High Flux Asymptotics and Critical Phenomena for the Magnetic Laplacian
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| Format: | Preprint |
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2026
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| author | Giacomelli, Emanuela L. Kachmar, Ayman Sundqvist, Mikael |
| author_facet | Giacomelli, Emanuela L. Kachmar, Ayman Sundqvist, Mikael |
| contents | We study the lowest eigenvalue of the Neumann magnetic Laplacian in a planar domain divided into two regions, with piecewise constant magnetic fields that may scale differently in the inner and outer parts. Our aim is to describe the high-flux limit and determine when the ground-state energy is eventually monotone and when it continues to oscillate.
We identify several asymptotic regimes according to the relative strength of the outer field. When the outer field is fixed, the lowest eigenvalue exhibits persistent oscillations and the low-energy states localize in the outer region. When the outer field grows more slowly, the behavior depends strongly on the geometry: it is eventually monotone for non-circular domains, while oscillations may persist for disks. In the critical regime, where the two fields are comparable, geometry and flux distribution both play a decisive role. When the outer field dominates, the problem reduces asymptotically to an effective operator on the inner region.
These results show how uneven magnetic scaling, topology, and geometry shape the high-flux spectral behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30793 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High Flux Asymptotics and Critical Phenomena for the Magnetic Laplacian Giacomelli, Emanuela L. Kachmar, Ayman Sundqvist, Mikael Spectral Theory Mathematical Physics Analysis of PDEs We study the lowest eigenvalue of the Neumann magnetic Laplacian in a planar domain divided into two regions, with piecewise constant magnetic fields that may scale differently in the inner and outer parts. Our aim is to describe the high-flux limit and determine when the ground-state energy is eventually monotone and when it continues to oscillate. We identify several asymptotic regimes according to the relative strength of the outer field. When the outer field is fixed, the lowest eigenvalue exhibits persistent oscillations and the low-energy states localize in the outer region. When the outer field grows more slowly, the behavior depends strongly on the geometry: it is eventually monotone for non-circular domains, while oscillations may persist for disks. In the critical regime, where the two fields are comparable, geometry and flux distribution both play a decisive role. When the outer field dominates, the problem reduces asymptotically to an effective operator on the inner region. These results show how uneven magnetic scaling, topology, and geometry shape the high-flux spectral behavior. |
| title | High Flux Asymptotics and Critical Phenomena for the Magnetic Laplacian |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2605.30793 |