Bound-State Spectra of a Lifshitz-Type Dirac Equation in (2+1) Dimensions

Abstract

We investigate a Dirac-type equation in (2+1) dimensions modified by Lifshitz spatial derivatives with dynamical exponent $z=2$, focusing on the spectral properties of bound states under radial confinement. Analytical solutions are obtained for constant backgrounds, hard-wall confinement, and harmonic potentials, while logarithmic confinement is treated numerically via the Numerov method and complemented by a semiclassical WKB analysis. The resulting spectra exhibit characteristic scaling laws governed by the Lifshitz parameter $b$, including $E - M \propto b/R_0^2$ for hard-wall confinement, $E - M \propto \sqrt{2b}\,\omega$ for harmonic trapping, and $E - M \sim \alpha \ln\sqrt{b}$ in the semiclassical regime of logarithmic confinement. These results provide a consistent characterization of how higher-order spatial derivatives modify bound-state spectra in two-dimensional Dirac systems and may be relevant for effective descriptions of materials with quadratic low-energy dispersion, such as bilayer graphene and related anisotropic 2D systems.