Bootstrapping the Kronig-Penney Model

TL;DR

This paper explores the use of bootstrap methods from conformal field theory to compute the energy spectrum of the Kronig-Penney model, highlighting both successes in band gap calculation and challenges in constraining minimum energy.

Contribution

It introduces a bootstrap approach tailored for the Schrödinger equation with periodic potentials, specifically applied to the Kronig-Penney model, and suggests enhancements for better spectral constraints.

Findings

Efficiently computes band gaps of the energy spectrum.

Struggles to effectively constrain the minimum energy.

Proposes using higher powers of momenta for improved constraints.

Abstract

Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.