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Resurgent deformations for an ordinary differential equation of order 2. (English) Zbl 1116.34068
The paper is the first of three papers to come. The authors consider the second order differential equation
\[ (d^2/dx^2)\Phi (x)=(P_m(x)/x^2)\Phi (x) \] in the complex domain, where the monic polynomial \(P_m\) is of degree \(m\). They investigate the asymptotic and resurgent properties of its solutions at infinity, in particular – the dependence of the Stokes-Sibuya multipliers (SSM) on the coefficients of \(P_m\). They derive a set of functional equations for the SSM (taking into account the nontrivial monodromy at the origin) and show how these equations can be used to compute the SSM for a class of polynomials \(P_m\). In particular, they obtain conditions for isomonodromic deformations when \(m=3\).

MSC:
34M37 Resurgence phenomena (MSC2000)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
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