Localization Landscapes as Operator-Induced Geometry for Heterogeneous Systems

Abstract

Heterogeneous operators often organize response through hidden barriers, wells, and weakly communicating compartments that are not faithfully described by Euclidean distance or raw graph connectivity alone. We study this structure through the localization landscape, obtained from a single source solve, and the associated effective potential $W_{\mathrm{eff}}=1/u$. Starting from both discrete Schrödinger-type operators and heterogeneous reaction-diffusion operators, we show how $W_{\mathrm{eff}}$ induces effective wells, basin partitions, and Agmon-type neighborhoods that serve as an operator-aware geometry for localization and transport. This geometry explains the spatial support of low-energy eigenmodes, produces adaptive receptive fields on a fixed graph, sharpens locality in attention-like interactions, and yields bottleneck-sensitive modulation of diffusion. We also include a finite-element reaction-diffusion example with strongly compartmentalized coefficients, demonstrating that the same framework extends beyond simple grid operators to heterogeneous continuum systems. The results support a precise conclusion: localization-landscape geometry is a strong inductive bias when relevance is controlled by confinement, barriers, and low-energy accessibility, but its benefits are task-dependent rather than universal.

14 pages, 8 numbered figures