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

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/R₀² for hard-wall confinement, E - M propto sqrt{2b} ω for harmonic trapping, and E - M sim αlnsqrt{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.