Abstract
We consider the spectrum of the family of one-dimensional self-adjoint operators$-{\mathrm{d}}^2/{\mathrm{d}}t^2+(t-\zeta)^2$, $\zeta\in \mathbb{R}$ on the half-line with Neumann boundary condition.It is well known that the first eigenvalue $\mu(\zeta)$ of this family of harmonic oscillators has a unique minimum when $\zeta\in\mathbb{R}$.This paper is devoted to the accurate computations of this minimum $\Theta_{0}$ and $\Phi(0)$ where $\Phi$ is the associated positive normalized eigenfunction.We propose an algorithm based on finite element method to determine this minimum and we give a sharp estimate of the numerical accuracy. We compare these results with a finite element method.
Highlights
Before motivating our analysis, we first define the parameters Θ0 and Φ(0)
Thanks to the model situation given by the analysis of the angular sector, we are able to determine the asymptotic expansion of the low-lying eigenmodes of the Schrodinger operator on curvilinear polygons: Proposition 2.4
√ For non constant magnetic field, the low-lying eigenvalues admit an asymptotic expansion in power of h
Summary
We first define the parameters Θ0 and Φ(0). We consider the operator −d2/dt2 + (t − ζ) on (0, +∞). In Theorem 3.3, we prove H1-estimate between the normalized eigenfunction Φ associated with Θ0 for the operator H(ζ0) and a normalized quasi-mode for H(ζ). We implement this method in Subsection 4.5 and obtain an accurate approximation of Θ0 and Φ(0): Theorem 1.2. From a numerical point of view, we mention papers [4, 3] which deal with the numerical computations for the bottom of the spectrum of −d2/dt2 + (t − ζ) on a symmetric interval using a finite difference method
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