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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.
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 Da
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 ordina
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 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-