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Register a new userAn Elliptic Garnier System from Interpolation
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Authors
Yasuhiko Yamada
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Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painleve equation.
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An interpolation problem related to the elliptic Painleve equation is formulated and solved. A simple form of the elliptic Painleve equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.
We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic $mathfrak{sl}_2$-connections on $mathbb{P}^1$ and argue that the topological recursion produces a $2g$-parameter family of associated tau functions, where $2g$ is the dimension of the moduli space considered. We apply this procedure to the 6 Painleve equations which correspond to $g=1$ and consider a $g=2$ example.
We construct solutions of analogues of the nonstationary Schrodinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of linear ordinary differential equations whose compatibility condition is the Garnier system. This solutions upto explicit transform also satisfy the Belavin --- Polyakov --- Zamolodchikov equations with four time variables and two space variables.
We introduce constellation ensembles, in which charged particles on a line (or circle) are linked with charged particles on parallel lines (or concentric circles). We present formulas for the partition functions of these ensembles in terms of either the Hyperpfaffian or the Berezin integral of an appropriate alternating tensor. Adjusting the distances between these lines (or circles) gives an interpolation between a pair of limiting ensembles, such as one-dimensional $beta$-ensembles with $beta=K$ and $beta=K^2$.
The 8-parameter elliptic Sakai difference Painleve equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrodinger equation for the $BC_1$ 8-parameter relativistic Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painleve-Calogero correspondence to the highest level in the two hierarchies.
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