  The model of free charged Bosons in an external constant magnetic field
inside a cylinder, one of the few locally gauge covariant systems amenable to
analytic treatment, is rigorously investigated in the semiclassical
approximation. The model was first studied by Schafroth and is suitable for the
description of quasi-bound electron pairs localized in physical space,
so-called Schafroth pairs, which occur in certain compounds. Under the
assumption of existence of a solution of the semiclassical problem for which
the ground state (g.s.) expectation value of the current $<\vec{j}(\vec{x})>$
is of the London form, i.e., $<\vec{j}(\vec{x})> = -c |\phi_{0}(\vec{x})|^{2}
\vec{A}(\vec{x})$, where c is a positive constant, $\vec{A}$ the vector
potential and $\phi_{0}$ the one-particle g.s. wave-function. as well as some
regularity assumptions, the magnetic induction may be proved to decay
exponentially from its value on the surface of the cylinder. An important role
is played by a theorem on the pointwise monotonicity of the ground state
wave-function on the potential.
