I worked for many years on non-perturbative methods in quantum mechanics and quantum field theory. Perturbation theory is only a valid option when the coupling constant is small, which leaves a large range of problems where it is not applicable. One particular alternative, variational perturbation theory, or the linear delta expansion, has been successful in a wide range of applications. It is a variant of perturbation theory in which the split between the bare Hamiltonian H0 and the interaction H1 is not fixed once and for all, but changes in each order of the expansion according to a well-defined principle, the principle of minimum sensitivity. The classic application of this method (A. Duncan and HFJ, Phys. Rev. D47 (1993) 2560) is to the quartic anharmonic oscillator, where it gives a rapidly convergent sequence of approximants, whereas the standard perturbative expansion is a divergent oscillating series.
A great deal of interest has been generated by a paper by Bender and Boettcher, which showed that a class of quantum-mechanical systems with a non-Hermitian, but PT-symmetric potential, possessed a real positive spectrum. The simplest case where this occurs is the potential V=x2+igx3. Originally, research concentrated on finding other examples of such potentials, but subsequently the focus moved to whether these theories could have a physical, probabilistic, interpretation on the lines of conventional quantum mechanics.
The problem arises because the natural Hilbert-space metric in such theories is not positive definite. However, in 2001 we (Bender, Brody and Jones) showed how another (dynamically determined) metric could be determined, implemented by an operator we called C, which was indeed positive definite, so that a physical interpretation could be regained. The operator C, or a closely-related operator Q, can be used to construct an equivalent, Hermitian Hamiltonian, h, with the same spectrum as the original H. In quant-ph/0411171 I carried out the construction of h in perturbation theory for the x2+igx3 theory, and exactly for another, soluble model.
In recent years there has been an explosion of interest in PT-symmetry, not in the context of quantum mechanics, but in the apparently unrelated area of classical optics. The connection arises because in a certain approximation, the paraxial approximation, the propagation of electromagnetic waves is governed by an analogue Schrödinger equation. In that context, a complex potential corresponds to a complex index of refraction, i.e. gain and/or loss. For a PT-symmetric potential the gain and loss are balanced in a particular way. That gives rise to a whole host of interesting optical phenomena, including unidirectional transparency, possibilities for optical switching, and control of lasing.