Abstract:
We consider the motion of a relativistic charged zero-mass fermion in Coulomb and Aharonov–Bohm potentials in $2+1$ dimensions. With these singular external potentials, we construct one-parameter self-adjoint Dirac Hamiltonians classified by self-adjoint boundary conditions. We show that if the so-called effective charge becomes overcritical, then virtual (quasistationary) bound states occur. The wave functions corresponding to these states have large amplitudes near the Coulomb center, and their energy spectrum is quasidiscrete and consists of a number of broadened levels of a width related to the inverse lifetime of the quasistationary state. We derive equations for the quasidiscrete spectra and quasistationary state lifetimes and solve these equations in physically interesting cases. We study the so-called local densities of state, which can be assessed in physical experiments, as functions of the energy and the problem parameters, investigating these densities both analytically and graphically.
Keywords:singular external potential, self-adjoint extension of a Hamiltonian, self-adjoint boundary condition, massless fermion, Coulomb potential in 2+1 dimensions, Aharonov–Bohm potential in 2+1 dimensions, virtual (quasistationary) bound state, quasidiscrete energy level, level width.
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