Uniform WKB, Multi-instantons, and Resurgent Trans-Series

TL;DR

This paper demonstrates that energy levels in quantum systems with degenerate harmonic minima admit a resurgent trans-series that unifies perturbative and non-perturbative physics. By employing a uniform WKB framework and global boundary conditions, the authors show that all non-perturbative information, including multi-instanton effects and quasi-zero modes, is encoded in the perturbative data through a simple relation between the perturbative energy and the non-perturbative factor. They establish explicit resurgence relations, connect their results to Zinn-Justin–Jentschura quantization, and verify exact cancellations of ambiguities across DW, SG, FP, and AHO potentials. The work highlights the broad applicability of resurgence to QM and its potential implications for quantum field theories with degenerate vacua, where trans-series provide a consistent, unambiguous framework for non-perturbative physics.

Abstract

We illustrate the physical significance and mathematical origin of resurgent trans-series expansions for energy eigenvalues in quantum mechanical problems with degenerate harmonic minima, by using the uniform WKB approach. We provide evidence that the perturbative expansion, combined with a global eigenvalue condition, contains all information needed to generate all orders of the non-perturbative multi-instanton expansion. This provides a dramatic realization of the concept of resurgence, whose structure is naturally encoded in the resurgence triangle. We explain the relation between the uniform WKB approach, multi-instantons, and resurgence theory. The essential idea applies to any perturbative expansion, and so is also relevant for quantum field theories with degenerate minima which can be continuously connected to quantum mechanical systems.

Figures (5)

• Figure 1: a) Dilute gas of 1-instantons for a periodic potential (as given in typical textbook treatment). b) Dilute gas of 1-instantons, 2-instantons, 3-instantons, etc. 2-instanton events (topological molecules) such as $[{\mathcal{I}} {\mathcal{I}}], [\bar{{\mathcal{I}}} \bar{{\mathcal{I}}}], [{\mathcal{I}} \bar{{\mathcal{I}}}]$ are rarer, but present. The amplitude associated with neutral 2-instantons or any other $k$-instanton with neutral 2-instanton subcomponent are multi-fold ambiguous. This ambiguity cures the ambiguity of perturbation theory around the perturbative vacuum. c) $n$-instanton events classified according to homotopy (columns) and resurgence (refined structure in each column.) This picture is the result of uniform WKB and multi-instanton approach.
• Figure 2: The global boundary condition for the lowest two states in the double-well potential $V(y)=y^2(1+y)^2$. The lower state wave function is nodeless and has vanishing derivative at the midpoint of the barrier. The upper state wave function has one node at the midpoint of the barrier.
• Figure 3: The global boundary condition for the band-edge states of the lowest band for the Sine-Gordon potential $V(y)=\sin^2 y$. The lower band-edge wave function is nodeless and has vanishing derivative at the midpoint of each barrier. The upper band-edge wave function has one node at the midpoint of each barrier.
• Figure 4: Same as Fig.1, for the double-well potential.
• Figure 5: A comparison of the exact band edges, and the center of the band, for the lowest band of the Sine-Gordon potential [solid lines], with the weak-coupling trans-series expansion [dotted lines], and the strong-coupling results [dashed lines]. The exact results are generated using the Mathematica functions MathieuCharacteristicA and MathieuCharacteristicB, which compute Mathieu band edges numerically. The weak-coupling expansions have been plotted here using the expression in (\ref{['weak']}), and the strong-coupling expansions have been plotted using (\ref{['strong']}). Note the excellent numerical agreement.