45 research outputs found
Transmutations and spectral parameter power series in eigenvalue problems
We give an overview of recent developments in Sturm-Liouville theory
concerning operators of transmutation (transformation) and spectral parameter
power series (SPPS). The possibility to write down the dispersion
(characteristic) equations corresponding to a variety of spectral problems
related to Sturm-Liouville equations in an analytic form is an attractive
feature of the SPPS method. It is based on a computation of certain systems of
recursive integrals. Considered as families of functions these systems are
complete in the -space and result to be the images of the nonnegative
integer powers of the independent variable under the action of a corresponding
transmutation operator. This recently revealed property of the Delsarte
transmutations opens the way to apply the transmutation operator even when its
integral kernel is unknown and gives the possibility to obtain further
interesting properties concerning the Darboux transformed Schr\"{o}dinger
operators.
We introduce the systems of recursive integrals and the SPPS approach,
explain some of its applications to spectral problems with numerical
illustrations, give the definition and basic properties of transmutation
operators, introduce a parametrized family of transmutation operators, study
their mapping properties and construct the transmutation operators for Darboux
transformed Schr\"{o}dinger operators.Comment: 30 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1111.444
Transmutations for Darboux transformed operators with applications
We solve the following problem. Given a continuous complex-valued potential
q_1 defined on a segment [-a,a] and let q_2 be the potential of a Darboux
transformed Schr\"odinger operator. Suppose a transmutation operator T_1 for
the potential q_1 is known such that the corresponding Schr\"odinger operator
is transmuted into the operator of second derivative. Find an analogous
transmutation operator T_2 for the potential q_2.
It is well known that the transmutation operators can be realized in the form
of Volterra integral operators with continuously differentiable kernels. Given
a kernel K_1 of the transmutation operator T_1 we find the kernel K_2 of T_2 in
a closed form in terms of K_1. As a corollary interesting commutation relations
between T_1 and T_2 are obtained which then are used in order to construct the
transmutation operator for the one-dimensional Dirac system with a scalar
potential
A Moving Magnetic Trap Decelerator: a New Source for Cold Atoms and Molecules
We present an experimental realization of a moving magnetic trap decelerator,
where paramagnetic particles entrained in a cold supersonic beam are
decelerated in a co-moving magnetic trap. Our method allows for an efficient
slowing of both paramagnetic atoms and molecules to near stopping velocities.
We show that under realistic conditions we will be able to trap and decelerate
a large fraction of the initial supersonic beam. We present our first results
on deceleration in a moving magnetic trap by bringing metastable neon atoms to
near rest. Our estimated phase space volume occupied by decelerated particles
at final velocity of 50 m/s shows an improvement of two orders of magnitude as
compared to currently available deceleration techniques