4,659 research outputs found
Factorisation of Macdonald polynomials
We discuss the problem of factorisation of the symmetric Macdonald
polynomials and present the obtained results for the cases of 2 and 3
variables.Comment: 13 pages, LaTex, no figure
Eigenproblem for Jacobi matrices: hypergeometric series solution
We study the perturbative power-series expansions of the eigenvalues and
eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d.
The(small) expansion parameters are being the entries of the two diagonals of
length d-1 sandwiching the principal diagonal, which gives the unperturbed
spectrum.
The solution is found explicitly in terms of multivariable (Horn-type)
hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables
for the eigenvalue growing from a corner matrix element. To derive the result,
we first rewrite the spectral problem for a Jacobi matrix as an equivalent
system of cubic equations, which are then resolved by the application of the
multivariable Lagrange inversion formula. The corresponding Jacobi determinant
is calculated explicitly. Explicit formulae are also found for any monomial
composed of eigenvector's components.Comment: Latex, 20 pages; v2: corrected typos, added section with example
Backlund transformations for the sl(2) Gaudin magnet
Elementary, one- and two-point, Backlund transformations are constructed for
the generic case of the sl(2) Gaudin magnet. The spectrality property is used
to construct these explicitly given, Poisson integrable maps which are
time-discretizations of the continuous flows with any Hamiltonian from the
spectral curve of the 2x2 Lax matrix.Comment: 17 pages, LaTeX, refs adde
Separation of variables for the Ruijsenaars system
We construct a separation of variables for the classical n-particle
Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser
system). The separated coordinates appear as the poles of the properly
normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax
matrix. Two different normalisations of the BA functions are analysed. The
canonicity of the separated variables is verified with the use of r-matrix
technique. The explicit expressions for the generating function of the
separating canonical transform are given in the simplest cases n=2 and n=3.
Taking nonrelativistic limit we also construct a separation of variables for
the elliptic Calogero-Moser system.Comment: 26 pages, LaTex, no figure
Q-operator and factorised separation chain for Jack polynomials
Applying Baxter's method of the Q-operator to the set of Sekiguchi's
commuting partial differential operators we show that Jack polynomials
P(x_1,...,x_n) are eigenfunctions of a one-parameter family of integral
operators Q_z. The operators Q_z are expressed in terms of the
Dirichlet-Liouville n-dimensional beta integral. From a composition of n
operators Q_{z_k} we construct an integral operator S_n factorising Jack
polynomials into products of hypergeometric polynomials of one variable. The
operator S_n admits a factorisation described in terms of restricted Jack
polynomials P(x_1,...,x_k,1,...,1). Using the operator Q_z for z=0 we give a
simple derivation of a previously known integral representation for Jack
polynomials.Comment: 26 page
Electrochemical behavior of a titanium electrode in hydrazine solutions
The kinetics of the establishment of the oxidation-reduction potential of a titanium electrode upon contact with hydrazine was studied in different media: H2SO4, NaOH, and Na2SO4. It was found that the nature of the potential shift depends little on the medium. The initial potential determines the rate of potential displacement upon contact with hydrazine, which is explained by the different condition of the electrode's surface
New boundary conditions for integrable lattices
New boundary conditions for integrable nonlinear lattices of the XXX type,
such as the Heisenberg chain and the Toda lattice are presented. These
integrable extensions are formulated in terms of a generic XXX Heisenberg
magnet interacting with two additional spins at each end of the chain. The
construction uses the most general rank 1 ansatz for the 2x2 L-operator
satisfying the reflection equation algebra with rational r-matrix. The
associated quadratic algebra is shown to be the one of dynamical symmetry for
the A1 and BC2 Calogero-Moser problems. Other physical realizations of our
quadratic algebra are also considered.Comment: 22 pages, latex, no figure
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