Quantum bootstrap for central potentials

TL;DR

This work extends the quantum-mechanical bootstrap to central potentials in three dimensions, including non-algebraic cases like Yukawa and Gaussian, and compares with the well-understood Coulomb and Cornell potentials as well as conformal quantum mechanics. Using semidefinite programming, the authors construct operator bases and enforce positivity, commutation, and equation-of-motion constraints to obtain rigorous bounds on ground-state energies, achieving exceptional precision in several cases (e.g., Cornell bounds better than $10^{-7}$ and critical coupling bounds near $10^{-8}$ accuracy). They report consistently tight lower bounds for Yukawa and Gaussian, along with exact or near-exact results for Coulomb and highly precise, two-sided bounds for the Cornell potential, while noting limitations in obtaining upper bounds for some models and highlighting boundary anomalies at r=0 in central potentials. The study also discusses convergence questions and the role of self-adjoint extensions in conformal quantum mechanics, signaling both the practical impact and remaining theoretical challenges of bootstrap methods in 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.

Figures (4)

• Figure 1: Bootstrap lower bounds on the ground-state energy of the Yukawa potential, across a range of couplings $\rho$. The bottom panel shows the absolute error in these calculations, which is typically better than one part in one million. The comparison points are obtained by direct diagonalization as described in the appendix. Direct diagonalization is trustworthy only below some binding energy; to be conservative, we have truncated the results to the case $E_0 < - 2 \times 10^{-2}$.
• Figure 2: Bootstrap lower bounds on the ground-state energy of the Gaussian potential, as a function of coupling. The relative error is characteristically around 1%.
• Figure 3: Upper and lower bounds (separated by less than one pixel) on the ground-state energy of the Cornell potential, as a function of the strong coupling $\alpha_s$. For all couplings investigated, the bounds constrain the energy to an interval of size less than $10^{-7}$ (in natural units). These bounds are consistent with numerical results obtained by direct diagonalization (not shown).
• Figure 4: Allowed regions in terms of energies and $\langle \hat{r}^{-1} \rangle$. The blue region uses two $2 \times 2$ bootstrap matrices, and the orange region uses two $3 \times 3$ matrices. The black dots are the exact results. The energy is unbounded below, but at each $\langle \hat{r}^{-1}\rangle$ non-trivial bounds are readily obtained.