Resonant Weighted Nonlocal Schr\"odinger Equation with Gauge Invariance, Conservation Laws and Measurable Phase Detuning

Abstract

We present a gauge-invariant Schr\"odinger-type evolution that combines (i) weighted local diffusion, (ii) symmetric nonlocal exchange through a kernel operator, and (iii) a mean-free phase-resonant drive. The resulting Resonant Weighted Nonlocal Schr\"odinger (RWNS) equation exactly conserves mass and, when the drive is absent, admits a Hamiltonian structure with energy conservation. Under standard assumptions on the weight, kernel, and nonlinearity, we establish local well-posedness in $H^1$ and provide defocusing conditions for global continuation. Linearization yields a dispersion relation in which the nonlocal kernel and the mean-free phase field contribute additively to a measurable spectral detuning. Building on this, we define two observables: a wavenumber-resolved detuning $\Delta\omega(k)$ and a kernel-contrast functional $\Xi[\psi]$ that isolates the nonlocal exchange. We outline feasible implementations in nonlinear-optical lattices and cavity-assisted cold-atom platforms, and discuss conceptual links to propagation-induced phase signatures in astrophysical media. The RWNS model thus offers a compact and analytically tractable framework that unifies weighted local dynamics, symmetric nonlocality, and a mean-free phase drive, yielding clear, testable predictions for laboratory measurements and, in principle, precision timing data.