107 research outputs found
Extension of Nikiforov-Uvarov Method for the Solution of Heun Equation
We report an alternative method to solve second order differential equations
which have at most four singular points. This method is developed by changing
the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU)
method. This is called extended NU method for this paper. The eigenvalue
solutions of Heun equation and confluent Heun equation are obtained via
extended NU method. Some quantum mechanical problems such as Coulomb problem on
a 3-sphere, two Coulombically repelling electrons on a sphere and hyperbolic
double-well potential are investigated by this method
The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations
We obtain isomonodromic transformations for Heun's equation by generalizing
Darboux transformation, and we find pairs and triplets of Heun's equation which
have the same monodromy structure. By composing generalized Darboux
transformations, we establish a new construction of the commuting operator
which ensures finite-gap property. As an application, we prove conjectures in
part III.Comment: 24 page
Time-sliced path integrals with stationary states
The path integral approach to the quantization of one degree-of-freedom
Newtonian particles is considered within the discrete time-slicing approach, as
in Feynman's original development. In the time-slicing approximation the
quantum mechanical evolution will generally not have any stationary states. We
look for conditions on the potential energy term such that the quantum
mechanical evolution may possess stationary states without having to perform a
continuum limit. When the stationary states are postulated to be solutions of a
second-order ordinary differential equation (ODE) eigenvalue problem it is
found that the potential is required to be a solution of a particular
first-order ODE. Similarly, when the stationary states are postulated to be
solutions of a second-order ordinary difference equation (OE)
eigenvalue problem the potential is required to be a solution of a particular
first-order OE. The classical limits (which are at times very
nontrivial) are integrable maps.Comment: 7 page
Behaviour of three charged particles on a plane under perpendicular magnetic field
We consider the problem of three identical charged particles on a plane under
a perpendicular magnetic field and interacting through Coulomb repulsion. This
problem is treated within Taut's framework, in the limit of vanishing center of
mass vector , which corresponds to the strong magnetic
field limit, occuring for example in the Fractional Quantum Hall Effect. Using
the solutions of the biconfluent Heun equation, we compute the eigenstates and
show that there is two sets of solutions. The first one corresponds to a system
of three independent anyons which have their angular momenta fixed by the value
of the magnetic field and specified by a dimensionless parameter , the ratio of , the magnetic length, over , the Bohr
radius. This anyonic character, consistent with quantum mechanics of identical
particles in two dimensions, is induced by competing physical forces. The
second one corresponds to the case of the Landau problem when .
Finally we compare these states with the quantum Hall states and find that the
Laughlin wave functions are special cases of our solutions under certains
conditions.Comment: 15 pages, 3 figures, Accepeted in JP
Ince's limits for confluent and double-confluent Heun equations
We find pairs of solutions to a differential equation which is obtained as a
special limit of a generalized spheroidal wave equation (this is also known as
confluent Heun equation). One solution in each pair is given by a series of
hypergeometric functions and converges for any finite value of the independent
variable , while the other is given by a series of modified Bessel functions
and converges for , where denotes a regular singularity.
For short, the preceding limit is called Ince's limit after Ince who have used
the same procedure to get the Mathieu equations from the Whittaker-Hill ones.
We find as well that, when tends to zero, the Ince limit of the
generalized spheroidal wave equation turns out to be the Ince limit of a
double-confluent Heun equation, for which solutions are provided. Finally, we
show that the Schr\"odinger equation for inverse fourth and sixth-power
potentials reduces to peculiar cases of the double-confluent Heun equation and
its Ince's limit, respectively.Comment: Submitted to Journal of Mathmatical Physic
Adult rat hepatocytes in primary monolayer culture. Ultrastructural characteristics of intercellular contacts and cell membrane differentiations.
Primary monolayer cultures were obtained in 60 mm petri dishes by incubating 3 x 106 isolated hepatocytes at 37°C in Dulbecco's medium supplemented with 17% fetal calf serum. The ultrastructure of monolayer cells was examined after various incubation periods. Within 4 h of plating, the isolated spherical cells adhere to the plastic surface, establish their first contacts by numerous intertwined microvilli, and form a new hemidesmosomes. After 12 h of culture, wide branched trabeculae of flattened polyhedral cells extend in all directions. Finally, after 24 h of culture, bile canaliculi are reconstituted, and a biliary polarity is recovered: the Golgi elements, which are scattered throughout the cytoplasm in the isolated cells, are reassembled in front of the newly formed bile canaliculi, symmetrically in the adjacent cells; lysosomes are concentrated in that region, and microtubules reappear. Concomitantly, plasma membrane differentiations, namely desmosomes and tight junctions, develop. Tight junctions sealing the bile ducts constitute a barrier to the passage of ruthenium red and horseradish peroxidase. De novo formation of these junctions was studied by the freeze etching technique: 10 nm particles compose a network of anastomosed linear arrays in the vicinity of the bile canalculi; in the next step of differentiation, the particles fuse, form short ridge segments and finally continuous branched smooth strands, characteristic of the mature tight junction.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
Transformations of Heun's equation and its integral relations
We find transformations of variables which preserve the form of the equation
for the kernels of integral relations among solutions of the Heun equation.
These transformations lead to new kernels for the Heun equation, given by
single hypergeometric functions (Lambe-Ward-type kernels) and by products of
two hypergeometric functions (Erd\'elyi-type). Such kernels, by a limiting
process, also afford new kernels for the confluent Heun equation.Comment: This version was published in J. Phys. A: Math. Theor. 44 (2011)
07520
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
The concept of Lyapunov exponent has long occupied a central place in the
theory of Anderson localisation; its interest in this particular context is
that it provides a reasonable measure of the localisation length. The Lyapunov
exponent also features prominently in the theory of products of random matrices
pioneered by Furstenberg. After a brief historical survey, we describe some
recent work that exploits the close connections between these topics. We review
the known solvable cases of disordered quantum mechanics involving random point
scatterers and discuss a new solvable case. Finally, we point out some
limitations of the Lyapunov exponent as a means of studying localisation
properties.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on
"Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change
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