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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.23156 |
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| _version_ | 1866908854683435008 |
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| author | Keller, Matthias Pettinari, Lorenzo van de Ven, Christiaan J. F. |
| author_facet | Keller, Matthias Pettinari, Lorenzo van de Ven, Christiaan J. F. |
| contents | We analyze the semiclassical $d$-dimensional Schrödinger operator in the continuum $ \frac{1}{2} Δ+ λ_N^2 V$ discretized on a mesh with spacing proportional to $1/N$. The semi-classical parameter $λ_N$ is chosen as $λ_N = N^{1 - γ}$, with $γ\in (-1,1)$, which ensures that $N$ governs both the semiclassical and continuum limit simultaneously. We prove that all eigenvalues of the discrete operator converge to those of the continuum, as $λ_N\to\infty$. Beyond this semi-classical domain, in the case of the harmonic oscillator, we further discuss the spectral asymptotics for $γ\in \mathbb{R} \setminus (-1,1)$, thereby fully characterizing the eigenvalue behavior across all possible values of $γ\in\mathbb{R}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23156 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coupling of the continuum and semiclassical limit. Part I: convergence of eigenvalues Keller, Matthias Pettinari, Lorenzo van de Ven, Christiaan J. F. Mathematical Physics We analyze the semiclassical $d$-dimensional Schrödinger operator in the continuum $ \frac{1}{2} Δ+ λ_N^2 V$ discretized on a mesh with spacing proportional to $1/N$. The semi-classical parameter $λ_N$ is chosen as $λ_N = N^{1 - γ}$, with $γ\in (-1,1)$, which ensures that $N$ governs both the semiclassical and continuum limit simultaneously. We prove that all eigenvalues of the discrete operator converge to those of the continuum, as $λ_N\to\infty$. Beyond this semi-classical domain, in the case of the harmonic oscillator, we further discuss the spectral asymptotics for $γ\in \mathbb{R} \setminus (-1,1)$, thereby fully characterizing the eigenvalue behavior across all possible values of $γ\in\mathbb{R}$. |
| title | Coupling of the continuum and semiclassical limit. Part I: convergence of eigenvalues |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2602.23156 |