Abstract
We discuss the strong field limit for the lowest eigenvalue of the magnetic Laplacian, in two dimensional domains, when the magnetic field has interior singular points. In particular, we consider a singular magnetic potential of unit length, occurring in the theory of liquid crystals, and determine the leading eigenvalue asymptotics. Moreover, we present examples of magnetic potentials having several singular points, whose cost is minimized by locating them near the boundary of the domain.
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Notes
- 1.
In that context this is called a director field [13].
- 2.
The Soboelv space \(W^{1,p}(\Omega ,\mathbb R^2)\) consists of functions in \(L^p(\Omega ,\mathbb R^2)\) with gradient in \(L^p(\Omega ,\mathbb R^2)\).
- 3.
The support of the function u is compact, hence contained in some disk D. The boundary term resulting from the integration by parts vanishes since u vanishes on the boundary of D.
- 4.
The lowest eigenvalues with magnetic potentials \(\mathbf A\) and \(\mathbf A^{\prime }:=\mathbf A-\nabla \chi \) are equal, by the unitary transformation \((u,\mathbf A)\mapsto (u^{\prime }:=e^{i\chi }u,\mathbf A^{\prime }=\mathbf A-\nabla \chi )\), for any function \(\chi \in H^1(\Omega )\).
- 5.
We can take the boundary neighborhood of the form \(V_{x_0}=D(x_0,r)\cap \Omega \).
- 6.
The only difference is that the remainder \(\mathcal O(|x-x_0|{ }^2)\) in (20) will be replaced by \(o(|x-x_0|)\).
- 7.
In general, when localizing in a disc of radius \(\sigma ^{-\rho }\), one encounters two types of errors, \(\mathcal O(\sigma ^{2\rho })\) resulting from the localization cut-off, and \(\mathcal O(\sigma ^{3-6\rho })\) resulting from the approximation of the magnetic field; optimizing we get \(2\rho =3-6\rho \) and therefore \(\rho =3/8\); see [16, Sec. 5.1 & Eq. (5.4)].
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Acknowledgements
AK acknowledges the support of the Istituto Nazionale di Alta Matematica “F. Severi”, through the Intensive Period “INdAM Quantum Meetings (IQM22)” and of the Center for Advanced Mathematical Sciences at the American University of Beirut. Pan was partially supported by the National Natural Science Foundation of China grant no. 12071142, the Natural Science Foundation of Guangdong Province grant no. 2023A1515012868, the Natural Science Foundation of Shenzhen grant no. JCYJ20220530143801004, and the Shenzhen Municipal Grant No. GXWD20201231105722002. The authors would like to thank the anonymous referee for their valuable suggestions.
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Kachmar, A., Pan, XB. (2023). Lowest Eigenvalue Asymptotics in Strong Magnetic Fields with Interior Singularities. In: Correggi, M., Falconi, M. (eds) Quantum Mathematics I. INdAM 2022. Springer INdAM Series, vol 57. Springer, Singapore. https://doi.org/10.1007/978-981-99-5894-8_11
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