Bootstrapping Simple QM Systems

Abstract

We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians, following the recent approach of Han, Hartnoll, and Kruthoff. We focus on comparing the bootstrap method data to known analytical predictions for the hydrogen atom and the harmonic oscillator. We resolve many energy levels for each, and more levels are resolved as the size of the matrices used to solve the problem increases. Using the bootstrap approach we find the spectrum of the Coulomb and harmonic potentials converge exponentially fast.