Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers

Abstract

We construct connected quasi-conical Euclidean domains for the ordinary Dirichlet Laplacian with prescribed weighted bounds on the entire space of inputs supported in a bounded region. For every dimension d≥2, locally essentially bounded weight w≥1, and prescribed positive Borel filter h≤1 uniformly vanishing at high energy, one adaptive cube-window tower has a complete eigenbasis, essential spectrum [0,∞), and a weighted bound uniform over bounded spectral multipliers of norm at most one. Consequences include arbitrarily slowly prescribed divergent energy-ceiling bounds and weighted recurrence uniform over the local source unit ball. Bounded all-energy control is impossible for uniformly diverging weights. With a rational first room, computable spatial upper bounds and a computable filter-tail modulus, a terminating procedure produces every finite geometry prefix and its spectral certificates. If an observable is additionally supplied as a computable continuous function of the resolvent, dominated by the filter, we construct finite-rank approximants to its actual infinite-domain local-source operator with certified weighted operator error. This includes half-smoothed evolution at each computable time on the corresponding domains. The geometry is chosen after the weight and filter; arbitrary Borel observables need not be computably evaluable. The proof uses local Neumann approximation, summable whole-cluster continuation, aperture bounds and crack-aware physical resolvent certification.