460 research outputs found
Orthogonal Polynomials from Hermitian Matrices
A unified theory of orthogonal polynomials of a discrete variable is
presented through the eigenvalue problem of hermitian matrices of finite or
infinite dimensions. It can be considered as a matrix version of exactly
solvable Schr\"odinger equations. The hermitian matrices (factorisable
Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding
to second order difference equations. By solving the eigenvalue problem in two
different ways, the duality relation of the eigenpolynomials and their dual
polynomials is explicitly established. Through the techniques of exact
Heisenberg operator solution and shape invariance, various quantities, the two
types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the
coefficients of the three term recurrence, the normalisation measures and the
normalisation constants etc. are determined explicitly.Comment: 53 pages, no figures. Several sentences and a reference are added. To
be published in J. Math. Phy
Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
We study the variance and the Laplace transform of the probability law of
linear eigenvalue statistics of unitary invariant Matrix Models of
n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test
function of statistics is smooth enough and using the asymptotic formulas by
Deift et al for orthogonal polynomials with varying weights, we show first that
if the support of the Density of States of the model consists of two or more
intervals, then in the global regime the variance of statistics is a
quasiperiodic function of n generically in the potential, determining the
model. We show next that the exponent of the Laplace transform of the
probability law is not in general 1/2variance, as it should be if the Central
Limit Theorem would be valid, and we find the asymptotic form of the Laplace
transform of the probability law in certain cases
Polynomial solutions of nonlinear integral equations
We analyze the polynomial solutions of a nonlinear integral equation,
generalizing the work of C. Bender and E. Ben-Naim. We show that, in some
cases, an orthogonal solution exists and we give its general form in terms of
kernel polynomials.Comment: 10 page
Multipartite minimum uncertainty products
In our previous work we have found a lower bound for the multipartite
uncertainty product of the position and momentum observables over all separable
states. In this work we are trying to minimize this uncertainty product over a
broader class of states to find the fundamental limits imposed by nature on the
observable quantites. We show that it is necessary to consider pure states only
and find the infimum of the uncertainty product over a special class of pure
states (states with spherically symmetric wave functions). It is shown that
this infimum is not attained. We also explicitly construct a parametrized
family of states that approaches the infimum by varying the parameter. Since
the constructed states beat the lower bound for separable states, they are
entangled. We thus show that there is a gap that separates the values of a
simple measurable quantity for separable states from entangled ones and we also
try to find the size of this gap.Comment: 18 pages, 5 figure
Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlev\'e II Equation
We consider the polynomials orthonormal with respect to the weight on the unit circle in the complex plane. The leading coefficient
is found to satisfy a difference-differential (spatially discrete)
equation which is further proved to approach a third order differential
equation by double scaling. The third order differential equation is equivalent
to the Painlev\'e II equation. The leading coefficient and second leading
coefficient of can be expressed asymptotically in terms of the
Painlev\'e II function.Comment: 16 page
Form factor expansion of the row and diagonal correlation functions of the two dimensional Ising model
We derive and prove exponential and form factor expansions of the row
correlation function and the diagonal correlation function of the two
dimensional Ising model
The Wigner function associated to the Rogers-Szego polynomials
We show here that besides the well known Hermite polynomials, the q-deformed
harmonic oscillator algebra admits another function space associated to a
particular family of q-polynomials, namely the Rogers-Szego polynomials. Their
main properties are presented, the associated Wigner function is calculated and
its properties are discussed. It is shown that the angle probability density
obtained from the Wigner function is a well-behaved function defined in the
interval [-Pi,Pi), while the action probability only assumes integer values
greater or equal than zero. It is emphasized the fact that the width of the
angle probability density is governed by the free parameter q characterizing
the polynomial.Comment: 12 pages, 2 (mathemathica) figure
Explicit Integration of the Full Symmetric Toda Hierarchy and the Sorting Property
We give an explicit formula for the solution to the initial value problem of
the full symmetric Toda hierarchy. The formula is obtained by the
orthogonalization procedure of Szeg\"{o}, and is also interpreted as a
consequence of the QR factorization method of Symes \cite{symes}. The sorting
property of the dynamics is also proved for the case of a generic symmetric
matrix in the sense described in the text, and generalizations of tridiagonal
formulae are given for the case of matrices with nonzero diagonals.Comment: 13 pages, Latex
A quantum exactly solvable non-linear oscillator related with the isotonic oscillator
A nonpolynomial one-dimensional quantum potential representing an oscillator,
that can be considered as placed in the middle between the harmonic oscillator
and the isotonic oscillator (harmonic oscillator with a centripetal barrier),
is studied. First the general case, that depends of a parameter , is
considered and then a particular case is studied with great detail. It is
proven that it is Schr\"odinger solvable and then the wave functions
and the energies of the bound states are explicitly obtained. Finally it
is proven that the solutions determine a family of orthogonal polynomials
related with the Hermite polynomials and such that: (i) Every
is a linear combination of three Hermite polynomials, and (ii)
They are orthogonal with respect to a new measure obtained by modifying the
classic Hermite measure.Comment: 11 pages, 11 figure
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
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