This paper concerns a boundary integral formulation for the two-dimensional massive Dirac equation. The mass term is assumed to jump across a one-dimensional interface, which models a transition between two insulating materials. This jump induces surface waves that propagate outward along the interface but decay exponentially in the transverse direction. After providing a derivation of our integral equation, we prove that it has a unique solution for almost all choices of parameters using holomorphic perturbation theory. We then extend these results to a Dirac equation with two interfaces. Finally, we implement a fast numerical method for solving our boundary integral equations and present several numerical examples of solutions and scattering effects.

翻译：本文研究二维有质量狄拉克方程的边界积分公式。假设质量项在一维界面上发生跳跃，该跳跃模拟了两种绝缘材料之间的过渡。这种跳跃会诱发沿界面传播但在横向方向上指数衰减的表面波。在推导出积分方程后，我们利用全纯扰动理论证明该方程在几乎所有参数选择下具有唯一解。随后将这些结果推广至具有两个界面的狄拉克方程。最后，我们实现了求解边界积分方程的快速数值方法，并展示了若干解与散射效应的数值算例。