When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit (Q2855130)

The paper addresses the elliptic equation in the three-dimensional space with coordinates \((x_1,x_2,x_3)\), which plays a fundamentally important role in the Ginzburg-Landau superconductivity theory, NEWLINE\[NEWLINE(-ih\nabla + \mathbf{A(x_1,x_2,x_3)})^2\psi=\lambda \psi,NEWLINE\]NEWLINE with the Neumann boundary conditions on a surface with a non-smooth edge. A uniform magnetic field is taken here, corresponding to the vector potential in the form of \(\mathbf{A(x_1,x_2,x_3)}=(-x_2,0,0)\). The objective is to find eigenvalues \(\lambda\) and the corresponding eigenmodes of the wave function, \(\psi(x_1,x_2,x_3)\). The particular problem addressed in this work is to construct an asymptotic expansion for the lowest eigenvalues in the semi-classical limit, \(h\to 0\) (\(h\) is the Planck's constant). The result is that the first and second terms of the expansion are proportional respectively, to \(h\) and \(h^{3/2}\).