595 research outputs found
How to construct spin chains with perfect state transfer
It is shown how to systematically construct the quantum spin chains with
nearest-neighbor interactions that allow perfect state transfer (PST). Sets of
orthogonal polynomials (OPs) are in correspondence with such systems. The key
observation is that for any admissible one-excitation energy spectrum, the
weight function of the associated OPs is uniquely prescribed. This entails the
complete characterization of these PST models with the mirror symmetry property
arising as a corollary. A simple and efficient algorithm to obtain the
corresponding Hamiltonians is presented. A new model connected to a special
case of the symmetric -Racah polynomials is offered. It is also explained
how additional models with PST can be derived from a parent system by removing
energy levels from the one-excitation spectrum of the latter. This is achieved
through Christoffel transformations and is also completely constructive in
regards to the Hamiltonians.Comment: 7 page
The Geodesic Deviation Method and Extreme Mass-Ratio Systems: Theoretical methods and application to the calculation of gravitational waves
Brand, J.F.J. van den [Promotor]Holten, J.W. van [Promotor
New connection formulae for some q-orthogonal polynomials in q-Askey scheme
New nonlinear connection formulae of the q-orthogonal polynomials, such
continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and
q-Gegenbauer polynomials, in terms of their respective classical analogues are
obtained using a special realization of the q-exponential function as infinite
multiplicative series of ordinary exponential function
Quantum state transfer in spin chains with q-deformed interaction terms
We study the time evolution of a single spin excitation state in certain
linear spin chains, as a model for quantum communication. Some years ago it was
discovered that when the spin chain data (the nearest neighbour interaction
strengths and the magnetic field strengths) are related to the Jacobi matrix
entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect
state transfer takes place. The extension of these ideas to other types of
discrete orthogonal polynomials did not lead to new models with perfect state
transfer, but did allow more insight in the general computation of the
correlation function. In the present paper, we extend the study to discrete
orthogonal polynomials of q-hypergeometric type. A remarkable result is a new
analytic model where perfect state transfer is achieved: this is when the spin
chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The
other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk
polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn
polynomials and q-Racah polynomials) do not give rise to models with perfect
state transfer. However, the computation of the correlation function itself is
quite interesting, leading to advanced q-series manipulations
Interpolation of SUSY quantum mechanics
Interpolation of two adjacent Hamiltonians in SUSY quantum mechanics
, is discussed together
with related operators. For a wide variety of shape-invariant degree one
quantum mechanics and their `discrete' counterparts, the interpolation
Hamiltonian is also shape-invariant, that is it takes the same form as the
original Hamiltonian with shifted coupling constant(s).Comment: 18 page
A study on the fourth q-Painlev\'e equation
A q-difference analogue of the fourth Painlev\'e equation is proposed. Its
symmetry structure and some particular solutions are investigated.Comment: 18 page
Epicycles and Poincar\'{e} Resonances in General Relativity
The method of geodesic deviations provides analytic approximations to
geodesics in arbitrary background space-times. As such the method is a useful
tool in many practical situations. In this note we point out some subtleties in
the application of the method related to secular motions, in first as well as
in higher order. In particular we work out the general second-order
contribution to bound orbits in Schwarzschild space-time and show that it
provides very good analytical results all the way up to the innermost stable
circular orbit.Comment: 24 pages, 4 figure
Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator
The SU(1,1) coherent states for a relativistic model of the linear singular
oscillator are considered. The corresponding partition function is evaluated.
The path integral for the transition amplitude between SU(1,1) coherent states
is given. Classical equations of the motion in the generalized curved phase
space are obtained. It is shown that the use of quasiclassical Bohr-Sommerfeld
quantization rule yields the exact expression for the energy spectrum.Comment: 14 pages, 2 figures, Uses RevTeX4 styl
More on the q-oscillator algebra and q-orthogonal polynomials
Properties of certain -orthogonal polynomials are connected to the
-oscillator algebra. The Wall and -Laguerre polynomials are shown to
arise as matrix elements of -exponentials of the generators in a
representation of this algebra. A realization is presented where the continuous
-Hermite polynomials form a basis of the representation space. Various
identities are interpreted within this model. In particular, the connection
formula between the continuous big -Hermite polynomials and the continuous
-Hermite polynomials is thus obtained, and two generating functions for
these last polynomials are algebraically derived
A relativistic model of the -dimensional singular oscillator
Exactly solvable -dimensional model of the quantum isotropic singular
oscillator in the relativistic configurational -space is proposed. It
is shown that through the simple substitutions the finite-difference equation
for the -dimensional singular oscillator can be reduced to the similar
finite-difference equation for the relativistic isotropic three-dimensional
singular oscillator. We have found the radial wavefunctions and energy spectrum
of the problem and constructed a dynamical symmetry algebra.Comment: 8 pages, accepted for publication in J. Phys.
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