Semiclassical expansion for exactly solvable differential operators

Below we study a linear differential equation $\text{MM}(v(z,\eta ))={\eta }^{M}v(z,\eta )$, where $\eta >0$ is a large spectral parameter and $\text{MM}={\sum }_{k=1}^M{\rho }_k(z)\frac{d^k}{d{z}^k},M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient ${\rho }_M(z)$ is a monic complex-valued polynomial with $\text{dgr}{\rho }_M=M$ and other ${\rho }_k(z)$'s are complex-valued polynomials with $\text{dgr}{\rho }_k\le k$. 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.