163 research outputs found
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
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
Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
In a previous paper (arXiv:math-ph/0604055) we introduced a very simple
PT-symmetric non-Hermitian Hamiltonian with real spectrum and derived a closed
formula for the metric operator relating the problem to a Hermitian one. In
this note we propose an alternative formula for the metric operator, which we
believe is more elegant and whose construction -- based on a backward use of
the spectral theorem for self-adjoint operators -- provides new insights into
the nature of the model.Comment: LaTeX, 6 page
Superevolution
Usually, in supersymmetric theories, it is assumed that the time-evolution of
states is determined by the Hamiltonian, through the Schr\"odinger equation.
Here we explore the superevolution of states in superspace, in which the
supercharges are the principal operators. The superevolution equation is
consistent with the Schr\"odinger equation, but it avoids the usual degeneracy
between bosonic and fermionic states. We discuss superevolution in
supersymmetric quantum mechanics and in a simple supersymmetric field theory.Comment: 23 page
Higher order Schrodinger and Hartree-Fock equations
The domain of validity of the higher-order Schrodinger equations is analyzed
for harmonic-oscillator and Coulomb potentials as typical examples. Then the
Cauchy theory for higher-order Hartree-Fock equations with bounded and Coulomb
potentials is developed. Finally, the existence of associated ground states for
the odd-order equations is proved. This renders these quantum equations
relevant for physics.Comment: 19 pages, to appear in J. Math. Phy
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