Adaptive Perturbation Theory I: Quantum Mechanics

TL;DR

Adaptive perturbation theory offers a novel approach to compute quantum eigenvalues and eigenstates by exactly capturing non-perturbative effects, demonstrated on anharmonic oscillators and tunneling, with extensions to quantum field theory.

Contribution

It introduces adaptive perturbation theory, a new method that decomposes Hamiltonians to extract non-perturbative behavior, advancing quantum mechanical and field theory calculations.

Findings

Successfully applied to anharmonic oscillator

Analyzed tunneling between minima

Outlined extension to quantum field theory

Abstract

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.