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## Bi-orbital states in hyperbolic traps

Russian Journal of Mathematical Physics. 2018. Vol. 25. No. 4. P. 500-508.

We consider a charge in a general electromagnetic trap near a hyperbolic stationary point. The two-dimensional trap Hamiltonian is a sum of hyperbolic harmonic part and higher order anharmonic corrections. We suppose that two frequencies of the harmonic part are under a resonance 1 : (-1). In this case, anharmonic terms define the dynamics and an effective Hamiltonian on the space of motion constants of the ideal harmonic operator. We show that if the anharmonic part is symmetric, then the effective Hamiltonian has unstable equilibriums and separatrix, witch define distinct classically allowed regions in the space of mouton constants of the ideal trap. The corresponding stationary  tates of the trapped charge can form a bi-orbital states, that is a state localized on two distinct classical trajectories. We obtain semiclassical asymptotics of the energy splitting corresponding to the charge tunneling between these two trajectories in the phase space and express it in terms of complex instantons.