Restricted unbounded transformations

A. Complex Scaling as an Example of a Restricted Unbounded Transformation... 

It should finally be observed that an unbounded transformation U which is restricted to satisfy the relations given by Eq. (1.57) or (2.51) must necessarily be complex. In the case of a real transformation, i.e., U = U, one has... 

In this section we will consider the method of complex scaling (2) as a typical example of an unbounded similarity transformation of the restricted type. It is here sufficient to consider a single one-dimensional particle with the real coordinate x( — oo < x < +qo), since the IV-particle operator U in a 3N-dimensional system may then be built up by using the product constructions given by Eqs. (2.23) and (2.25). 

In conclusion, the method of complex scaling as an unbounded similarity transformation of the restricted type is briefly discussed, and some numerical applications containing complex eigenvalues - which may be related to resonance states... 

The Laplace transform for fo < 0 is unbounded if 5- < 6 therefore, the real part of 5- must be restricted to be larger than -b for the integral to be finite. This condition is satisfied for all problems we consider in this book. 

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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