6 research outputs found
Eigenvectors of open Bazhanov-Stroganov quantum chain
In this contribution we give an explicit formula for the eigenvectors of
Hamiltonians of open Bazhanov-Stroganov quantum chain. The Hamiltonians of this
quantum chain is defined by the generation polynomial which is
upper-left matrix element of monodromy matrix built from the cyclic
-operators. The formulas for the eigenvectors are derived using iterative
procedure by Kharchev and Lebedev and given in terms of -function which
is a root of unity analogue of -function.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Wave Functions of the Toda Chain with Boundary Interaction
In this contribution, we give an integral representation of the wave
functions of the quantum N-particle Toda chain with boundary interaction. In
the case of the Toda chain with one-boundary interaction, we obtain the wave
function by an integral transformation from the wave functions of the open Toda
chain. The kernel of this transformation is given explicitly in terms of
\Gamma-functions. The wave function of the Toda chain with two-boundary
interaction is obtained from the previous wave functions by an integral
transformation. In this case, the difference equation for the kernel of the
integral transformation admits separation of variables. The separated
difference equations coincide with the Baxter equation.Comment: 14 pages, based on the talk given at the Workshop ``Classical and
quantum integrable systems'' (Dubna, January 2004
Relativistic Toda Chain with Boundary Interaction at Root of Unity
We apply the Separation of Variables method to obtain eigenvectors of
commuting Hamiltonians in the quantum relativistic Toda chain at a root of
unity with boundary interaction.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Ising correlations and elliptic determinants
Correlation functions of the two-dimensional Ising model on the periodic
lattice can be expressed in terms of form factors - matrix elements of the spin
operator in the basis of common eigenstates of the transfer matrix and
translation operator. Free-fermion structure of the model implies that any
multiparticle form factor is given by the pfaffian of a matrix constructed from
the two-particle ones. Crossed two-particle form factors can be obtained by
inverting a block of the matrix of linear transformation induced on fermions by
the spin conjugation. We show that the corresponding matrix is of elliptic
Cauchy type and use this observation to solve the inversion problem explicitly.
Non-crossed two-particle form factors are then obtained using theta functional
interpolation formulas. This gives a new simple proof of the factorized
formulas for periodic Ising form factors, conjectured by A. Bugrij and one of
the authors.Comment: 31 pages; v2: added references, final version to appear in J. Stat.
Phy
On solutions of the Fuji-Suzuki-Tsuda system
International audienceWe derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painleve VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for