8,608 research outputs found
On a factorization of second order elliptic operators and applications
We show that given a nonvanishing particular solution of the equation
(divpgrad+q)u=0 (1) the corresponding differential operator can be factorized
into a product of two first order operators. The factorization allows us to
reduce the equation (1) to a first order equation which in a two-dimensional
case is the Vekua equation of a special form. Under quite general conditions on
the coefficients p and q we obtain an algorithm which allows us to construct in
explicit form the positive formal powers (solutions of the Vekua equation
generalizing the usual powers of the variable z). This result means that under
quite general conditions one can construct an infinite system of exact
solutions of (1) explicitly, and moreover, at least when p and q are real
valued this system will be complete in ker(divpgrad+q) in the sense that any
solution of (1) in a simply connected domain can be represented as an infinite
series of obtained exact solutions which converges uniformly on any compact
subset of . Finally we give a similar factorization of the operator
(divpgrad+q) in a multidimensional case and obtain a natural generalization of
the Vekua equation which is related to second order operators in a similar way
as its two-dimensional prototype does
Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics
We consider the following first order systems of mathematical physics.
1.The Dirac equation with scalar potential. 2.The Dirac equation with
electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The
system describing non-linear force free magnetic fields or Beltrami fields with
nonconstant proportionality factor. 5.The Maxwell equations for slowly changing
media. 6.The static Maxwell system.
We show that all this variety of first order systems reduces to a single
quaternionic equation the analysis of which in its turn reduces to the solution
of a Schroedinger equation with biquaternionic potential. In some important
situations the biquaternionic potential can be diagonalized and converted into
scalar potentials
Formation of three-particle clusters in hetero-junctions and MOSFET structures
A novel interaction mechanism in MOSFET structures and
hetero-junctions between the zone electrons of the two-dimensional (2D) gas and
the charged traps on the insulator side is considered. By applying a canonical
transformation, off-diagonal terms in the Hamiltonian due to the trapped level
subsystem are excluded. This yields an effective three-particle attractive
interaction as well as a pairing interaction inside the 2D electronic band. A
type of Bethe- Goldstone equation for three particles is studied to clarify the
character of the binding and the energy of the three-particle bound states. The
results are used to offer a possible explanation of the Metal-Insulator
transition recently observed in MOSFET and hetero-junctions.Comment: 4 page
Quaternion Analysis for Generalized Electromagnetic Fields of Dyons in Isotropic Medium
Quaternion analysis of time dependent Maxwell's equations in presence of
electric and magnetic charges has been developed and the solutions for the
classical problem of moving charges (electric and magnetic) are obtained in
unique, simple and consistent manner
Magnetoresistance of a two-dimensional electron gas in a parallel magnetic field
The conductivity of a two-dimensional electron gas in a parallel magnetic
field is calculated. We take into account the magnetic field induced
spin-splitting, which changes the density of states, the Fermi momentum and the
screening behavior of the electron gas. For impurity scattering we predict a
positive magnetoresistance for low electron density and a negative
magnetoresistance for high electron density. The theory is in qualitative
agreement with recent experimental results found for Si inversion layers and Si
quantum wells.Comment: 4 pages, figures included, PDF onl
Thermodynamic Signature of a Two-Dimensional Metal-Insulator Transition
We present a study of the compressibility, K, of a two-dimensional hole
system which exhibits a metal-insulator phase transition at zero magnetic
field. It has been observed that dK/dp changes sign at the critical density for
the metal-insulator transition. Measurements also indicate that the insulating
phase is incompressible for all values of B. Finally, we show how the phase
transition evolves as the magnetic field is varied and construct a phase
diagram in the density-magnetic field plane for this system.Comment: 4 pages, 4 figures, submitted to Physical Review Letters; version 1
is identical to version 2 but didn't compile properl
Hall Coefficient of a Dilute 2D Electron System in Parallel Magnetic Field
Measurements in magnetic fields applied at a small angle with respect to the
2D plane of the electrons of a low-density silicon MOSFET indicate that the
Hall coefficient is independent of parallel field from H=0 to , the
field above which the longitudinal resistance saturates and the electrons have
reached full spin-polarization. This implies that the mobilities of the spin-up
and spin-down electrons remain comparable at all magnetic fields, and suggests
there is strong mixing of spin-up and spin-down electron states.Comment: 4 pages, 2 figure
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