218 research outputs found
Singular components of spectral measures for ergodic Jacobi matrices
For ergodic 1d Jacobi operators we prove that the random singular components
of any spectral measure are almost surely mutually disjoint as long as one
restricts to the set of positive Lyapunov exponent. In the context of extended
Harper's equation this yields the first rigorous proof of the Thouless' formula
for the Lyapunov exponent in the dual regions.Comment: to appear in the Journal of Mathematical Physics, vol 52 (2011
Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two
We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two
with , being an integer, and a quasi-periodic potential V(\x).
We prove that the absolutely continuous spectrum of contains a semiaxis and
there is a family of generalized eigenfunctions at every point of this semiaxis
with the following properties. First, the eigenfunctions are close to plane
waves at the high energy region. Second, the isoenergetic
curves in the space of momenta \k corresponding to these eigenfunctions have
a form of slightly distorted circles with holes (Cantor type structure). A new
method of multiscale analysis in the momentum space is developed to prove these
results.Comment: 125 pages, 4 figures. arXiv admin note: incorporates arXiv:1205.118
Edge Currents for Quantum Hall Systems, I. One-Edge, 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 unbounded subsets
of the plane by confining potential barriers. The edges of the confining
potential barrier create edge currents. In this, the first of two papers, we
prove explicit lower bounds on the edge currents associated with one-edge,
unbounded geometries formed by various confining potentials. This work extends
some known results that we review. The edge currents are carried by states with
energy localized between any two Landau levels. These one-edge geometries
describe the electron confined to certain unbounded regions in the plane
obtained by deforming half-plane regions. We prove that the currents are stable
under various potential perturbations, provided the perturbations are suitably
small relative to the magnetic field strength, including perturbations by
random potentials. For these cases of one-edge geometries, the existence of,
and the estimates on, the edge currents imply that the corresponding
Hamiltonian has intervals of absolutely continuous spectrum. In the second
paper of this series, we consider the edge currents associated with two-edge
geometries describing bounded, cylinder-like regions, and unbounded,
strip-like, regions.Comment: 68 page
Bound States at Threshold resulting from Coulomb Repulsion
The eigenvalue absorption for a many-particle Hamiltonian depending on a
parameter is analyzed in the framework of non-relativistic quantum mechanics.
The long-range part of pair potentials is assumed to be pure Coulomb and no
restriction on the particle statistics is imposed. It is proved that if the
lowest dissociation threshold corresponds to the decay into two likewise
non-zero charged clusters then the bound state, which approaches the threshold,
does not spread and eventually becomes the bound state at threshold. The
obtained results have applications in atomic and nuclear physics. In
particular, we prove that atomic ion with atomic critical charge and
electrons has a bound state at threshold given that , whereby the electrons are treated as fermions and the mass of the
nucleus is finite.Comment: This is a combined and updated version of the manuscripts
arXiv:math-ph/0611075v2 and arXiv:math-ph/0610058v
Effect of quasi-bound states on coherent electron transport in twisted nanowires
Quantum transmission spectra of a twisted electron waveguide expose the
coupling between traveling and quasi-bound states. Through a direct numerical
solution of the open-boundary Schr\"odinger equation we single out the effects
of the twist and show how the presence of a localized state leads to a
Breit-Wigner or a Fano resonance in the transmission. We also find that the
energy of quasi-bound states is increased by the twist, in spite of the
constant section area along the waveguide. While the mixing of different
transmission channels is expected to reduce the conductance, the shift of
localized levels into the traveling-states energy range can reduce their
detrimental effects on coherent transport.Comment: 8 pages, 9 color figures, submitte
Pauli-Fierz model with Kato-class potentials and exponential decays
Generalized Pauli-Fierz Hamiltonian with Kato-class potential \KPF in
nonrelativistic quantum electrodynamics is defined and studied by a path
measure. \KPF is defined as the self-adjoint generator of a strongly
continuous one-parameter symmetric semigroup and it is shown that its bound
states spatially exponentially decay pointwise and the ground state is unique.Comment: We deleted Lemma 3.1 in vol.
The Schr\"odinger operator on an infinite wedge with a tangent magnetic field
We study a model Schr\"odinger operator with constant magnetic field on an
infinite wedge with Neumann boundary condition. The magnetic field is assumed
to be tangent to a face. We compare the bottom of the spectrum to the model
spectral quantities coming from the regular case. We are particularly motivated
by the influence of the magnetic field and the opening angle of the wedge on
the spectrum of the model operator and we exhibit cases where the bottom of the
spectrum is smaller than in the regular case. Numerical computations enlighten
the theoretical approach
On the Geometry of Supersymmetric Quantum Mechanical Systems
We consider some simple examples of supersymmetric quantum mechanical systems
and explore their possible geometric interpretation with the help of geometric
aspects of real Clifford algebras. This leads to natural extensions of the
considered systems to higher dimensions and more complicated potentials.Comment: 18 page
Existence of the Stark-Wannier quantum resonances
In this paper we prove the existence of the Stark-Wannier quantum resonances
for one-dimensional Schrodinger operators with smooth periodic potential and
small external homogeneous electric field. Such a result extends the existence
result previously obtained in the case of periodic potentials with a finite
number of open gaps.Comment: 30 pages, 1 figur
Extended States for Polyharmonic Operators with Quasi-periodic Potentials in Dimension Two
We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two
with , being an integer, and a quasi-periodic potential V(\x).
We prove that the spectrum of contains a semiaxis and there is a family of
generalized eigenfunctions at every point of this semiaxis with the following
properties. First, the eigenfunctions are close to plane waves
at the high energy region. Second, the isoenergetic curves in the space of
momenta \k corresponding to these eigenfunctions have a form of slightly
distorted circles with holes (Cantor type structure). A new method of
multiscale analysis in the momentum space is developed to prove these results.Comment: This is an announcement only. Text with the detailed proof is under
preparation. 11 pages, 4 figures. arXiv admin note: text overlap with
arXiv:math-ph/0601008, arXiv:0711.4404, arXiv:1008.463
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