Volume growth and bounds for the essential spectrum for Dirichlet forms

Abstract

We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases, we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric), we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth.

Graphs of finite measure

“…They have been brought forward and first studied systematically in [12] (see [51] for earlier appearances as well). Subsequently, they have become a main tool in certain geometric and spectral theoretic considerations, see, e.g., [2,3,10,11,18,21,24,44,45]. They are always defined with respect to some measure m on X.…”

Intrinsic Metrics on Graphs: A Survey

“…Theorem 10.1 (Corollary 4.2 in [38]). Let b be a connected graph over (X, m) and ρ be an intrinsic metric such that the balls are finite (B).…”

“…Such a result fails for unbounded graph Laplacians and the combinatorial graph distance, since there are graphs with only little more than cubic volume growth that admit a spectral gap, [55]. Again, using intrinsic metrics the result that holds for Riemannian manifolds can be recovered for graphs and even for general Dirichlet forms, [38].…”

Spectral Theory of Infinite Quantum Graphs

“…Let B r (u) be a distance ball with respect to the natural path metric ̺ 0 . Following [54] (see also [59]), we define…”

“…The proof follows from the growth volume estimates on the spectrum of h 0 . More precisely, the following bounds were established in [54] (see also [43,59]):…”