Fractional Green-Operator Methods for a Schrödinger–Poisson-Type System with Nonlocal Self-Consistent Fields

We study a fractional Schrödinger–Poisson system involving the spectral fractional Laplacian on a bounded domain Ω⊂RN(N>2s) subject to homogeneous Dirichlet boundary conditions. The model consists of a fractional Schrödinger equation coupled with a fractional Poisson equation through a self-consistent potential. Using the spectral Green operator associated with (−Δ)t, the coupled system is reduced to a single nonlocal integro-differential equation. The associated Green kernel admits a spectral representation in terms of the Dirichlet eigenpairs of the Laplacian. Under suitable assumptions on the Green kernel and Lipschitz conditions on the nonlinearities, we establish the existence of weak solutions together with local uniqueness within the contraction framework via the Banach fixed point theorem for sufficiently small coupling parameters. We further investigate the regularity of the self-consistent potential, continuous dependence on the model parameters, and a conditional convergence to the classical Schrödinger–Poisson system as s,t→1−. A numerical illustration based on truncated spectral expansions is presented to demonstrate the practical implementation of the proposed framework. Unlike the predominantly variational methods available in the literature, the proposed framework combines a spectral Green-operator reduction with an operator-theoretic fixed-point analysis, providing a constructive formulation that is directly amenable to numerical implementation.