1,789 research outputs found
Effective inverse spectral problem for rational Lax matrices and applications
We reconstruct a rational Lax matrix of size R+1 from its spectral curve (the
desingularization of the characteristic polynomial) and some additional data.
Using a twisted Cauchy--like kernel (a bi-differential of bi-weight (1-nu,nu))
we provide a residue-formula for the entries of the Lax matrix in terms of
bases of dual differentials of weights nu and 1-nu respectively. All objects
are described in the most explicit terms using Theta functions. Via a sequence
of ``elementary twists'', we construct sequences of Lax matrices sharing the
same spectral curve and polar structure and related by conjugations by rational
matrices. Particular choices of elementary twists lead to construction of
sequences of Lax matrices related to finite--band recurrence relations (i.e.
difference operators) sharing the same shape. Recurrences of this kind are
satisfied by several types of orthogonal and biorthogonal polynomials. The
relevance of formulae obtained to the study of the large degree asymptotics for
these polynomials is indicated.Comment: 33 pages. Version 2 with added references suggested by the refere
Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions
We consider the class of biorthogonal polynomials that are used to solve the
inverse spectral problem associated to elementary co-adjoint orbits of the
Borel group of upper triangular matrices; these orbits are the phase space of
generalized integrable lattices of Toda type. Such polynomials naturally
interpolate between the theory of orthogonal polynomials on the line and
orthogonal polynomials on the unit circle and tie together the theory of Toda,
relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish
corresponding Christoffel-Darboux formulae . For all these classes of
polynomials a 2x2 system of Differential-Difference-Deformation equations is
analyzed in the most general setting of pseudo measures with arbitrary rational
logarithmic derivative. They provide particular classes of isomonodromic
deformations of rational connections on the Riemann sphere. The corresponding
isomonodromic tau function is explicitly related to the shifted Toeplitz
determinants of the moments of the pseudo-measure. In particular the results
imply that any (shifted) Toeplitz (Hankel) determinant of a symbol (measure)
with arbitrary rational logarithmic derivative is an isomonodromic tau
function.Comment: 35 pages, 1 figur
Cauchy Biorthogonal Polynomials
The paper investigates the properties of certain biorthogonal polynomials
appearing in a specific simultaneous Hermite-Pade' approximation scheme.
Associated to any totally positive kernel and a pair of positive measures on
the positive axis we define biorthogonal polynomials and prove that their
zeroes are simple and positive. We then specialize the kernel to the Cauchy
kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a
four-term recurrence relation, have relevant Christoffel-Darboux generalized
formulae, and their zeroes are interlaced. In addition, these polynomial solve
a combination of Hermite-Pade' approximation problems to a Nikishin system of
order 2. The motivation arises from two distant areas; on one side, in the
study of the inverse spectral problem for the peakon solution of the
Degasperis-Procesi equation; on the other side, from a random matrix model
involving two positive definite random Hermitian matrices. Finally, we show how
to characterize these polynomials in term of a Riemann-Hilbert problem.Comment: 38 pages, partially replaces arXiv:0711.408
The PDEs of biorthogonal polynomials arising in the two-matrix model
The two-matrix model can be solved by introducing bi-orthogonal polynomials.
In the case the potentials in the measure are polynomials, finite sequences of
bi-orthogonal polynomials (called
"windows") satisfy polynomial ODEs as well as deformation equations (PDEs)
and finite difference equations (Delta-E) which are all Frobenius compatible
and define discrete and continuous isomonodromic deformations for the irregular
ODE, as shown in previous works of ours.
In the one matrix model an explicit and concise expression for the
coefficients of these systems is known and it allows to relate the partition
function with the isomonodromic tau-function of the overdetermined system.
Here, we provide the generalization of those expressions to the case of
bi-orthogonal polynomials, which enables us to compute the determinant of the
fundamental solution of the overdetermined system of ODE+PDEs+Delta-E.Comment: 20 pages v1 18 Nov 2003; v2 9 Jan 2004: trivial Latex mistake
correcte
Free Energy of the Two-Matrix Model/dToda Tau-Function
We provide an integral formula for the free energy of the two-matrix model
with polynomial potentials of arbitrary degree (or formal power series). This
is known to coincide with the tau-function of the dispersionless
two--dimensional Toda hierarchy. The formula generalizes the case studied by
Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately
Takhtajan in the case of conformal maps of Jordan curves. Finally we generalize
the formula found in genus zero to the case of spectral curves of arbitrary
genus with certain fixed data.Comment: Ver 2: 18 pages added important formulas for higher genus spectral
curves, few typos removed (and few added). Ver 3: 19 pages (minor changes).
Typos removed, added appendix and improved exposition Ver 4: 19 pages, minor
corrections. Version submitted Ver 4; corrections prompted by referee and
accepted in Nuclear Phys.
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