Outer projections

Regardless of the nature of the problem, approximate upper bounds, follow from the variation principle, i. e. from outer projections. So, introducing the projection operator (41) we define ... 

The precise connection with finite dimensional matrix formulas obtains simply from Lowdin s inner and outer projections [21, 22], see more below, or equivalently from the corresponding Hylleraas-Lippmann-Schwinger-type variational principles [24, 25]. For instance, if we restrict our operator representations to an n-dimensional linear manifold (orthonormal for simplicity) defined by... 

If the sum over states is incomplete Eq. (4) is an outer projection of the resolvent R onto the subspace spanned by the states k). 

It should be observed that one can conveniently describe the calculation of approximate eigenfunctions in terms of truncated basis sets by using the concept of outer projections (7) of the operators involved. If F = F(t and d> = < > for k = 1,2,., m are the dual bases chosen, the operator... 

Introducing the outer projection T of the operator T with respect to the projector Q through the relation... 

It is interesting to observe that the outer projection T = OTO of the transformed operator T = UTU-1 may always be written as a similarity transformation with respect to the operator U of an outer projection T = QTQ of the original operator T, so that... 

It is clear that the concept of outer projections permits us to formulate the approximate treatment of the eigenfunctions and eigenvalues based on the bivariational principle in an exact manner. 

In the formulation of the approximate theory based on outer projections of the type given by Eq. (1.43), it is interesting to observe that the fundamental projector Q defined by Eq. (1.40) in the bivariational treatment of the original Hamiltonian takes the special form... 

This makes the physical body a kind of vessel which "contains" higher dimensional constructs. In three-dimensional terms it is logically absurd for a vessel to be smaller than its contents, yet this is not an inevitable conclusion if other dimensions are factored into the equation. Another way to conceptualize this is to think of space/time as the "outer" projection of an inner infinity. Imagine what it would be like to be a two-dimensional entity living on one of the six faces of a cube. 

Let us now consider the connection with perturbation theory in somewhat greater detail. For the resolvent of the outer projection PHqP, one obtains the conventional spectral resolution (a - PHqP) = Sfc I Pk 1 1where the first sum goes over the... 

In the limited basis, the outer projection Green s function... 

Fig. 1. Spectrum of the 2-0 GO bandhead in WL 16. The smooth line is a Keploian disk model with the indicated inner and outer projected velocities and a r " radial intensity gradient.

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