Can we find a new deformation of \((\text{SL}_ J)\) with respect to the parameters contained in \((\text{P}_ J)\) (Q2744096)

This is substantially a short summary of the article [\textit{T. Kawai} and \textit{Y. Takei}, RIMS Kôkyûroku 1001, 39--63 (1997; Zbl 0940.34072)].NEWLINENEWLINE The authors have been studying Painlevé equations with large parameter \(\eta\) by WKB method. The \(J\)th Painlevé equation \((\text{P}_J)\) with parameter \(\eta\) is a compatibility condition of a particular Schrödinger equation \((\text{SL}_J)\) \(L\psi\equiv (\partial^2/\partial x^2- \eta^2 Q_J)\psi= 0\) and its deformation (in \(t\)) equation NEWLINE\[NEWLINEM\psi\equiv (\partial/\partial t- A_J\partial/\partial x+ (1/2)\partial A_J/\partial x)\psi= 0.NEWLINE\]NEWLINE The equations \((\text{P}_J)\) and \((\text{SL}_J)\) contain explicitly several parameters (complex constants). The authors propose a problem if \((\text{SL}_J)\) with a pure and single-valued solution of \((\text{P}_J)\) substituted in its coefficients can be isomonodromically deformed with respect to these parameters.NEWLINENEWLINE In this paper, the consideration is restricted to the case \(J= II\) and five conjectures are given as well as the relations among them in order to confirm the proposed problem.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].