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Feshbach resonance: A one dimensional example
2. Advances in Atomic Physics, edited by C. Cohen-Tannoudji and D. Guéry-Odelin (World Scientific, Singapore, 2011), part 6.
8.The Zeeman states are B independent in our toy-model, but in general this is not the case. This can be seen by repeating the calculations after adding, say, a splitting term to Eqs. (2) and (3) of the form . It is not difficult to find that now the Zeeman states depend on B. (The potentials depend on B as well.)
9.The dissociation threshold of a given potential is the energy above which the two atoms are not bounded by it.
11. Theoretical Nuclear Physics: Nuclear Reactions, edited by H. Feshbach (John Wiley & Sons, New York, 1992).
12.This picture is valid when the coupling among the channels is weak and may be viewed as a perturbation of the scenario with the channels uncoupled. However, multichannel physics is in general richer than what perturbation theory provides and the picture given above is not necessarily valid in situations when the coupling is strong. The exact results we presented in Secs. III and IV do not rely on perturbation theory.
13.A general procedure to reduce a multichannel problem to a single-channel-equivalent problem with an effective potential (the so called Optical Potential) follows from Feshbach's analysis (see Ref. 10). This procedure also covers the case with the two channels open; in our example this boils down to the substitution in Eq. (18). The effective potential in Eq. (20) becomes complex with an imaginary part that reflects the inelastic transition to the v-channel that is now allowed, with the absorption of an energy . Indeed, we find that . Being non-Hermitian, this potential leads by itself to probability non-conservation. The physical interpretation is clear and reflects the fact that part of the probability at the entrance channel leaks through the other channel that is also open. It should be mentioned that the correspondence of a two-channel problem and a single-channel-equivalent one is not always free from ambiguities (see Refs. 14 and 15). Several examples on how some of such difficulties have been dealt with can be found (see Refs. 6, 16, and 17).
18.The correction can only contain even powers of w. There is one power of w for each channel swap and the number of swaps cannot be odd because the process has to end up in the original channel.
19.At we find , which coincides with the bare v-channel bound-state wavefunction, except for normalization. In general one expects a further contribution from the bare v-channel continuum states too, which turns out to vanish in our example.
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