Numerical analysis of spectral properties of coupled oscillator Schrödinger operators. I. Single and double well anharmonic oscillators

Abstract

We describe several methods for computing many eigenvalues and eigenfunctions of a single anharmonic oscillator Schrödinger operator whose potential may have one or two minima. One of the methods requires the solution of an ill-conditioned generalized eigenvalue problem. This method has the virtue of using a bounded amount of work to achieve a given accuracy in both the single and double well regions. We give rigorous bounds, and we prove that the approximations converge faster than any inverse power of the size of the matrices needed to compute them. We present the results of our computations for the g : ϕ 4 : 1 g:{\phi ^4}{:_1} theory. These results indicate that the methods actually converge exponentially fast. We conjecture why this is so.