|XXXII Workshop on Geometric Methods in Physics||30.06-6.07.2013|
Participants of Workshop
Participants of School
Aspects of the quantum separation of variables for the Toda chain
The quantum separation of variables method consists in mapping the original Hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler "separated" form. In order to realise such a program, one should first construct the so-called SoV map explicitly and then show that it is unitary. Second, one should "translate" the local operators on the original Hilbert onto operators on the "separated" space, ie solve the so-called quantum inverse scattering problem. In the present talk, we shall discuss several progress that we have made in respect to the two point points mentioned previously. Namely, in the case of the quantum Toda chain,
we shall describe a technique which allows one to prove the unitarity of the SoV map. Then we shall discuss the progress we have made in the resolution of the quantum inverse scattering problem for this model. Our approach to the proof of unitarity solely builds on objects and relations naturally arising in the framework of the so-called quantum inverse scattering method. Hence, with minor modifications, it appears readily transposable to other quantum integrable models solvable by the quantum separation of variables method. As such, it provides an important alternative, in what concerns studying unitarity of the SoV map, in respect to results obtained previously within the group theoretical interpretation of the model, which is absent for more complex quantum integrable models.
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