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Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians. (English) Zbl 1286.35180

Summary: Let \(H_{0,D}\) (respectively, \(H_{0,N}\)) be the Schrödinger operator in constant magnetic field on the half-plane with Dirichlet (respectively, 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 behavior of the discrete spectrum of \(H_{\ell}\) near \(\inf \sigma_{\text{ess}}(H_{\ell}) = \inf \sigma(H_{0,\ell}), \;\ell = D, N\). Applying these Hamiltonians, we show that \(\sigma_{\text{disc}}(H_D)\) is infinite even if \(V\) has a compact support, while \(\sigma_{\text{disc}}(H_N)\) could be finite or infinite depending on the decay rate of \(V\).

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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