Cauchy data for 1D singular Schrödinger operators

Abstract

We study semiclassical 1D Schrödinger operators of the form $Pu=-h^2{u}^{\prime \prime }+x^{\gamma }W(x)u$ 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 $[h^{1-\epsilon },b]$ 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.