We consider the Schrödinger operator, with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H(V ) near the boundary points of its essential spectrum. If the decay of V is Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasiclassical formulas known for the case where V admits a power-like decay.

No abstract

We consider the 3D Pauli operator with nonconstant magnetic field B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy asymptotics of the corresponding spectral shift function. As a corollary, for generic negative V , we obtain a generalized Levinson formula, relating the low-energy asymptotics of the eigenvalue counting function and of the scattering phase of the perturbed operator.

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

No abstract