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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.13545 |
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Table of Contents:
- The particle in a well in dimension one is a classical problem in quantum mechanics. We study higher-dimensional analogues of the problem, where the well is a smooth domain in $\mathbb{R}^d$. We show that simple eigenvalues and eigenfunctions of the corresponding Schrödinger operator depend smoothly on the square root $h$ of the inverse depth of the well and provide an explicit first-order expansion of the eigenvalues at $h = 0$. Our proof consists of two steps. In the first step, we construct $\mathcal{O}(h^\infty)$ quasimodes (approximate eigenfunctions) on a resolution of $[0, 1)_h\times\mathbb{R}^d$ which allows us to capture fine structure near the boundary of the well. The second step corrects these quasimodes to true eigenfunctions via a fixed point argument.