We consider a Schrödinger operator with a constant magnetic field in a one-halfthree-dimensional space, with Neumann-type boundary conditions. It is knownfrom the works by Lu-Pan and Helffer-Morame that the lower bound of its spectrumis less than b, the intensity of the magnetic field, provided that the magneticfield is not normal to the boundary. We prove that the spectrum under b is a finiteset of eigenvalues (each of infinite multiplicity). In the case when the angle betweenthe magnetic field and the boundary is small, we give a sharp asymptoticexpansion of the number of these eigenvalues.

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