Low-energy eigenstates in a vanishing magnetic field
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909620023328768 |
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| author | Benedetto, Lino |
| author_facet | Benedetto, Lino |
| contents | This paper is dedicated to the spectral analysis of the semiclassical purely magnetic Laplacian on the plane in the situation where the magnetic field $B$ vanishes nondegenerately on an open smooth curve $Γ$. We prove the existence of a discrete spectrum for energy windows of the scale $h^{4/3}$ and give complete asymptotics in the semiclassical paramater $h$ for eigenvalues in such windows. Our strategy relies on the microlocalization of the corresponding eigenfunctions close to the zero locus $Γ$ and on the implementation of a Born-Oppenheimer strategy through the use of operator-valued pseudodifferential calculus and superadiatic projectors. This allows us to reduce our spectral analysis to that of effective semiclassical pseudodifferential operators in dimension 1 and apply the well-known semiclassical techniques à la Helffer-Sjöstrand. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low-energy eigenstates in a vanishing magnetic field Benedetto, Lino Spectral Theory Analysis of PDEs This paper is dedicated to the spectral analysis of the semiclassical purely magnetic Laplacian on the plane in the situation where the magnetic field $B$ vanishes nondegenerately on an open smooth curve $Γ$. We prove the existence of a discrete spectrum for energy windows of the scale $h^{4/3}$ and give complete asymptotics in the semiclassical paramater $h$ for eigenvalues in such windows. Our strategy relies on the microlocalization of the corresponding eigenfunctions close to the zero locus $Γ$ and on the implementation of a Born-Oppenheimer strategy through the use of operator-valued pseudodifferential calculus and superadiatic projectors. This allows us to reduce our spectral analysis to that of effective semiclassical pseudodifferential operators in dimension 1 and apply the well-known semiclassical techniques à la Helffer-Sjöstrand. |
| title | Low-energy eigenstates in a vanishing magnetic field |
| topic | Spectral Theory Analysis of PDEs |
| url | https://arxiv.org/abs/2505.16671 |