Effective Inverse Spectral Problem for Rational Lax Matrices and Applications

Abstract

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 − ν, ν)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights ν, 1−ν 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.

Geometry of Spectral Curves and All Order Dispersive Integrable System

“…The only difference is that, we reformulate it intrinsically in terms of 1-forms ω, instead of using time coordinates ω = k t k ω k . For this purpose, instead of Baker-Akhiezer functions, we prefer to use a "spinor kernel", which is a convenient special case of Baker-Akhiezer function, which turns out to be a more intrinsic object for our formulation (see also [13,59,60]).…”

Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model

“…x a0 ρ 1 (s)ds − 1 . From (4-2)-(4-9), Vice versa, for z ∈ A (and outside of the right lenses) one has p n (z) = e N g (1) Γ 11 (z) (4-61) (4)(5)(6)(7)(8)(9)(10)(11). Then one can obtain uniform asymptotic information on the behaviour of p n in any compact set of the complex plane.…”

“…y−z dy and all integrals are oriented integrals. We will study the asymptotic behavior of the solution of Problem 2.1 in various regions of the complex plane for N N → ∞ , n := N + r, r ∈ Z , (2)(3)(4)(5)(6)(7)(8)(9)(10) where the integer r is bounded.…”

“…The expressions in terms of Theta functions for the columns of the outer parametrix Ψ possess other interesting relations that were investigated in [5].…”

“…We solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle in the spirit of [8,5]). Much of the effort goes towards showing that the model problem for the outer parametrix always admits a solution (Proposition 4.4, Theorem 4.1).…”