Semiclassical expansion for exactly solvable differential operators

Abstract

Below we study a linear differential equation $\MM (v(z,\eta))=\eta^M{v(z,\eta)}$, where $\eta>0$ is a large spectral parameter and $\MM=\sum_{k=1}^{M}\rho_{k}(z)\frac{d^k}{dz^k},\; M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient $\rho_M(z)$ is a monic complex-valued polynomial with $\dgr{\rho_M }=M$ and other $\rho_k(z)$'s are complex-valued polynomials with $\dgr{\rho_k }\leq k$. We prove the Borel summability of its WKB-solutions in the Stokes regions. For $M=3$ under the assumption that $\rho_M$ has simple zeros, we give the full description of the Stokes complex (i.e. the union of all Stokes curves) of this equation. Finally, we show that for the Euler-Cauchy equations, their WKB-solutions converge in the usual sense.