The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann surface provides complete information on the topology of complex paths, which allows us to enumerate all the possible candidates contributing to the semiclassical sum formula for tunnelling splittings. This naturally leads to action relations among classically disjoined regions, revealing entirely non-local nature in the quantization condition. The importance of the proper treatment of Stokes phenomena is also discussed in Hamiltonians in the normal form.
The Pais-Uhlenbeck(PU) oscillator is the simplest model with higher time derivatives. Its properties were studied for a long time. In this paper, we extend the 4th order free PU oscillator to a more non-trivial case, dubbed the 4th order time dependent PU oscillator, which has time dependent frequencies. We show that this model cannot be decomposed into two harmonic oscillators in contrast to the original PU oscillator. An interaction is added by the coordinate transformation of Smilga.
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