Cauchy data for 1D singular Schrödinger operators

We study semiclassical 1D Schrödinger operators of the form Pu = -h^{2} u'' + x^{\gamma} W(x) u on a finite interval [0,b] for 0 < \gamma \in \mathbb{R}\setminus \mathbb{Q} . We show that the WKB expansions of solution can be extended on [h^{1-\varepsilon},b] for any \varepsilon>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.