  Adaptive perturbation is a new method for perturbatively computing the
eigenvalues and eigenstates of quantum mechanical Hamiltonians that heretofore
were not believed to be obtainable by such methods. The novel feature of
adaptive perturbation theory is that it decomposes a given Hamiltonian, $H$,
into an unperturbed part and a perturbation in a way which extracts the leading
non-perturbative behavior of the problem exactly. This paper introduces the
method in the context of the pure anharmonic oscillator and then goes on to
apply it to the case of tunneling between both symmetric and asymmetric minima.
It concludes with an introduction to the extension of these methods to the
discussion of a quantum field theory. A more complete discussion of this issue
will be given in the second paper in this series. This paper will show how to
use the method of adaptive perturbation theory to non-perturbatively extract
the structure of mass, wavefunction and coupling constant renormalization.
