### 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*L* _{ z }. Let λ (*B*) [λ (*m*,*B*)] denote the smallest eigenvalue of *H* (*B*) [*H* (*m*,*B*)]. For *V*=−*e* ^{2}/*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*) =*e* *B*‖*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*/*r* ^{2}+*r* ^{2} for *B*=0, λ (0) =λ (0,0) and the ground statewavefunction is an almost everywhere positive function with *L* _{ z }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.

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