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 $H_\ell : = H_{0, \ell} - V$, $\ell =D,N$, 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 $H_\ell$ near $\inf \sigma_{ess}(H_\ell) = \inf \sigma(H_{0,\ell})$, $\ell = D,N$. Applying these Hamiltonians, we show that $\sigma_{disc}(H_D)$ is infinite even if $V$ has a compact support, while $\sigma_{disc}(H_N)$ could be finite or infinite depending on the decay rate of $V$.