5,092 research outputs found
Supersymmetric quantum mechanics and Painleve equations
In these lecture notes we shall study first the supersymmetric quantum
mechanics (SUSY QM), specially when applied to the harmonic and radial
oscillators. In addition, we will define the polynomial Heisenberg algebras
(PHA), and we will study the general systems ruled by them: for zero and first
order we obtain the harmonic and radial oscillators, respectively; for second
and third order PHA the potential is determined by solutions to Painleve IV
(PIV) and Painleve V (PV) equations. Taking advantage of this connection, later
on we will find solutions to PIV and PV equations expressed in terms of
confluent hypergeometric functions. Furthermore, we will classify them into
several solution hierarchies, according to the specific special functions they
are connected with.Comment: 38 pages, 20 figures. Lecture presented at the XLIII Latin American
School of Physics: ELAF 2013 in Mexico Cit
Integrability of one degree of freedom symplectic maps with polar singularities
In this paper, we treat symplectic difference equations with one degree of
freedom. For such cases, we resolve the relation between that the dynamics on
the two dimensional phase space is reduced to on one dimensional level sets by
a conserved quantity and that the dynamics is integrable, under some
assumptions. The process which we introduce is related to interval exchange
transformations.Comment: 10 pages, 2 figure
Solutions to the Painlev\'e V equation through supersymmetric quantum mechanics
In this paper we shall use the algebraic method known as supersymmetric
quantum mechanics (SUSY QM) to obtain solutions to the Painlev\'e V (PV)
equation, a second-order non-linear ordinary differential equation. For this
purpose, we will apply first the SUSY QM treatment to the radial oscillator. In
addition, we will revisit the polynomial Heisenberg algebras (PHAs) and we will
study the general systems ruled by them: for first-order PHAs we obtain the
radial oscillator, while for third-order PHAs the potential will be determined
by solutions to the PV equation. This connection allows us to introduce a
simple technique for generating solutions of the PV equation expressed in terms
of confluent hypergeometric functions. Finally, we will classify them into
several solution hierarchies.Comment: 39 pages, 18 figures, 4 tables, 70 reference
Backlund Transformations, D-Branes, and Fluxes in Minimal Type 0 Strings
We study the Type 0A string theory in the (2,4k) superconformal minimal model
backgrounds, focusing on the fully non-perturbative string equations which
define the partition function of the model. The equations admit a parameter,
Gamma, which in the spacetime interpretation controls the number of background
D-branes, or R-R flux units, depending upon which weak coupling regime is
taken. We study the properties of the string equations (often focusing on the
(2,4) model in particular) and their physical solutions. The solutions are the
potential for an associated Schrodinger problem whose wavefunction is that of
an extended D-brane probe. We perform a numerical study of the spectrum of this
system for varying Gamma and establish that when Gamma is a positive integer
the equations' solutions have special properties consistent with the spacetime
interpretation. We also show that a natural solution-generating transformation
(that changes Gamma by an integer) is the Backlund transformation of the KdV
hierarchy specialized to (scale invariant) solitons at zero velocity. Our
results suggest that the localized D-branes of the minimal string theories are
directly related to the solitons of the KdV hierarchy. Further, we observe an
interesting transition when Gamma=-1.Comment: 17 pages, 3 figure
Asymptotic behaviour of two-point functions in multi-species models
We extract the long-distance asymptotic behaviour of two-point correlation
functions in massless quantum integrable models containing multi-species
excitations. For such a purpose, we extend to these models the method of a
large-distance regime re-summation of the form factor expansion of correlation
functions. The key feature of our analysis is a technical hypothesis on the
large-volume behaviour of the form factors of local operators in such models.
We check the validity of this hypothesis on the example of the
-invariant XXX magnet by means of the determinant representations for
the form factors of local operators in this model. Our approach confirms the
structure of the critical exponents obtained previously for numerous models
solvable by the nested Bethe Ansatz.Comment: 45 pages, 1 figur
On Asymptotics of Polynomial Eigenfunctions for Exactly-Solvable Differential Operators
In this paper we study the asymptotic zero distribution of eigenpolynomials
for degenerate exactly-solvable operators. We present an explicit conjecture
and partial results on the growth of the largest modulus of the roots of the
unique and monic n:th degree eigenpolynomial of any such operator as the degree
n tends to infinity. Based on this conjecture we deduce the algebraic equation
satified by the Cauchy transform of the asymptotic root measure of the properly
scaled eigenpolynomials, for which the union of all roots is conjecturally
contained in a compact set.Comment: 36 pages, 37 figures, to appear in Journal of Approximation Theor
The effect of uniaxial crystal-field anisotropy on magnetic properties of the superexchange antiferromagnetic Ising model
The generalized Fisher super-exchange antiferromagnetic model with uniaxial
crystal-field anisotropy is exactly investigated using an extended mapping
technique. An exact relation between partition function of the studied system
and that one of the standard zero-field spin-1/2 Ising model on the
corresponding lattice is obtained applying the decoration-iteration
transformation. Consequently, exact results for all physical quantities are
derived for arbitrary spin values S of decorating atoms. Particular attention
is paid to the investigation of the effect of crystal-field anisotropy and
external longitudinal magnetic field on magnetic properties of the system under
investigation. The most interesting numerical results for ground-state and
finite-temperature phase diagrams, thermal dependences of the sublattice
magnetization and other thermodynamic quantities are discussed.Comment: 9 pages, 6 figures, submitted to Condens. Matter Phy
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