Volume growth and bounds for the essential spectrum for Dirichlet forms (Q2869835)

In [Math. Z. 178, 501--508 (1981; Zbl 0458.58024)], \textit{R. Brooks} proved that the bottom of the essential spectrum of the Laplace-Beltrami operator on a complete non-compact Riemannian manifold with infinite measure can be bounded by the exponential volume growth rate of the manifold. The paper under review is devoted to the proof of Brooks-type theorems for positive self-adjoint operators \(L\) on the Hilbert space \(L^2(X,m)\) arising from regular Dirichlet forms with vanishing killing term. The authors use the corresponding concept of an intrinsic metric \(\rho\) proposed by \textit{R. L. Frank}, \textit{D. Lenz} and \textit{D. Wingert} [``Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory'', J. Funct. Anal., to appear, \url{arXiv:1012.5050}].NEWLINENEWLINEDefine the distance ball \(B_r = B_r(x_0) = \{x\in X: \rho(x, x_0) \leq r\}\). Let the \textit{exponential volume growth} be defined as NEWLINE\[NEWLINE\mu=\liminf\limits_{r\to\infty}\frac{1}{r}\log m(B_r(x_0)). NEWLINE\]NEWLINE The \textit{minimal exponential volume growth} is defined as NEWLINE\[NEWLINE\tilde{\mu}=\liminf\limits_{r\to\infty}\frac{1}{r}\inf\limits_{x\in X} \frac{m(B_r(x))}{m(B_1(x))}. NEWLINE\]NEWLINE The main results are the following: if all distance balls are compact, then NEWLINE\[NEWLINE\inf \sigma(L)\leq \frac{\tilde{\mu}^2}{4}; NEWLINE\]NEWLINE if, additionally, \(m\left(\bigcup\limits_r B_r(x_0)\right)=\infty\) for some \(x_0\), then NEWLINE\[NEWLINE\inf \sigma_{\text{ess}}(L)\leq \frac{\mu^2}{4}. NEWLINE\]NEWLINE As special cases, the authors discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric), the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian is investigated. This threshold is shown to lie at cubic polynomial growth.