New raising and lowering operators for highly anharmonic coupled oscillators

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Abstract
New raising and lowering operators A+, A are proposed for vibrations in the highly anharmonic regime (roughly corresponding to the zone of overlapping nonlinear resonances and the transition to classical chaos). All matrix elements 〈ν+k‖0‖ν〉 of p and powers xλ of x=2Ce−aq between bound states ‖ν〉 of the Morse oscillator can be written exactly in terms of powers of the new operators, (A+)k and (A)k, and a supplementary operator α̂. In terms of α̂, A+, A, the momentum and coordinate operators take a form similar to that of the harmonic oscillator in terms of a+, a. It is shown that it is the operator α̂ that is crucial for representing the novel qualitative behavior of dynamical operators in the highly anharmonic regime. It is therefore suggested that the operators α̂(A+)k and α̂(A)k be used in place of the conventional (a+)k and (a)k in fits and dynamical models for systems of coupled anharmonic vibrations.