A Convergent Continuum Strong Coupling Expansion For Quantum Mechanics & Quantum Field Theory

We generalize the notion of an asymptotic weak coupling expansion about an exactly solvable model in quantum mechanics and quantum field theory to an all positive value coupling convergent expansion. This is done by rescaling the variables available in the theory by free parameters, then adding and subtracting the exactly solvable model. The rest (initial rescaled theory by free parameters + the subtracted exactly solvable model) is expanded about the added exactly solvable model. Evaluating finite orders of this expansion at its extremum points with respect to the free parameter(s) gives a sequence that converges to the result of the previous asymptotic expansion, with a good convergence rate, at relative strong coupling. We solve for the eigenenergies of the anharmonic, pure anharmonic and double well potential problems using this method by expanding about the symmetrical point of these potentials. Accurate results for the eigenenergies can be obtained for all positive values of the coupling for the anharmonic and pure anharmonic oscillators and at strong coupling for the double well potential. To provide confirmation for the convergent formalism developed for $\phi^4$ theory and QED we improve the electron g-factor calculation at the one loop level using the convergent formalism. Applications of this method are not limited to quantum mechanics or quantum field theory, for example it can also have applications in the context of differential equations.