Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers

Abstract

For every dimension d >= 2 and every measurable spatial weight w >= 1 essentially bounded on bounded sets, we construct a connected tower of enlarging cubes whose ordinary Dirichlet Laplacian has a complete orthonormal eigenbasis and essential spectrum [0,infinity). Given any finite nondecreasing r(E) >= 1 tending to infinity, the tower can be chosen so that weighted local-source evolution below energy E is bounded uniformly in time by C_K min(r(E),sqrt(E+1)). More generally, any prescribed Borel spectral filter 0 < h <= 1 uniformly vanishing at high energy admits a uniform weighted local-source bound. Local Neumann approximation targets and a summable spectral-projection budget supply the uniform source estimate. With a rational initial room, computable local weight upper bounds and a supplied computable filter-tail modulus, the geometry and its finite spectral certificates can be produced by a procedure terminating at every stage. Crack-aware primal and flux approximation certify whole-resolvent compressions and strict whole-cluster spectral tests. Explicit sufficient aperture bounds are logarithmic in the plane and linear in higher dimensions. For uniformly diverging spatial weights, a uniformly bounded all-energy factor is impossible; the geometry depends on the prescribed rate.