Quantum potential with no perturbative series, and nonperturbative vacuum dominated by complex classical paths

Abstract

Spectra of standard 1d potentials (double-well, sin-Gordon etc) are given by trans-series in coupling, including (badly divergent) perturbative series (PS), and nonperturbative terms. All of them are badly defined (e.g. PS are badly divergent) but in sum supposed to be good. In this paper we discuss an example of a potential with specially defined couplings making PS completely absent. We calculate its nonperturbative vacuum energy and show that they are reproduced by the action of certain complex solutions to holomorphic Newton equation.

Figures (7)

• Figure 1: The splitting of the negative-parity state from the vacuum, $gap=E_- - E_+$, shown as data points. The three curves correspond to one-, two-, and three-loop semiclassical approximations.
• Figure 2: Potentials for $a=0,0.2,0.3,0.4$ (red,blue, orange,green).
• Figure 3: Upper left corner of matrix form of the generic Hamiltonian.
• Figure 4: Real and imaginary parts of the extrema, as a function of parameter $a$. Lower plot to complex conjugate pair only.
• Figure 5: The ground state (vacuum) energy versus parameter $a$. The upper (red) and lower (blue) points are the same data, just plotted in linear and logarithmic plots. Red circles on the lower plot represent effects of complex bions according to (\ref{['eqn_contribution_bions']}) derived below.