  This paper presents a nonperturbative method for solving eigenproblems. This
method applies to almost all potentials and provides nonperturbative
approximations for any energy level. The method converts an eigenproblem into a
perturbation problem, obtains perturbation solutions through standard
perturbation theory, and then analytically continues the perturbative solution
into a nonperturbative solution. Concretely, we follow three main steps: (1)
Introduce an auxiliary potential that can be solved exactly and treat the
potential to be solved as a perturbation on this auxiliary system. (2) Use
perturbation theory to obtain an approximate polynomial of the eigenproblem.
(3) Use a rational approximation to analytically continue this approximate
polynomial into the nonperturbative region.
