31,630 research outputs found
Supersymmetric Quantum Mechanics, Engineered Hierarchies of Integrable Potentials, and the Generalised Laguerre Polynomials
Within the context of Supersymmetric Quantum Mechanics and its related
hierarchies of integrable quantum Hamiltonians and potentials, a general
programme is outlined and applied to its first two simplest illustrations.
Going beyond the usual restriction of shape invariance for intertwined
potentials, it is suggested to require a similar relation for Hamiltonians in
the hierarchy separated by an arbitrary number of levels, N. By requiring
further that these two Hamiltonians be in fact identical up to an overall shift
in energy, a periodic structure is installed in the hierarchy of quantum
systems which should allow for its solution. Specific classes of orthogonal
polynomials characteristic of such periodic hierarchies are thereby generated,
while the methods of Supersymmetric Quantum Mechanics then lead to generalised
Rodrigues formulae and recursion relations for such polynomials. The approach
also offers the practical prospect of quantum modelling through the engineering
of quantum potentials from experimental energy spectra. In this paper these
ideas are presented and solved explicitly for the cases N=1 and N=2. The latter
case is related to the generalised Laguerre polynomials, for which indeed new
results are thereby obtained. At the same time new classes of integrable
quantum potentials which generalise that of the harmonic oscillator and which
are characterised by two arbitrary energy gaps are identified, for which a
complete solution is achieved algebraically.Comment: 1+19 page
A note on first-order spectra with binary relations
The spectrum of a first-order sentence is the set of the cardinalities of its
finite models. In this paper, we consider the spectra of sentences over binary
relations that use at least three variables. We show that for every such
sentence , there is a sentence that uses the same number of
variables, but only one symmetric binary relation, such that its spectrum is
linearly proportional to the spectrum of . Moreover, the models of
are all bipartite graphs. As a corollary, we obtain that to settle
Asser's conjecture, i.e., whether the class of spectra is closed under
complement, it is sufficient to consider only sentences using only three
variables whose models are restricted to undirected bipartite graphs
Linearized flavor-stability analysis of dense neutrino streams
Neutrino-neutrino interactions in dense neutrino streams, like those emitted
by a core-collapse supernova, can lead to self-induced neutrino flavor
conversions. While this is a nonlinear phenomenon, the onset of these
conversions can be examined through a standard stability analysis of the
linearized equations of motion. The problem is reduced to a linear eigenvalue
equation that involves the neutrino density, energy spectrum, angular
distribution, and matter density. In the single-angle case, we reproduce
previous results and use them to identify two generic instabilities: The system
is stable above a cutoff density ("cutoff mode"), or can approach an asymptotic
instability for increasing density ("saturation mode"). We analyze multi-angle
effects on these generic types of instabilities and find that even the
saturation mode is suppressed at large densities. For both types of modes, a
given multi-angle spectrum typically is unstable when the neutrino and electron
densities are comparable, but stable when the neutrino density is much smaller
or much larger than the electron density. The role of an instability in the SN
context depends on the available growth time and on the range of affected
modes. At large matter density, most modes are off-resonance even when the
system is unstable.Comment: 19 pages, 8 figures, revtex4 forma
Prompt photon in association with a heavy-quark jet in Pb-Pb collisions at the LHC
We present a phenomenological study of the associated production of a prompt
photon and a heavy quark jet (charm or bottom) in Pb-Pb collisions at the LHC.
This channel allows for estimating the amount of energy loss experienced by the
charm and bottom quarks propagating in the dense QCD medium produced in those
collisions. Calculations are carried out at next-to-leading order (NLO)
accuracy using the BDMPS-Z heavy-quark quenching weights. The quenching of the
single heavy-quark jet spectrum reflects fairly the hierarchy in the heavy
quark energy loss assumed in the perturbative calculation. On the contrary, the
single photon spectrum in heavy-ion collisions is only modified at low momenta,
for which less heavy-quark jets pass the kinematic cuts. On top of single
particle spectra, the two-particle final state provides a range of observables
(photon-jet pair momentum, jet asymmetry, among others) which are studied in
detail. The comparison of the photon-jet pair momentum, from p-p to Pb-Pb
collisions, is sensitive to the amount of energy lost by the heavy-quarks and
could therefore be used in order to better understand parton energy loss
processes in the heavy quark sector.Comment: 22 pages, 18 figures, gzipped tar file. Minor changes, 2 references
added. Version published in JHE
Orthogonalized smoothing for rescaled spike and slab models
Rescaled spike and slab models are a new Bayesian variable selection method
for linear regression models. In high dimensional orthogonal settings such
models have been shown to possess optimal model selection properties. We review
background theory and discuss applications of rescaled spike and slab models to
prediction problems involving orthogonal polynomials. We first consider global
smoothing and discuss potential weaknesses. Some of these deficiencies are
remedied by using local regression. The local regression approach relies on an
intimate connection between local weighted regression and weighted generalized
ridge regression. An important implication is that one can trace the effective
degrees of freedom of a curve as a way to visualize and classify curvature.
Several motivating examples are presented.Comment: Published in at http://dx.doi.org/10.1214/074921708000000192 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
SUSY-hierarchy of one-dimensional reflectionless potentials
A class of one-dimensional reflectionless potentials, an absolute
transparency of which is concerned with their belonging to one SUSY-hierarchy
with a constant potential, is studied. An approach for determination of a
general form of the reflectionless potential on the basis of construction of
such a hierarchy by the recurrent method is proposed. A general form of
interdependence between superpotentials with neighboring numbers of this
hierarchy, opening a possibility to find new reflectionless potentials, have a
simple analytical view and are expressed through finite number of elementary
functions (unlike some reflectionless potentials, which are constructed on the
basis of soliton solutions or are shape invariant in one or many steps with
involving scaling of parameters, and are expressed through series), is
obtained. An analysis of absolute transparency existence for the potential
which has the inverse power dependence on space coordinate (and here tunneling
is possible), i.e. which has the form
(where and are constants, is natural number), is
fulfilled. It is shown that such a potential can be reflectionless at n = 2
only. A SUSY-hierarchy of the inverse power reflectionless potentials is
constructed. Isospectral expansions of this hierarchy is analyzed.Comment: 33 pages, 10 files of figures in EPS format, LaTeX v.2e, ElsArt styl
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