Werner Kirsch

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We prove pure point spectrum with exponentially decaying eigenfunctions at all band edges for Schrödinger Operators with a periodic potential plus a random potential of the form Vω(x) = ∑ qi(ω)f (x − i), where f decays at infinity like |x|−m form > 4d resp.m > 3d depending on the regularity of f . The random variables qi are supposed to be independent and(More)
Suppose V is a potential decaying near infinity. Let us denote by NJ V) the number of eigenvalues of H = -A + V below -E. The finiteness (resp. infiniteness) of N,,(V) is determined by the rate of decay of V at infinity (see Reed-Simon [4, X111.31) In fact, N,(V) < co if V(x) B -c/~xI*+~, while N,(V) = cc for V(x) < -c/\x~*~‘. For the borderline case(More)
We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is(More)
We prove localization for Anderson-type random perturbations of periodic Schr dinger operators on R near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schr dinger operators(More)
We investigate the integrated density of states of the Schrr odinger operator in the Euclidean plane with a perpendicular constant magnetic eld and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the(More)
Let V^ and V^ be two ergodic random potentials on KA We consider the Schrόdinger operator Hω = H0 + Vω, with Ho= —A and for x = (x1,...,xd) if xt<0 if x^O ' We prove certain ergodic properties of the spectrum for this model, and express the integrated density of states in terms of the density of states of the stationary potentials V^ and V^\ Finally we(More)
We provide lower bounds on the eigenvalue splitting for — d/dx + V(x) depending only on qualitative properties of V. For example, if V is C on [α, b~] and £„, £„_ 1 are two successive eigenvalues of — d /dx + V with u(a) = u(b) = 0 boundary conditions, and if λ = max \E— V(x)\, then Ee(En_ ^EJ xem The exponential factor in such bounds are saturated(More)
We consider the three-dimensional Schrödinger operator with constant magnetic field and bounded random electric potential. We investigate the asymptotic behaviour of the integrated density of states for this operator as the norm of the magnetic field tends to infinity. Résumé On considère l’opérateur de Schrödinger tridimensionnel avec un champ magnétique(More)