Simple diamagnetic monotonicities for Schr$ouml$dinger operators with inhomogeneous magnetic fields of constant direction

Abstract

Under certain simplifying conditions we detect monotonicity properties of the ground-state energy and the canonical-equilibrium density matrix of a spinless charged particle in the Euclidean plane subject to a perpendicular, possibly inhomogeneous magnetic field and an additional scalar potential. Firstly, we point out a simple condition warranting that the ground-state energy does not decrease when the magnetic field and/or the potential is increased pointwise. Secondly, we consider the case in which both the magnetic field and the potential are constant along one direction in the plane and give a genuine path-integral argument for corresponding monotonicities of the density-matrix diagonal and the absolute value of certain off-diagonals. Our results complement to some degree results of M. Loss and B. Thaller [Commun. Math. Phys. 186 (1997) 95] and L. Erdos [J. Math. Phys. 38 (1997) 1289].