Communications on Pure and Applied Analysis (CPAA)

Harmonic oscillators with Neumann condition on the half-line

Pages: 2221 - 2237, Volume 11, Issue 6, November 2012      doi:10.3934/cpaa.2012.11.2221

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Virginie Bonnaillie-Noël - IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av Robert Schuman, F-35170 Bruz, France (email)

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.

Keywords:  Harmornic oscillator, Eigenvalue, finite difference method, error estimate, finite element method
Mathematics Subject Classification:  Primary: 35P15, 65F15, 65N25; Secondary: 35J10, 65N30

Received: November 2010;      Revised: September 2011;      Available Online: April 2012.