Harmonic oscillators with Neumann condition on the half-line

Virginie Bonnaillie-Noël

doi:10.3934/cpaa.2012.11.2221

Article Contents

2012, Volume 11, Issue 6: 2221-2237. Doi: 10.3934/cpaa.2012.11.2221

This issue Previous Article The maximal regularity operator on tent spaces Next Article Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates

Harmonic oscillators with Neumann condition on the half-line

Virginie Bonnaillie-Noël^1,

1.
IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av Robert Schuman, F-35170 Bruz

Received: October 31, 2010

Revised: August 31, 2011

Published: October 2012

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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.

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Mathematics Subject Classification: Primary: 35P15, 65F15, 65N25; Secondary: 35J10, 65N30.

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