  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 Schr\"{o}dinger operators, called shape-invariant operators, and its
simpler subclass, namely reflectionless P\"{o}schl-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.
