Wave Packet Propagation in Anisotropic Media: Interplay of Comb Geometry and a Dirac δ-Barrier

We investigate numerically the time-dependent Schrödinger equation on a two-dimensional comb in the presence of a localized Dirac delta potential on the backbone. The geometric constraint and the point interaction are both regularized via narrow Gaussian approximations, and the resulting equation is discretized with second-order finite differences and integrated with a fourth-order Runge–Kutta scheme on a finite domain. Two initial states are considered—a Gaussian wave packet and a comb-like wave function—and two barrier configurations are studied: a potential centered at the origin and one displaced along the backbone. In both cases, the Dirac delta interaction suppresses the probability density at the barrier position and reorganizes the wave packet into a persistent split-peak profile, in contrast to the single spreading maximum observed in the free case. When the barrier is displaced, due to reflection from the barrier, the left–right symmetry is broken and the density remains asymmetrically concentrated near the interaction site. To elucidate the results further, we also analyze the second moment, which undoubtedly confirms that the anomalous dynamics features, such as the subdiffusive behavior on comb-like structures in classical systems are retained in the quantum case as well.