Quantum bootstrap for central potentials

TL;DR

This paper applies the quantum bootstrap method to various central potentials, demonstrating high-precision calculations of bound states and critical couplings, and extends the approach to non-algebraic potentials like Yukawa and Gaussian.

Contribution

It introduces the bootstrap approach to non-algebraic potentials and achieves unprecedented precision in calculating ground-state energies and critical couplings.

Findings

Determined the critical coupling in the Cornell potential to better than 10^{-7}.

Achieved high-precision lower bounds on ground-state energies, sometimes exceeding 10^{-8}.

Extended bootstrap methods to Yukawa, Gaussian, and conformal quantum mechanics.

Abstract

We study the quantum-mechanical bootstrap as it applies to the bound states of several central potentials in three dimensions. As part of this effort, we show how the bootstrap approach may be applied to ``non-algebraic'' potentials, such as the Yukawa potential (which asymptotically decays as an exponential) and a Gaussian potential. We additionally review the bootstrap of the Coulomb potential, demonstrate a high-precision bootstrap of the Cornell potential, and study conformal quantum mechanics. These results further recommend the bootstrap as a numerical method for high-precision calculations of ground-state physics, where applicable: for example, we are able to determine the critical coupling in the Cornell potential to better than one part in $10^7$, the most precise determination to date. Lower bounds on energies are also of high precision, occasionally one part in greater than $10^8$. Finally, we discuss the circumstances under which we are able to obtain meaningful upper bounds on ground-state energies.