Volume growth, big jump, and essential spectrum for regular Dirichlet forms
Yuichi Shiozawa
TL;DR
This work derives a Persson-type upper bound for the bottom of the essential spectrum of generators of regular Dirichlet forms, separating small and big jumps via adapted length ρ_r and jump-height F_r. It then connects this spectral bound to volume-growth regimes (polynomial and exponential) and coefficient-growth effects (in the jump kernel and via time-change), showing that polynomial volume growth typically forces the essential spectrum to be non-positively bounded (λ_e = 0), while exponential growth can yield finite positive bounds. The paper also analyzes Ornstein–Uhlenbeck–type non-local operators, establishing regimes where the bottom of the essential spectrum is strictly positive even in finite volume and highlighting the sharpness of the bounds through concrete models on R^d and hyperbolic-type settings. Overall, the results extend Brooks–type spectral bounds to general regular Dirichlet forms with non-locality and no graph structure, tying spectral noncompactness to geometric-volume growth and jump-structure features.
Abstract
We establish an upper bound of the bottom of the essential spectrum for the generator associated with a regular Dirichlet form in terms of the rates of the volume growth/decay and big jump. Using this bound, we discuss how the bottom of the essential spectrum is affected by the volume growth and coefficient growth.