On the magnetic laplacian with a piecewise constant magnetic field in ${\mathbb{R}}_{+}^3$

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.