Cauchy Data for 1D singular Schrödinger operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915660522586112 |
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| author | Hillairet, Luc Marzuola, Jeremy L. |
| author_facet | Hillairet, Luc Marzuola, Jeremy L. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cauchy Data for 1D singular Schrödinger operators Hillairet, Luc Marzuola, Jeremy L. Mathematical Physics Classical Analysis and ODEs Spectral Theory 81Q20, 34E20, 34E13 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. |
| title | Cauchy Data for 1D singular Schrödinger operators |
| topic | Mathematical Physics Classical Analysis and ODEs Spectral Theory 81Q20, 34E20, 34E13 |
| url | https://arxiv.org/abs/2507.05772 |