Cauchy Data for 1D singular Schrödinger operators

We study semiclassical 1-D Schrödinger operators of the form $Pu=-h^2{u}^{\mathrm{\prime }\mathrm{\prime }}+x^{\gamma }W(x)u$ on a finite interval $[0,b]$ for $0<\gamma \in \mathbb{R}\setminus \mathbb{Q}$. We show that that the WKB expansions of solution can be extended on $[h^{1-\epsilon },b]$, for any $\epsilon >0$. Using a different approximation near $0$ and a matching procedure, we obtain the Cauchy Data at $0$ of such WKB solutions. This allows us to derive singular Bohr-Sommerfeld rules. We also pay special attention to uniformity in $W$ for our expansions.