Bootstrapping More QM Systems

TL;DR

This paper evaluates the bootstrap method for calculating spectra of 1D Hamiltonians, focusing on double well and Mathieu potentials, showing promising results but also highlighting current limitations in capturing band structures.

Contribution

It applies the bootstrap approach to new classes of quantum systems, specifically double well and Mathieu potentials, and analyzes its effectiveness and limitations.

Findings

Bootstrap accurately reproduces energies for double well potentials.

The method captures band structures but struggles with quasi-momentum determination.

Further techniques are needed for complete dispersion relation analysis.

Abstract

We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians. In this paper we focus on problems that have a two parameter search space in the bootstrap approach: the double well and a periodic potential associated to the Mathieu equation. For the double well, we compare the energies with contributions from perturbative and non-perturbative results, finding good agreement. For the periodic potentials, we notice that the bootstrap approach gives the band structure of the periodic potential, but it has trouble finding the quasi-momentum of the system. To make further progress on the dispersion relation of the bands, new techniques are needed.