Unbounded similarity transformations

On the Change of Spectra Associated with Unbounded Similarity Transformations of a Many-Particle Hamiltonian and the Occurrence of Resonance States in the Method of Complex Scaling. Part I. General Theory... 

III. Unbounded Similarity Transformations and the Associated Change of Boundary Conditions... 

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). 

It is hardly necessary to emphasize that the present review is a fairly simple exercise in linear algebra and is intended to familiarized theoretical physicists and chemists working on the quantum theory of matter with the fundamental properties of the unbounded similarity transformations as applied to N-electron systems. Special attention has been given to the change of the spectra and how it is related to the domain of the transformation applied and to the fact that the eigenfunctions may be transformed not only within the L2 Hilbert space, but also out of and into this space (see Fig. 1). 

On the Change of Spectra Associated with Unbounded Similarity Transformations of a Many-Particle Hamiltonian... 

The theory of the change of spectra of a many-particle operator associated with an unbounded similarity transformation... 

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... 

For such a bounded operator, the spectral and general stability properties of the linear operator T stay essentially invariant under the transformation Eq. (1.2), and, for self-adjoint and normal operators, one has a series of well-known theorems. Before proceeding to the unbounded transformations, we will briefly review some additional properties of the bounded similarity transformations. 

We have seen in the previous section that the combination of variables (similarity transform) is applicable only to initial value problems, with unbounded independent variables. Nonetheless, we gave an example where such an approach could be applied under conditions of short contact time (which is the same as short penetration for the length variable). Thus, we pretended that a boundary was infinite in extent, when in fact it was physically finite. For large contact time (deep penetration) such an approach breaks down, and the solution becomes invalid. 

A similar result holds true for the case shown in Figure 13-4b. Applying the inverse Fourier transform to both sides of the last expression, we cirrive at the Kirchhoff integral formula for an unbounded domain ... 

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