A 3D-Schrödinger operator under magnetic steps with semiclassical applications

Figure 1.  Let $ \alpha \in (0,\pi) $, $ \gamma\in[0,\pi/2] $ and $ a\in[-1,1)\setminus\{0\} $. The magnetic field $ {\mathbf{B}}_{ \alpha,\gamma,a} $ in $ \mathbb{R}_+^3 $ can have different directions in the two regions $ \mathcal{D}_\alpha^1 $ and $ \mathcal{D}_\alpha^2 $, according to the sign of $ a $. The strength of the magnetic field is $ \mathsf s_{ \alpha,a} = 1 $ in $ \mathcal D^1_\alpha $ and $ \mathsf s_{ \alpha,a} = a $ in $ \mathcal D^2_\alpha $. The transition of the strength occurs at the plane $ P_\alpha $ of equation $ x_1\sin\alpha-x_2\cos\alpha = 0 $, referred to as the discontinuity plane. The angle $ \gamma $ (modulo $ -\pi $) represents the angle that $ {\mathbf{B}}_{ \alpha,\gamma,a} $ makes with the $ x_3 $-axis

Figure 2.  For the triplets $ (\alpha,\gamma,a) $ in the colored region, $ \lambda_{\alpha, \gamma,a} $ is an eigenvalue of an operator $ \mathcal L_{\underline {\mathbf{A}}_{\alpha,\gamma,a}}+V_{\underline{{\mathbf{B}}}_{\alpha,\gamma,a},\tau_*} $

Figure 3.  Illustration of the domain $ \Omega $, the discontinuity surface $ S $ (shaded), and the discontinuity edge $ \Gamma: = \partial S $. The magnetic field $ {\mathbf{B}} $ is tangent to $ S $ and its strength exhibits a jump of discontinuity at $ S $

Figure 4.  For $ \alpha\in(0,\pi) $, $ \gamma\in(0,\pi/2] $ and $ a\in[-1,0) $, the set $ \Upsilon_{\alpha, \gamma, a,\tau} $ is drawn in blue. For $ \tau\geq0 $ (at right), $ \Upsilon_{\alpha, \gamma,a,\tau} = \Upsilon^{(1)}_{\alpha, \gamma, \tau}\cup\Upsilon^{(2)}_{\alpha, \gamma, a,\tau} $. For $ \tau<0 $ (at left), $ \Upsilon_{\alpha, \gamma, a,\tau} = l_\alpha $

Figure 5.  For $ \alpha\in(0,\pi) $, $ \gamma\in(0,\pi/2] $ and $ a\in(0,1) $, the set $ \Upsilon_{\alpha, \gamma, a,\tau} $ is drawn in blue. For $ \tau\geq0 $ (at right), $ \Upsilon_{\alpha, \gamma, a,\tau} = \Upsilon^{(1)}_{\alpha, \gamma, \tau} $. For $ \tau<0 $ (at left), $ \Upsilon_{\alpha, \gamma, a,\tau} = \Upsilon^{(2)}_{\alpha, \gamma, a,\tau} $

Figure 6.  $ \mathrm{dist}(\mathrm{supp}\chi_1 u,\Upsilon^{(1)}_{\alpha,\gamma,\tau})\geq d_1 $ and $ \mathrm{dist}(\mathrm{supp}\chi_3 u,\Upsilon^{(2)}_{\alpha,\gamma,a,\tau})\geq d_2 $

Figure 7.  An illustration of the coordinates-transformation $ \Phi = \Phi_2\circ \Phi_1 $ as a composition of the local diffeomorphisms $ \Phi_1 $ and $ \Phi_2 $, as defined in Section 5.1.1. The shaded regions respectively represent (from the left to the right) the surfaces $ S $, $ \breve{S} $ and $ P_{\alpha_0} $ near $ (0,0,0) $