61 research outputs found
Anomalous electron trapping by localized magnetic fields
We consider an electron with an anomalous magnetic moment g>2 confined to a
plane and interacting with a nonzero magnetic field B perpendicular to the
plane. We show that if B has compact support and the magnetic flux in the
natural units is F\ge 0, the corresponding Pauli Hamiltonian has at least 1+[F]
bound states, without making any assumptions about the field profile.
Furthermore, in the zero-flux case there is a pair of bound states with
opposite spin orientations. Using a Birman-Schwinger technique, we extend the
last claim to a weak rotationally symmetric field with B(r) = O(r^{-2-\delta})
correcting thus a recent result. Finally, we show that under mild regularity
assumptions the existence can be proved for non-symmetric fields with tails as
well.Comment: A LaTeX file, 12 pages; to appear in J. Phys. A: Math. Ge
One-Dimensional Discrete Stark Hamiltonian and Resonance Scattering by Impurities
A one-dimensional discrete Stark Hamiltonian with a continuous electric field
is constructed by extension theory methods. In absence of the impurities the
model is proved to be exactly solvable, the spectrum is shown to be simple,
continuous, filling the real axis; the eigenfunctions, the resolvent and the
spectral measure are constructed explicitly. For this (unperturbed) system the
resonance spectrum is shown to be empty. The model considering impurity in a
single node is also constructed using the operator extension theory methods.
The spectral analysis is performed and the dispersion equation for the
resolvent singularities is obtained. The resonance spectrum is shown to contain
infinite discrete set of resonances. One-to-one correspondence of the
constructed Hamiltonian to some Lee-Friedrichs model is established.Comment: 20 pages, Latex, no figure
Spectral flow and level spacing of edge states for quantum Hall hamiltonians
We consider a non relativistic particle on the surface of a semi-infinite
cylinder of circumference submitted to a perpendicular magnetic field of
strength and to the potential of impurities of maximal amplitude . This
model is of importance in the context of the integer quantum Hall effect. In
the regime of strong magnetic field or weak disorder it is known that
there are chiral edge states, which are localised within a few magnetic lengths
close to, and extended along the boundary of the cylinder, and whose energy
levels lie in the gaps of the bulk system. These energy levels have a spectral
flow, uniform in , as a function of a magnetic flux which threads the
cylinder along its axis. Through a detailed study of this spectral flow we
prove that the spacing between two consecutive levels of edge states is bounded
below by with , independent of , and of the
configuration of impurities. This implies that the level repulsion of the
chiral edge states is much stronger than that of extended states in the usual
Anderson model and their statistics cannot obey one of the Gaussian ensembles.
Our analysis uses the notion of relative index between two projections and
indicates that the level repulsion is connected to topological aspects of
quantum Hall systems.Comment: 22 pages, no figure
Magnetic strip waveguides
We analyze the spectrum of the "local" Iwatsuka model, i.e. a two-dimensional
charged particle interacting with a magnetic field which is homogeneous outside
a finite strip and translationally invariant along it. We derive two new
sufficient conditions for absolute continuity of the spectrum. We also show
that in most cases the number of open spectral gaps of the model is finite. To
illustrate these results we investigate numerically the situation when the
field is zero in the strip being screened, e.g. by a superconducting mask.Comment: 22 pages, a LaTeX source file with three eps figure
Bound States in Mildly Curved Layers
It has been shown recently that a nonrelativistic quantum particle
constrained to a hard-wall layer of constant width built over a geodesically
complete simply connected noncompact curved surface can have bound states
provided the surface is not a plane. In this paper we study the weak-coupling
asymptotics of these bound states, i.e. the situation when the surface is a
mildly curved plane. Under suitable assumptions about regularity and decay of
surface curvatures we derive the leading order in the ground-state eigenvalue
expansion. The argument is based on Birman-Schwinger analysis of Schroedinger
operators in a planar hard-wall layer.Comment: LaTeX 2e, 23 page
Multiple bound states in scissor-shaped waveguides
We study bound states of the two-dimensional Helmholtz equations with
Dirichlet boundary conditions in an open geometry given by two straight leads
of the same width which cross at an angle . Such a four-terminal
junction with a tunable can realized experimentally if a right-angle
structure is filled by a ferrite. It is known that for there is
one proper bound state and one eigenvalue embedded in the continuum. We show
that the number of eigenvalues becomes larger with increasing asymmetry and the
bound-state energies are increasing as functions of in the interval
. Moreover, states which are sufficiently strongly bent exist in
pairs with a small energy difference and opposite parities. Finally, we discuss
how with increasing the bound states transform into the quasi-bound
states with a complex wave vector.Comment: 6 pages, 6 figure
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