Cauchy Data for 1D singular Schrödinger operators

TL;DR

This work develops a two-region semiclassical analysis for 1D Schrödinger operators with a singular potential $V(x)=x^{\gamma}W(x)$ near $x=0$, where $0<\gamma\notin\mathbb{Q}$. Exterior WKB expansions are extended to $[h^{1-\varepsilon},b]$ and combined with interior-region constructions via a matching in an $h$-dependent overlap, yielding a precise relation between Cauchy data at $0$ and $b$. The authors establish singular Bohr–Sommerfeld rules and provide uniform, $W$-robust asymptotics for the resulting eigenstructure, including the half-line case via a Maslov-type ansatz. The results offer a systematic framework for spectral information in rough potentials and highlight how the irrational singular exponent shapes the spectral data through discrete exponent sets.

Abstract

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

Theorems & Definitions (29)

• Theorem 1
• Remark 1.1
• Definition 3.1
• Remark 3.1
• EXAMPLE 3.1
• Proposition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof