Cauchy Data for 1D singular Schrödinger operators

Abstract

We study semiclassical 1-D Schrödinger operators of the form $Pu=-h^2{u}^{\prime \prime }+x^{3}W(x)u$ on a finite interval $[0,b]$ for $0<{\hat{I}}^3\in \mathbb{R}\setminus \mathbb{Q}$. We show that that the WKB expansions of solution can be extended on $[h^{1-\hat{I}\mu },b]$, for any $\hat{I}\mu >0$. 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.

25 pages, comments welcome!