662 research outputs found
Morse potential and its relationship with the Coulomb in a position-dependent mass background
We provide some explicit examples wherein the Schr\"odinger equation for the
Morse potential remains exactly solvable in a position-dependent mass
background. Furthermore, we show how in such a context, the map from the full
line to the half line may convert an exactly
solvable Morse potential into an exactly solvable Coulomb one. This generalizes
a well-known property of constant-mass problems.Comment: 9 pages, no figure; final published versio
Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials
We present four types of infinitely many exactly solvable Fokker-Planck
equations, which are related to the newly discovered exceptional orthogonal
polynomials. They represent the deformed versions of the Rayleigh process and
the Jacobi process.Comment: 17 pages, 4 figure
PT-symmetric supersymmetry in a solvable short-range model
The simplest purely imaginary and piecewise constant -symmetric
potential located inside a larger box is studied. Unless its strength exceeds a
certain critical value, all the spectrum of its bound states remains real and
discrete. We interpret such a model as an initial element of the generalized
non-Hermitian Witten's hierarchy of solvable Hamiltonians and construct its
first supersymmetric (SUSY) partner in closed form.Comment: 3 figures, 1 tabl
A new family of shape invariantly deformed Darboux-P\"oschl-Teller potentials with continuous \ell
We present a new family of shape invariant potentials which could be called a
``continuous \ell version" of the potentials corresponding to the exceptional
(X_{\ell}) J1 Jacobi polynomials constructed recently by the present authors.
In a certain limit, it reduces to a continuous \ell family of shape invariant
potentials related to the exceptional (X_{\ell}) L1 Laguerre polynomials. The
latter was known as one example of the `conditionally exactly solvable
potentials' on a half line.Comment: 19 pages. Sec.5(Summary and Comments): one sentence added in the
first paragraph, several sentences modified in the last paragraph.
References: one reference ([25]) adde
PT-symmetric square well and the associated SUSY hierarchies
The PT-symmetric square well problem is considered in a SUSY framework. When
the coupling strength lies below the critical value
where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY
partner potentials, depicting an unbroken SUSY situation and reducing to the
family of -like potentials in the limit. For above
, there is a rich diversity of SUSY hierarchies, including
some with PT-symmetry breaking and some with partial PT-symmetry restoration.Comment: LaTeX, 18 pages, no figure; broken PT-symmetry case added (Sec. 6
Evaluation and management implications of uncertainty in a multispecies size-structured model of population and community responses to fishing
1. Implementation of an ecosystem approach to fisheries requires advice on trade-offs among fished species and between fisheries yields and biodiversity or food web properties. However, the lack of explicit representation, analysis and consideration of uncertainty in most multispecies models has limited their application in analyses that could support management advice. 2. We assessed the consequences of parameter uncertainty by developing 78 125 multispecies size-structured fish community models, with all combinations of parameters drawn from ranges that spanned parameter values estimated from data and literature. This unfiltered ensemble was reduced to 188 plausible models, the filtered ensemble (FE), by screening outputs against fish abundance data and ecological principles such as requiring species' persistence. 3. Effects of parameter uncertainty on estimates of single-species management reference points for fishing mortality (FMSY, fishing mortality rate providing MSY, the maximum sustainable yield) and biomass (BMSY, biomass at MSY) were evaluated by calculating probability distributions of estimated reference points with the FE. There was a 50% probability that multispecies FMSY could be estimated to within ±25% of its actual value, and a 50% probability that BMSY could be estimated to within ±40% of its actual value. 4. Signal-to-noise ratio was assessed for four community indicators when mortality rates were reduced from current rates to FMSY. The slope of the community size spectrum showed the greatest signal-to-noise ratio, indicating that it would be the most responsive indicator to the change in fishing mortality F. Further, the power of an ongoing international monitoring survey to detect predicted responses of size spectrum slope was higher than for other size-based metrics. 5. Synthesis and applications: Application of the ensemble model approach allows explicit representation of parameter uncertainty and supports advice and management by (i) providing uncertainty intervals for management reference points, (ii) estimating working values of reference points that achieve a defined reduction in risk of not breaching the true reference point, (iii) estimating the responsiveness of population, community, food web and biodiversity indicators to changes in F, (iv) assessing the performance of indicators and monitoring programmes and (v) identifying priorities for data collection and changes to model structure to reduce uncertainty
The Localization of -Wave and Quantum Effective Potential of a Quasi-Free Particle with Position-Dependent Mass
The properties of the s-wave for a quasi-free particle with
position-dependent mass(PDM) have been discussed in details. Differed from the
system with constant mass in which the localization of the s-wave for the free
quantum particle around the origin only occurs in two dimensions, the
quasi-free particle with PDM can experience attractive forces in dimensions
except D=1 when its mass function satisfies some conditions. The effective mass
of a particle varying with its position can induce effective interaction which
may be attractive in some cases. The analytical expressions of the
eigenfunctions and the corresponding probability densities for the s-waves of
the two- and three-dimensional systems with a special PDM are given, and the
existences of localization around the origin for these systems are shown.Comment: 12pages, 8 figure
Kepler problem in Dirac theory for a particle with position-dependent mass
Exact solution of Dirac equation for a particle whose potential energy and
mass are inversely proportional to the distance from the force centre has been
found. The bound states exist provided the length scale which appears in
the expression for the mass is smaller than the classical electron radius
. Furthermore, bound states also exist for negative values of
even in the absence of the Coulomb interaction. Quasirelativistic expansion of
the energy has been carried out, and a modified expression for the fine
structure of energy levels has been obtained. The problem of kinetic energy
operator in the Schr\"odinger equation is discussed for the case of
position-dependent mass. In particular, we have found that for highly excited
states the mutual ordering of the inverse mass and momentum operator in the
non-relativistic theory is not important.Comment: 9 page
Multi-indexed (q-)Racah Polynomials
As the second stage of the project multi-indexed orthogonal polynomials, we
present, in the framework of `discrete quantum mechanics' with real shifts in
one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from
the (q-)Racah polynomials by multiple application of the discrete analogue of
the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state'
vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials
reported earlier. The virtual state vectors are the `solutions' of the matrix
Schr\"odinger equation with negative `eigenvalues', except for one of the two
boundary points.Comment: 29 pages. The type II (q-)Racah polynomials are deleted because they
can be obtained from the type I polynomials. To appear in J.Phys.
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