Oscillatory endpoint with a quadratically vanishing drift ($s_t\sim \sqrt{2}{ϵ}^{1/4},\Delta s=O({ϵ}^{5/12})$)

For $ϵy^{\prime \prime }+s^2{y}^{\prime }+y=0$ on $s\ge 0$, removing the drift by $y=e^{-s^3/(6ϵ)}v$ givesThe transformed potential changes sign at $s_t\sim \sqrt{2}{ϵ}^{1/4}$. WKB approximation is oscillatory inside and exponential outside this turning region; an Airy turning-point connection formula applies over width ${ϵ}^{5/12}$. At the endpoint the wavelength is $\sqrt{ϵ}$, while the physical exponential prefactor changes on the envelope scale ${ϵ}^{1/3}$. Balancing only diffusion and drift and declaring a single ${ϵ}^{1/3}$ layer misses the zeroth-order term and the turning region. Matching constants may be singular at homogeneous boundary-value resonances.