  We consider the Neumann realization of the magnetic laplacian in
$\mathbb{R}^3_+$, in the case in which the magnetic field has a piecewise
constant strength and a uniform direction. This operator is expected to be an
effective model in studying the threshold to the normal state for a 3D
superconductor exposed to a discontinuous magnetic field. We review some recent
results above the infimum of the spectrum of the aforementioned operator.
