Rational Symplectic Coordinates on the Space of Fuchs Equations m × m-Case

Abstract

Abstract. A method of constructing of Darboux coordinates on a space that is a symplectic reduction with respect to a diagonal action of GL(m, C) on a Cartesian product of N orbits of coadjoint representation of GL(m, C) is presented. The method gives an explicit symplectic birational isomorphism between the reduced space on the one hand and a Cartesian product of N − 3 coadjoint orbits of dimension m(m−1) on an orbit of dimension (m−1)(m−2) on the other hand. In a generic case of the diagonalizable matrices it gives just the isomorphism that is birational and symplectic between some open, in a Zariski topology, domain of the reduced space and the Cartesian product of the orbits in question.The method is based on a Gauss decomposition of a matrix on a product of upper-triangular, lower-triangular and diagonal matrices. It works uniformly for the orbits formed by diagonalizable or not-diagonalizable matrices. It is elaborated for the orbits of maximal dimension that is m(m − 1). IntroductionA set of Fuchs equationsmay be considered as a submanifold N n=1 A (n) = 0 of the space gl(m) × · · · × gl(m) that has a natural Poisson structure coming from gl(m).Some important problems like a problem of isomonodromic deformations may be restricted on a "symplectic leaf" of this Poisson manifold, that is a submanifold on which the Poisson structure induces the symplectic structure. It is the symplectic structure that we mean in the title.The symplectic manifold in question is closely related with the well known, "standard" symplectic manifold, the orbit of coadjoint representation of the Lie group. It is well investigated class of manifolds, and the problem of their canonical parametrization dates back to Archimedes who found a symplectomorphism between sphere, that can be considered as an orbit of SU(2), and a circumscribed cylinder.In the present article we construct an explicit birational symplectic isomorphism between the symplectic leaf of the space of Fuchs equations and a Cartesian product of the coadjoint orbits (for the version of the method for 2 × 2 matrices see [1]).

An Explicit Formula for the Nonstationary Diffracted Wave Scattered on a NN-Wedge

“…In this section we prove that the amplitude of the solution u(ρ, θ, t) to the nonsationary NN-scattering problem (14) converges to a limiting amplitude as t → ∞. This limiting amplitude is a well known solution to the stationary diffracted problem (see for example [27]).…”

“…Hence, the statement i) follows from Lemma 4.3 i) and (25), the estimate (43) follows from (40) and (27). The existence of the limit (44) follows from (37) and (28).…”

“…i) The first statement follows from the Paley-Wiener type Theorem for cones (Theorem I.5.2 in [26]) since w d (·, ·, ω) ∈ H(C + ) by (39) and it is bounded in C + by the second inequality in (40) and (27). The representation (47) follows from the fact that w d (·, ·, ω 1 ), ω 1 ∈ R is the S ′ -limit of the function w d (·, ·, ω 1 + iω 2 ), when ω 2 → 0+ by (39), (40) and (27), (28). ii) Substituting expression (26) in (46), and then plugging the result for w d into the integral (47), we obtain…”

“…It means that J d (·, ·, ω) ∈ C C + . iii) Estimate (27) follows from (26), (32), the fact that e iρ(ω 1 +iω 2 ) cosh β = e −ρω 2 cosh β for any β ∈ R and ω 2 ≥ 0. iv) Statement (28) follows from (33) and (27).…”

Isomonodromic deformations: Confluence, Reduction $\&$ Quantisation

“…The autonomous Gaudin model (5) can be generalised in two directions: by allowing higher order singularities at the marked points u i ∈ C thus giving rise to Gaudin models with irregular singularities in [22] or by taking an element µ ∈ g * that is not semi-simple (i.e. has non-trivial Jordan blocks).…”