$
He studies the analogue of the theorem of A. Persson that is proved in [Math. Scand. 8, 143-153 (1960; Zbl 0145.14901)]to obtain a version, in his context, of his variational characterization of the bottom of the essential spectrum of elliptic differential operators in divergence form.
Then, the author uses his result and those in cooperation with F. Cipriani [J. Reine Angew. Math. 496, 163-179 (1998; Zbl 0891.35016)] to prove suitable coercitivity conditions and weighted $$L^p$$ bounds for those eigenfunctions of the self-adjoint operator associated to $$\varepsilon$$ whose eigenvalue lies below the essential spectrum of the “small eigenfunctions”.
To obtain this result the author uses a process contained in the book of S. A. Agmon [Lectures on exponential decay of solutions of second-order elliptic equations, Princeton Univ. Press (1982; Zbl 0503.35001)].

##### MSC:
 31C25 Dirichlet forms 47A11 Local spectral properties of linear operators 35P20 Asymptotic distributions of eigenvalues in context of PDEs 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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##### References:
 [1] Agmon, S.: Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equa- tions. Princeton: Univ. Press 1982. · Zbl 0503.35001 [2] Biroli, M. and N. A. Tchou: Asymptotic behaviour of relaxed Dirichlet problems involving a Dirichlet-Poincaré form. Z. Anal. Anw. 16 (1997), 281 - 309. · Zbl 0885.35011 [3] Cipriani, F. and G. Grub: Poiniwise properties of eigenfunctions and heat kernels of Dirichlet-Schriidinger operators. Potential Anal. (to appear). · Zbl 0955.35018 [4] [6] Edmunds, D. E. and W. D. Evans: Spectral Theory for Differential , Operators. Oxford: Clarendon Press 1987. · Zbl 0628.47017 [5] [8] Fukushima, M., Oshida, Y. M. and Takeda: Dirichlet Forms and Symmetric Markov Processes. Berlin - New York: de Cruyter 1995. [6] Persson, A: Bounds for the discrete part of a semi-bounded Schrödinger operator. Math. Scand. 8 (1960), 143 - 153. 196. · Zbl 0145.14901
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