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arXiv · 2507.05772

Cauchy Data for 1D singular Schr\"odinger operators

Luc Hillairet·Jeremy L. Marzuola
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Abstract

We study semiclassical 1-D Schr\"odinger 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 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.

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Luc Hillairet, Jeremy L. Marzuola. 2025-07-08. Cauchy Data for 1D singular Schr\"odinger operators. https://arxiv.org/abs/2507.05772

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