HARMONIC OSCILLATORS WITH NEUMANN CONDITION ON THE HALF-LINE

We consider the spectrum of the family of one-dimensional self-adjoint operators -d(2)/dt(2) + (t - zeta)(2), zeta is an element of 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 is an element of 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.