110 research outputs found
The spectral shift function for compactly supported perturbations of Schr\"odinger operators on large bounded domains
We study the asymptotic behavior as L \to \infty of the finite-volume
spectral shift function for a positive, compactly-supported perturbation of a
Schr\"odinger operator in d-dimensional Euclidean space, restricted to a cube
of side length L with Dirichlet boundary conditions. The size of the support of
the perturbation is fixed and independent of L. We prove that the Ces\`aro mean
of finite-volume spectral shift functions remains pointwise bounded along
certain sequences L_n \to \infty for Lebesgue-almost every energy. In deriving
this result, we give a short proof of the vague convergence of the
finite-volume spectral shift functions to the infinite-volume spectral shift
function as L \to\infty . Our findings complement earlier results of W. Kirsch
[Proc. Amer. Math. Soc. 101, 509 - 512 (1987), Int. Eqns. Op. Th. 12, 383 - 391
(1989)] who gave examples of positive, compactly-supported perturbations of
finite-volume Dirichlet Laplacians for which the pointwise limit of the
spectral shift function does not exist for any given positive energy. Our
methods also provide a new proof of the Birman--Solomyak formula for the
spectral shift function that may be used to express the measure given by the
infinite-volume spectral shift function directly in terms of the potential.Comment: Minor changes and some rearrangements; version as publishe
Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries
Devices exhibiting the integer quantum Hall effect can be modeled by
one-electron Schroedinger operators describing the planar motion of an electron
in a perpendicular, constant magnetic field, and under the influence of an
electrostatic potential. The electron motion is confined to bounded or
unbounded subsets of the plane by confining potential barriers. The edges of
the confining potential barriers create edge currents. This is the second of
two papers in which we review recent progress and prove explicit lower bounds
on the edge currents associated with one- and two-edge geometries. In this
paper, we study various unbounded and bounded, two-edge geometries with soft
and hard confining potentials. These two-edge geometries describe the electron
confined to unbounded regions in the plane, such as a strip, or to bounded
regions, such as a finite length cylinder. We prove that the edge currents are
stable under various perturbations, provided they are suitably small relative
to the magnetic field strength, including perturbations by random potentials.
The existence of, and the estimates on, the edge currents are independent of
the spectral type of the operator.Comment: 57 page
Smoothness of Correlations in the Anderson Model at Strong Disorder
We study the higher-order correlation functions of covariant families of
observables associated with random Schr\"odinger operators on the lattice in
the strong disorder regime. We prove that if the distribution of the random
variables has a density analytic in a strip about the real axis, then these
correlation functions are analytic functions of the energy outside of the
planes corresponding to coincident energies. In particular, this implies the
analyticity of the density of states, and of the current-current correlation
function outside of the diagonal. Consequently, this proves that the
current-current correlation function has an analytic density outside of the
diagonal at strong disorder
CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary
The purpose of this paper is to describe certain CR-covariant differential
operators on a strictly pseudoconvex CR manifold as residues of the
scattering operator for the Laplacian on an ambient complex K\"{a}hler manifold
having as a `CR-infinity.' We also characterize the CR -curvature in
terms of the scattering operator. Our results parallel earlier results of
Graham and Zworski \cite{GZ:2003}, who showed that if is an asymptotically
hyperbolic manifold carrying a Poincar\'{e}-Einstein metric, the -curvature
and certain conformally covariant differential operators on the `conformal
infinity' of can be recovered from the scattering operator on . The
results in this paper were announced in \cite{HPT:2006}.Comment: 32 page
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