Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations

TL;DR

This work develops a comprehensive, unified framework for multi-instanton effects in quantum mechanics with degenerate minima, applying it to asymmetric, periodic, radial, and Fokker–Planck–type potentials. By deriving perturbative $B$-functions and instanton $A$-functions to high orders (up to $O(g^4)$ for several potentials and up to $O(g^8)$ for the double-well) and forming spectral determinants like $Δ(E)$ and $Δ(E,φ)$, the authors reveal a consistent resurgent structure linking perturbative and nonperturbative sectors. They provide extensive analytic and numerical verifications, including three- and higher-instanton corrections, large-order behavior, and reference benchmarks, illustrating how instanton physics completes the spectrum where perturbation theory alone fails. The results offer a blueprint for extending exact semi-classical methods to broader analytic potentials and potentially to higher-dimensional field theories, highlighting both the power and remaining questions in resurgent quantum mechanics.

Abstract

In this second part of the treatment of instantons in quantum mechanics, the focus is on specific calculations related to a number of quantum mechanical potentials with degenerate minima. We calculate the leading multi-instanton constributions to the partition function, using the formalism introduced in the first part of the treatise [J. Zinn-Justin and U. D. Jentschura, e-print quant-ph/0501136]. The following potentials are considered: (i) asymmetric potentials with degenerate minima, (ii) the periodic cosine potential, (iii) anharmonic oscillators with radial symmetry, and (iv) a specific potential which bears an analogy with the Fokker-Planck equation. The latter potential has the peculiar property that the perturbation series for the ground-state energy vanishes to all orders and is thus formally convergent (the ground-state energy, however, is nonzero and positive). For the potentials (ii), (iii), and (iv), we calculate the perturbative B-function as well as the instanton A-function to fourth order in g. We also consider the double-well potential in detail, and present some higher-order analytic as well as numerical calculations to verify explicitly the related conjectures up to the order of three instantons. Strategies analogous to those outlined here could result in new conjectures for problems where our present understanding is more limited.

Figures (3)

• Figure 1: Double-well potential: Comparison of numerical data obtained for the function $\Delta(g)$ with the sum of the terms of order $g^j$ (where $j \leq 10$) of its asymptotic expansion for $g$ small, where we express both the numerator as well as the denominator of (\ref{['defDelta']}) as a power series in $g$. [When the resulting fraction is further expanded in $g$, then the leading terms of order $g$ and $g^2$ are given in equation (\ref{['asympDelta']}) and represented graphically in figure \ref{['figdelta']}]. The higher-order terms employed here are easily derived using equations (\ref{['eE1N0']}) and (\ref{['eE2N0']}). There is good agreement between numerically determined ("exact") values (data points) and the smooth curve given by the analytic asymptotics.
• Figure 2: Double-well potential: Comparison of numerical data obtained for the function $\Delta_1(g)$ defined in (\ref{['defDelta1']}) with the sum of the terms up to the order of $g^{10}$ of its asymptotic expansion for $g$ small, where we express both the numerator as well as the denominator of (\ref{['defDelta1']}) as a power series in $g$. There is good agreement between numerically determined ("exact") values (data points) and the smooth curve given by the analytic asymptotics. See also Table \ref{['tabcirmi1']}.
• Figure 3: Double-well potential: Comparison of numerical data obtained for the function $D(g)$ defined in (\ref{['defD']}) with the sum of the terms up to the order of $g^{10}$ of its asymptotic expansion for $g$ small. Again, we express both the numerator as well as the denominator of the expression defining $D(g)$ [see equation (\ref{['defD']})] as a power series in $g$.