Computations of the first eigenpairs for the Schrödinger operator with magnetic field

Abstract

International audienceThis paper is devoted to computations of eigenvalues and eigenvectors for the Schrödinger operator with constant magnetic field in a domain with corners, as the semi-classical parameter $h$ tends to $0$. The eigenvectors corresponding to the smallest eigenvalues concentrate in the corners: They have a two-scale structure, consisting of a corner layer at scale $\sqrt h$ and an oscillatory term at scale $h$. The high frequency oscillations make the numerical computations particularly delicate. We propose a high order finite element method to overcome this difficulty. Relying on such a discretization, we illustrate theoretical results on plane sectors, squares, and other straight or curved polygons. We conclude by discussing convergence issues

Asymptotics for the Low-Lying Eigenstates of the Schrödinger Operator with Magnetic Field near Corners

“…We can see from our formulas, see (29) and (59), that eigenmodes have a two-scale structure: they concentrate near corners with the scale √ h but they have a strongly oscillating term of the form exp(iΦ(x)/h) and their approximation is very delicate. We investigate a finite element method using high degree polynomials in [6]. Numerical computations do exhibit tunneling effect with multiple crossings between eigenvalues when the domain presents some symmetry (for example, in a square).…”

“…We obtain asymptotics series in powers of √ h for a finite number of low-lying eigenstates of P h . In Section 8, we conclude our paper by commenting on numerical approximation issues: The eigenmodes have a two-scale structure, in the form of the product of a corner layer at scale √ h with an oscillatory term at scale h. The latter makes the numerical approximation delicate, see [1,2,6]. A finite element method using high degree polynomials is being investigated by the authors, together with the tunneling effect in presence of symmetries.…”

“…Based on the asymptotics of μ 1 (α) as α → 0 and numerical computations [1,2,6], we conjecture that μ 1 is increasing from (0, π] onto (0, Θ 0 ] and equal to Θ 0 on [π, 2π): See on Figure 1 the result of a series of computations of μ 1 (α) for α = kπ/200, k = 1, . .…”

“…. , 200, performed by a high order finite element method based on the library Mélina [23,6]. In Figure 1, the theoretical lower and upper bounds α/ √ 3 ≤ μ 1 (α) ≤ Θ 0 α/π are also plotted.…”

Breaking a magnetic zero locus: model operators and numerical approach

“…The discretization space for the finite element method is included in the form domain of the operator and thus the computed eigenvalue provides a rigorous upper-bound (see [1,Sect. 2] and [3, Sect.…”