68 research outputs found
A new realization of rational functions, with applications to linear combination interpolation
We introduce the following linear combination interpolation problem (LCI):
Given distinct numbers and complex numbers
and , find all functions analytic in a simply
connected set (depending on ) containing the points such
that To this end we prove a representation
theorem for such functions in terms of an associated polynomial . We
first introduce the following two operations, substitution of , and
multiplication by monomials . Then let be the
module generated by these two operations, acting on functions analytic near
. We prove that every function , analytic in a neighborhood of the roots
of , is in . In fact, this representation of is unique. To solve the
above interpolation problem, we employ an adapted systems theoretic
realization, as well as an associated representation of the Cuntz relations
(from multi-variable operator theory.) We study these operations in reproducing
kernel Hilbert space): We give necessary and sufficient condition for existence
of realizations of these representation of the Cuntz relations by operators in
certain reproducing kernel Hilbert spaces, and offer infinite product
factorizations of the corresponding kernels
Unified Approach to Thermodynamic Bethe Ansatz and Finite Size Corrections for Lattice Models and Field Theories
We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for
magnetic chains and field theories that includes the finite size (and zero
temperature) calculations for lattice BA models. In all cases, the free energy
follows by quadratures from the solution of a {\bf single} non-linear integral
equation (NLIE). [A system of NLIE appears for nested BA]. We derive the NLIE
for: a) the six-vertex model with twisted boundary conditions; b) the XXZ chain
in an external magnetic field and c) the sine-Gordon-massive Thirring
model (sG-mT) in a periodic box of size \b \equiv 1/T using the light-cone
approach. This NLIE is solved by iteration in one regime (high in the XXZ
chain and low in the sG-mT model). In the opposite (conformal) regime, the
leading behaviors are obtained in closed form. Higher corrections can be
derived from the Riemann-Hilbert form of the NLIE that we present.Comment: Expanded Introduction. Version to appear in Nucl. Phys. B. 60 pages,
TeX, Uses phyzz
New critical matrix models and generalized universality
We study a class of one-matrix models with an action containing nonpolynomial terms. By tuning the coupling constants in the action to criticality we obtain that the eigenvalue density vanishes as an arbitrary real power at the origin, thus defining a new class of multicritical matrix models. The corresponding microscopic scaling law is given and possible applications to the chiral phase transition in QCD are discussed. For generic coupling constants off-criticality we prove that all microscopic correlation functions at the origin of the spectrum remain in the known Bessel universality class. An arbitrary number of Dirac mass terms can be included and the corresponding massive universality is maintained as well. We also investigate the critical behavior at the edge of the spectrum: there, in contrast to the behavior at the origin, we find the same critical exponents as derived from matrix models with a polynomial action
Q-operators in the six-vertex model
In this paper we continue the study of -operators in the six-vertex model
and its higher spin generalizations. In [1] we derived a new expression for the
higher spin -matrix associated with the affine quantum algebra
. Taking a special limit in this -matrix we obtained
new formulas for the -operators acting in the tensor product of
representation spaces with arbitrary complex spin.
Here we use a different strategy and construct -operators as integral
operators with factorized kernels based on the original Baxter's method used in
the solution of the eight-vertex model. We compare this approach with the
method developed in [1] and find the explicit connection between two
constructions. We also discuss a reduction to the case of finite-dimensional
representations with (half-) integer spins.Comment: 18 pages, no figure
Nonsemisimple Fusion Algebras and the Verlinde Formula
We find a nonsemisimple fusion algebra F_p associated with each (1,p)
Virasoro model. We present a nonsemisimple generalization of the Verlinde
formula which allows us to derive F_p from modular transformations of
characters.Comment: LaTeX (amsart, xypic, times), 35p
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