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Magnetic Laplacian in sharp three dimensional cones

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Virginie Bonnaillie-Noël,
Monique Dauge,
Nicolas Popoff,
Nicolas Raymond
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The core result of this paper is an upper bound for the ground state energy of the magnetic Laplacian with constant magnetic field $\mathbf{B}=(B_1,B_2,B_3)$ on cones that are contained in a half-space. This bound involves a weighted norm $e(\mathbf{B},\omega)$ of the magnetic field related to moments $m_j$ on a plane section $\omega$ of the cone (Theorem 1.2):
$$
e(\mathbf{B},\omega)
=\left(B_{3}^2\frac{m_{0}m_{2}-m_{1}^2}{m_{0}+m_{2}}
+ B_{2}^2m_{2} + B_{1}^2m_{0} -2 B_{1}B_{2}m_{1} \right)^{1/2}.
$$
When the cone is sharp, i.e. when its section is small, this upper bound tends to 0. A lower bound on the essential spectrum is proved for families of sharp cones, implying that if the section is small enough the ground state energy is an eigenvalue. This circumstance produces corner concentration in the semi-classical limit for the magnetic Schrödinger operator when such sharp cones are involved.

12 mai 2015

Prépublication IRMAR ** 2015-39**

In Operator Theory Advances and Application (Birkhäuser).

Proceedings of the Conference "Spectral Theory and Mathematical Physics''. Santiago 2014.