Ground state properties and lower bounds for energy levels of a particle in a uniform magnetic field and external potential

Abstract

The Hamiltonian H (B), for a particle of mass μ and charge e in a uniform magnetic field of strength B in the z direction and an external axially symmetric potential V, is a direct sum of operators H (m,B) acting in the subspace of eigenvalue m of the z component of angular momentum Lz. Let λ (B) [λ (m,B)] denote the smallest eigenvalue of H (B) [H (m,B)]. For V=−e2/r (r=‖x‖), the attractive Coulomb potential, we obtain lower bounds l (m,B), for the spectrum of H (m,B) such that l (0,B) ≳l (0,0) =λ (0,0) for B≳0, l (m,B) ≳l (0,B) for m≠0, and l (‖m‖,B)−l (−‖m‖,B) =eB‖m‖/μ, l (m′,B) ≳l (m,B) if m′<m?0. We show at least for an interval [0,B′≳0] of B that the ground state of H (B) is the lowest eigenvalue, λ (0,B), of H (0,B) and is an almost everywhere positive function. If V=A/r2+r2 for B=0, λ (0) =λ (0,0) and the ground state wavefunction is an almost everywhere positive function with Lz eigenvalue zero. However, for large A, we prove that for an interval of B away from B=0, the lowest eigenvalue, λ (−1,B), of H (−1,B) is below the lowest eigenvalue, λ (0,B), of H (0,B) and that the ground state of H (B) is not an almost everywhere positive function.