The ground state of long-range Schrodinger equations and static $q\bar{q}$ potential

TL;DR

This paper develops analytical methods to compute the weak-coupling expansion of the ground state energy for long-range Schrödinger operators, revealing non-analytic behavior and infrared logarithm exponentiation, with applications to quark-antiquark potentials.

Contribution

It introduces a systematic approach to derive the asymptotic expansion of bound state energies in long-range potentials, including non-analytic and logarithmic corrections, and proves infrared logarithm exponentiation at all orders.

Findings

Derived the general third-order expansion of binding energy.

Confirmed infrared logarithm exponentiation in Schrödinger equations.

Validated methods on soluble and non-soluble potentials, including supersymmetric cases.

Abstract

Motivated by the recent results in arXiv:1601.05679 about the quark-antiquark potential in $\mathcal N=4$ SYM, we reconsider the problem of computing the asymptotic weak-coupling expansion of the ground state energy of a certain class of 1d Schr\"odinger operators $-\frac{d^{2}}{dx^{2}}+\lambda\,V(x)$ with long-range potential $V(x)$. In particular, we consider even potentials obeying $\int_{\mathbb R}dx\, V(x)<0$ with large $x$ asymptotics $V\sim -a/x^{2}-b/x^{3}+\cdots$. The associated Schr\"odinger operator is known to admit a bound state for $\lambda\to 0^{+}$, but the binding energy is rigorously non-analytic at $\lambda=0$. Its asymptotic expansion starts at order $\mathcal O(\lambda)$, but contains higher corrections $\lambda^{n}\,\log^{m}\lambda$ with all $0\le m\le n-1$ and standard Rayleigh-Schr\"odinger perturbation theory fails order by order in $\lambda$. We discuss various analytical tools to tame this problem and provide the general expansion of the binding energy at $\mathcal O(\lambda^{3})$ in terms of quadratures. The method is tested on a soluble potential that is fully under control, and on various non-soluble cases as well. A supersymmetric case, arising in the study of the quark-antiquark potential in $\mathcal N=6$ ABJ(M) theory, is also exploited to provide a further non-trivial consistency check. Our analytical results confirm at third order a remarkable exponentiation of the leading infrared logarithms, first noticed in $\mathcal N=4$ SYM where it may be proved by Renormalization Group arguments. We prove this interesting feature at all orders at the level of the Schr\"odinger equation for general potentials in the considered class.