We consider a Schrödinger operator with a constant magnetic field in a one-half
three-dimensional space, with Neumann-type boundary conditions. It is known
from the works by Lu–Pan and Helffer–Morame that the lower bound of its spectrum
is less than b, the intensity of the magnetic field, provided that the magnetic
field is not normal to the boundary. We prove that the spectrum under b is a finite
set of eigenvalues (each of infinite multiplicity). In the case when the angle between
the magnetic field and the boundary is small, we give a sharp asymptotic
expansion of the number of these eigenvalues.