Dirichlet and Neumann Eigenvalues for Half-Plane Magnetic Hamiltonians

Let $H_{0,D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_{\mathrm{\ell }}:=H_{0,\mathrm{\ell }}-V$, $\mathrm{\ell }=D,N$, where the scalar potential $V$ is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of $H_D$ and $H_N$ below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of $H_{\mathrm{\ell }}$ near $\inf \sigma_{ess}(H_\ell) = \inf \sigma(H_{0,\ell})$,$\ell = D,N$\mathrm{.}ApplyingtheseHamiltonians,weshowthat$\sigma_{disc}(H_D)$isinfiniteevenif$V$hasacompactsupport,while$\sigma_{disc}(H_N)$couldbefiniteorinfinitedependingonthedecayrateof$V$.