The transport and excitation characteristics of electrons in disordered systems represent a fundamental theme in condensed matter physics. In realistic materials such as semiconductor heterostructures and ultracold atomic lattices, the presence of impurities significantly alters electronic quantum states, leading to phenomena like Anderson localization and spectral line broadening. Traditionally, the zero-range δ-function potential (point-scatterer model) has been widely adopted due to its simplicity. However, this model suffers from severe theoretical deficiencies in higher dimensions (d ≥ 2), most notably the pathological ultraviolet (UV) divergence in self-energy and spectral function calculations. To circumvent these issues, researchers often resort to non-physical hard cutoffs or phenomenological parameterization, which compromises theoretical precision. This study aims to establish a unified analytical framework using a Gaussian-type impurity potential V (r) ~ e-^r2/2a2 to describe smooth disordered fields. By introducing a finite range a, we provide a natural physical scale that regularizes the UV behavior and suppresses Anderson localization through dominant forward scattering.
We employ the imaginary-time Green’s function formalism within the framework of the first Born approximation. To address the mathematical complexity of convolution integrals in momentum space, we introduce a novel analytical technique based on the Weierstrass operator. By transforming the integral equations into a differential form, we bypass the traditional diffculties associated with momentum-space integration. The diagrammatic technique involving non-crossing wigwam diagrams is utilized to derive the self-energy Σ_k(ω) and the spectral function A_k(ω) for both two-dimensional (2D) and three-dimensional (3D) systems.
Our results provide closed-form analytical expressions for the self-energy and spectral function across the entire momentum space. (1) Resolution of UV Divergence: We demonstrate that the exponential decay of the Gaussian potential in momentum space acts as a natural renormalization operator. This successfully eliminates the logarithmic divergence in 2D and the linear divergence in 3D that plague the δ-function model, ensuring the analytical completeness of the spectral theory. (2) Operator-based Regularization: The application of the Weierstrass operator reveals a “Gaussian filtering effect” that smooths the poles of the Green’s function, effectively regularizing both the real and imaginary parts of the self-energy. (3) Momentum and Dimensional Dependency: The derived analytical solutions reveal a significant non-local effect in the quasiparticle lifetime (or momentum relaxation time). In the high-energy (large momentum) limit, the spectral broadening decreases rapidly, indicating a transition to a “forwardscattering dominated” regime. This behavior is highly consistent with experimental observations in doped semiconductor transport.
This work provides a rigorous theoretical foundation for understanding electron transport and disorder effects. By incorporating the correlation length a, we bridge the gap between point-scattering and smooth disorder fields, offering a unified framework for refining the Drude transport picture. Furthermore, the Weierstrass transformation technique developed herein is versatile; it can be extended to systems with spin-orbit coupling, magnetic fields, or electron-electron interactions, and can simplify the calculation of higher-order self-energies and vertex corrections in response functions. The universal dependence of physical corrections on the potential strength V₀ and range a suggests a generalized approach to impurity scattering problems, paving the way for future research in complex lattice systems and Bloch energy bands.