Post #2854384

#paperOfTheDay "Convergent Strong-Coupling Expansions from divergent weak-coupling perturbation theory" from 1995 studies the anharmonic oscillator, one of the most popular models for perturbation theory in theoretical #physics . Recall that with a potential V(x)= w/2*x^2, an object performs harmonic oscillations with frequency w, i.e. the trajectory (of a classical system) is a sine, or the #quantum oscillations are a complex exponential function. One now makes this potential "anharmonic" by adding an additional term +g/4*x^4. The potential is still bounded from below, and the quantum system does some sort of oscillations, but no longer sine/cosine ones. One is interested in the energies, in particular the ground state energy, as a function of the parameter g. An obvious idea is to use perturbation theory, i.e. construct a power series expansion around the point g=0. It came as a big surprise that this power series is divergent for all non-zero g. This can be understood by considering small negative g: The potential would then be unbounded, and the quantum system instable, hence the ground state energy can not be finite (or real) in that case, hence the power series can not diverge in any radius around g. The present paper constructs a different type of perturbation series, essentially using the frequency w instead of the coupling g as an expansion parameter. Eventually, this turns out to be equivalent to a perturbation series in the parameter 1/g, hence it is an expansion in strong coupling. This new expansion is convergent, and it can be generated from the divergent small-coupling expansion simply by reordering terms. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.75.2787