Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions

TL;DR

The paper provides a unified framework to treat quantum-mechanical problems with degenerate minima where standard perturbation theory fails. By combining path-integral instanton calculations with generalized Bohr–Sommerfeld quantization and WKB analysis, it constructs resurgent expansions that incorporate both perturbative and nonperturbative (instanton) contributions, including their interactions and logarithmic structures. It derives explicit quantization conditions and large-order relations for a variety of potentials (double-well, symmetric/asymmetric wells, periodic cosine, and O(ν) models) and connects these to the spectral problem via Fredholm determinants and resolvent methods. The results offer precise predictions for energy splittings, decay amplitudes, and resonance properties, and demonstrate how leading multi-instanton contributions reproduce and extend the spectrum beyond perturbation theory, with analytic continuations ensuring real spectra after accounting for nonperturbative ambiguities. The work thus advances the understanding of nonperturbative quantum phenomena and provides tools potentially extendable to more complex systems.

Abstract

We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to "cure" the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary which include instanton configurations, characterized by nonanalytic factors exp(-a/g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a nonanalytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schroedinger equation, or alternatively of the Fredholm determinant det(H-E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function. The same strategy could result in new conjectures for problems where our present understanding is more limited.

Figures (7)

• Figure 1: Double-well potential for the case $g = 0.08$ with the two lowest eigenvalues and the unperturbed eigenstate. The energy of the unperturbed ground state is the energy of the ($N=0$)-state of the harmonic oscillator ground state with $E=1/2$. The quantum numbers of the ground state in the potential $V_s(g=0.08,q)$ are $(+,0)$ (positive parity and unperturbed oscillator occupation quantum number $N=0$). The potential $V_s$ is defined in (\ref{['origpot']}). The energy eigenvalue corresponding to the state $(+,0)$ is $E_{+,0} = 0.317\,851\,364\,6\dots$. The first excited state $(-,0)$ has an energy of $E_{-,0} = 0.566\,114\,759\,4\dots$ The arithmetic mean of the energies of the two states $(\pm,0)$ is $0.441\,983\,062\,0\dots$ and thus different from the value $1/2$. This is a consequence of the two-instanton (more general: even-instanton) shift of the energies.
• Figure 2: Double-well potential: Comparison of numerical data obtained for the function $\Delta(g)$ with the sum of the terms up to the order of $g^2$ of its asymptotic expansion for $g$ small [see equation (\ref{['asympDelta']})], solid line. If we express both the numerator and the denominator of (\ref{['defDelta']}) as a power series in $g$ [with the help of the instanton coefficients (\ref{['epsiloncoeff']})] and keep all terms up to the order of $g^2$, the dashed curve results. This latter approach is also used for the curves in figures \ref{['figdelta2']}, \ref{['figdelta3']}, and \ref{['figd']} below.
• Figure 3: The four turning points.
• Figure 4: The instanton configuration.
• Figure 5: The two-instanton configuration.