Dilatation analytic potentials

To summarize, we have considered a quantum mechanical N-body system with dilation analytic potentials, Ey, and its dependence on the scaling parameter i] = t] (for some 0 < < 0, depending on V). To be more detailed, we need to restrict Hilbert space to a dense subspace [54], the so-called Nelson class N, which provides the domain over which the unbounded complex scaling is well-defined. Closing the subset < - D T) in ft, see [9] and references therein for a more detailed expose, one obtains the scaled version of the original partial differential equation... 

B. Simon, Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. Math. 97 (2) (1973) 247. 

In order to appreciate the fine points in this analysis, we therefore return to the domain issues, i.e. how to define the operator and the basis functions so that the scaling operation above becomes meaningful. Following Balslev and Combes [3], we introduce the N-body (molecular) Hamiltonian as H = T + V, where T is the kinetic energy operator and V is the (dilatation analytic) interaction potential (expressed as sum of two-body potentials Vy bounded relative Ty = Ay, where the indices i and j refers to particles i and j respectively). As a first crucial point we realize that the complex scaling transformation is unbounded, which necessitates a restriction of the domain of H note that H is normally bounded from below. Hence we need to specify the domain of H as... 

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