Let H 0,D (resp., H 0,N ) be the Schrödinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let H ℓ := H 0,ℓ − V , ℓ = 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 ℓ near inf σ ess (H ℓ ) = inf σ(H 0,ℓ ), ℓ = D, N . Applying these Hamiltonians, we show that σ disc (H D ) is infinite even if V has a compact support, while σ disc (H N ) could be finite or infinite depending on the decay rate of V .Keywords: magnetic Schrödinger operators, Dirichlet and Neumann boundary conditions, eigenvalue distribution 2010 AMS Mathematics Subject Classification: 35P20, 35J10, 47F05, 81Q10 [8]) and E N ∈ (0, b) (see e.g [7]). Further, assume that 0 ≤ V ∈ L ∞ 0 (O) := u ∈ L ∞ (O) | lim |x|→∞ u(x) = 0 . Set

Abstract. We consider the unperturbed operatorHere A is a magnetic potential which generates a constant magnetic field b > 0, and the edge potential W is a non-decreasing non-constant bounded function depending only on the first coordinate x 2 R of .x; y/ 2 R 2 . Then the spectrum of H 0 has a band structure and is absolutely continuous; moreover, the assumption lim x!1 .W .x/ W . x// < 2b implies the existence of infinitely many spectral gaps for H 0 . We consider the perturbed operators H˙D H 0˙V where the electric potential V 2 L 1 .R 2 / is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H˙in the spectral gaps of H 0 . We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to V . Further, we restrict our attention on perturbations V of compact support and constant sign. We establish a geometric condition on the support of V which guarantees the finiteness of the number of the eigenvalues of H˙in any spectral gap of H 0 . In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of H C (resp. H ) to the lower (resp. upper) edge of a given spectral gap, is Gaussian.Mathematics Subject Classification (2010). 35P20, 35J10, 47F05, 81Q10.

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We consider the Schrödinger operator with constant magnetic field defined on the halfplane with a Dirichlet boundary condition, H 0 , and a decaying electric perturbation V . We analyze the spectral density near the Landau levels, which are thresholds in the spectrum of H 0 , by studying the Spectral Shift Function (SSF) associated to the pair (H 0 + V, H 0 ). For perturbations of a fixed sign, we estimate the SSF in terms of the eigenvalue counting function for certain compact operators. If the decay of V is power-like, then using pseudodifferential analysis, we deduce that there are singularities at the thresholds and we obtain the corresponding asymptotic behavior of the SSF. Our technique gives also results for the Neumann boundary condition.

We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator, and show the existence of exactly one resonance near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to the upper eigenvalues of the transversal problem, showing that the number of resonances is bounded by the multiplicity of the eigenvalue and obtaining the corresponding asymptotic behavior.