22 research outputs found
Exact Propagators for Soliton Potentials
Using the method of Darboux transformations (or equivalently supersymmetric
quantum mechanics) we obtain an explicit expression for the propagator for the
one-dimensional Schr\"odinger equation with a multi-soliton potential.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Frenkel electron on an arbitrary electromagnetic background and magnetic Zitterbewegung
We present Lagrangian which implies both necessary constraints and dynamical
equations for position and spin of relativistic spin one-half particle. The
model is consistent for any value of magnetic moment and for arbitrary
electromagnetic background. Our equations coincide with those of Frenkel in the
approximation in which the latter have been obtained by Frenkel. Transition
from approximate to exact equations yields two structural modifications of the
theory. First, Frenkel condition on spin-tensor turns into the Pirani
condition. Second, canonical momentum is no more proportional to velocity. Due
to this, even when (Frenkel case), the complete and approximate
equations predict different behavior of particle. The difference between
momentum and velocity means extra contribution into spin-orbit interaction. To
estimate the contribution, we found exact solution to complete equations for
the case of uniform magnetic field. While Frenkel electron moves around the
circle, our particle experiences magnetic {\it Zitterbewegung}, that is
oscillates in the direction of magnetic field with amplitude of order of
Compton wavelength for the fast particle. Besides, the particle has dipole
electric moment.Comment: 20 pages, 1 figure, close to published versio
Geometric Constructions Underlying Relativistic Description of Spin on the Base of Non-Grassmann Vector-Like Variable
Basic notions of Dirac theory of constrained systems have their analogs in
differential geometry. Combination of the two approaches gives more clear
understanding of both classical and quantum mechanics, when we deal with a
model with complicated structure of constraints. In this work we describe and
discuss the spin fiber bundle which appeared in various mechanical models where
spin is described by vector-like variable
Coupling between scattering channels with SUSY transformations for equal thresholds
Supersymmetric (SUSY) transformations of the multi-channel Schr\"odinger
equation with equal thresholds and arbitrary partial waves in all channels are
studied. The structures of the transformation function and the superpotential
are analyzed. Relations between Jost and scattering matrices of superpartner
potentials are obtained. In particular, we show that a special type of SUSY
transformation allows us to introduce a coupling between scattering channels
starting from a potential with an uncoupled scattering matrix. The possibility
for this coupling to be trivial is discussed. We show that the transformation
introduces bound and virtual states with a definite degeneracy at the
factorization energy. A detailed study of the potential and scattering matrices
is given for the case. The possibility of inverting coupled-channel
scattering data by such a SUSY transformation is demonstrated by several
examples (, and partial waves)
Exactly-solvable coupled-channel potential models of atom-atom magnetic Feshbach resonances from supersymmetric quantum mechanics
Starting from a system of radial Schr\"odinger equations with a vanishing
potential and finite threshold differences between the channels, a coupled exactly-solvable potential model is obtained with the help of a
single non-conservative supersymmetric transformation. The obtained potential
matrix, which subsumes a result obtained in the literature, has a compact
analytical form, as well as its Jost matrix. It depends on
unconstrained parameters and on one upper-bounded parameter, the factorization
energy. A detailed study of the model is done for the case: a
geometrical analysis of the zeros of the Jost-matrix determinant shows that the
model has 0, 1 or 2 bound states, and 0 or 1 resonance; the potential
parameters are explicitly expressed in terms of its bound-state energies, of
its resonance energy and width, or of the open-channel scattering length, which
solves schematic inverse problems. As a first physical application,
exactly-solvable atom-atom interaction potentials are constructed,
for cases where a magnetic Feshbach resonance interplays with a bound or
virtual state close to threshold, which results in a large background
scattering length.Comment: 19 pages, 15 figure
Eigenphase preserving two-channel SUSY transformations
We propose a new kind of supersymmetric (SUSY) transformation in the case of
the two-channel scattering problem with equal thresholds, for partial waves of
the same parity. This two-fold transformation is based on two imaginary
factorization energies with opposite signs and with mutually conjugated
factorization solutions. We call it an eigenphase preserving SUSY
transformation as it relates two Hamiltonians, the scattering matrices of which
have identical eigenphase shifts. In contrast to known phase-equivalent
transformations, the mixing parameter is modified by the eigenphase preserving
transformation.Comment: 16 pages, 1 figur
Spectral properties of non-conservative multichannel SUSY partners of the zero potential
Spectral properties of a coupled potential model obtained with
the help of a single non-conservative supersymmetric (SUSY) transformation
starting from a system of radial Schr\"odinger equations with the zero
potential and finite threshold differences between the channels are studied.
The structure of the system of polynomial equations which determine the zeros
of the Jost-matrix determinant is analyzed. In particular, we show that the
Jost-matrix determinant has zeros which may all correspond to
virtual states. The number of bound states satisfies . The
maximal number of resonances is . A perturbation technique
for a small coupling approximation is developed. A detailed study of the
inverse spectral problem is given for the case.Comment: 17 pages, 4 figure