Numerical Methods for Partial Differential Equations Seminar

Traditionally, the divergent perturbation series of a quartic anharmonic oscillator has been used as an example of a factorially divergent, alternating perturbation series describing an energy level of a quantum mechanical system. The Bender-Wu formulas have been used in order to connect the "stable" domain of positive coupling, where the resonance energy eigenvalues are real rather than complex, with the "unstable" domain of negative coupling, where the resonance energy eigenvalues are complex; the latter describe unstable states whose resonance energy has a nonvanishing imaginary part. The imaginary part of the resonance energy describes the decay width. In recent years, the concept of a perturbation series has been generalized to include series with perturbative (power) terms and nonpertubative exponential factors of the form exp(-A/g), where A is the so-called instanton action and g is the coupling parameter. These generalized perturbation series ("transseries") are able to describe, analytically, the manifestly complex resonance energies in the unstable domain. Such generalized perturbation series are even able to describe anharmonic oscillator energies in cases where the perturbation series vanishes to all orders, but the ground-state energy is manifestly nonzero.