Unperturbation theory: reconstructing Lagrangians from instanton fluctuations

Abstract

Instantons present a deep insight into non-perturbative effects both in physics and mathematics. While leading instanton effects can be calculated simply as an exponent of the instanton action, the calculation of subleading contributions usually requires the spectrum of fluctuation operator on the instanton background and its Green's function, explicit knowledge of which is rare and a great success. Thus, we propose an inverse problem, namely, the reconstruction of the nonlinear action of the theory admitting instantons from the given fluctuation operator with a known Green's function. We constructively build the solution for this problem and apply it to a wide class of exactly solvable Schrödinger operators, called shape-invariant operators, and its simpler subclass, namely reflectionless Pöschl-Teller operators. In the latter case, we found that for the most values of parameters the reconstructed potentials are naturally defined not on the real line, but on some special multisheet covering of the complex plane, and discuss its physical interpretation. For the wider but less simple class of shape-invariant operators, we derive the set of parameters leading to the new infinite families of analytic potentials.

Figures (15)

• Figure 1: Potentials, leading to the two types of typical instanton configurations: (a) tunneling solution, corresponding to infinite-time motion from the one local maximum of the inverted potential $-V(x)$ to another, and (b) bounce solution, consisting of infinite-time motion from the local maximum of the inverted potential, bounce off the turning point, and infinite-time motion back to the local maximum.
• Figure 2: Shapes of typical instanton configurations: (a) tunneling solution and (b) bounce solution, corresponding to the infinite Euclidean time motion in the potentials depicted at Fig. \ref{['fig:tun_bnc']}.
• Figure 3: (a) Shape of the instanton trajectory (\ref{['eq:dp_x_of_tau']}). (b) Two natural extensions (\ref{['eq:dp_pot_cont']}) of the potential (\ref{['eq:dp_pot_non_cont']}) beyond the image of instanton solution: "analytic" continuation $V_1(x)$ (orange line), and continuation by infinite walls $V_2(x)$ (violet line).
• Figure 4: Typical shapes of reconstructed instanton trajectories. (i) even $\ell-m$, (ii) odd $\ell-m$.
• Figure 5: Potentials, whose fluctuation operators are reflectionless Pöschl-Teller operators. For $\ell - m > 1$ these are multivalued and have intermediate turning points. (i) even $\ell-m$, (ii) odd $\ell-m$.