Complementary subspace

The perturbed space p falls naturally into two complementary subspaces q and q, such that... 

Taking a complementary subspace S, we decompose kP as S 0 S. Then we have... 

Set up a global basis xi)X2,— of basis functions and divide the global space into three subspaces (i) functions of a core space (ii) iVvai functions of a valence sp>ace and Na,m functions of a complementary subspace. 

The physical idea behind the PT is the fragmentation of the Hilbert space H of the problem under study into two subspaces, usually termed Q and P, for which solutions of the corresponding SE are obtainable, while the solution of the SE for the full H is computationally formidable. By the partitioning of 7i into two subspaces, one is able to write, in each subspace independently, the SE for the projection of the unknown solution of the SE in H, while the dynamical effect of the complementary subspace is fully incorporated. In this way, one ultimately can construct the solution of the SE in H after solving for its projections in Q and P independently. 

In Euclidean space we sometimes talk of complementary subspaces. For example, the z-axis is the complementary subspace to the xy-plane inside... 

We define complementary subspaces of complex scalar product spaces. 

Figure 3.3. Complementary subspaces, a.) A literal picture of a real example, b.) A schematic picture of the general situation.

When we consider a subrepresentation pyy of a representation p it is often useful to consider the leftovers, that is, the part of p that is not captured by Pw- If the original representation p is unitary, then there is a particularly nice way to package those leftovers we can put the complex scalar structure to work. Recall the notion (Detiitition 3.6) of the complementary subspace VP- -. 

These are nothing but the conditions of orthogonality of the subspace of interest lm/ (lm/ - image P - stands here for the set of vectors of a linear space which are obtained by action of the linear operator P upon all vectors of the linear vector space) and its complementary subspace IrriQ. 

The various contributions to the effective Hamiltonian operator on the right-hand side of equation (7.43) can be represented diagramatically in a way which is very helpful in practice. These diagrams are shown in figure 7.1. For each diagram, the position at the bottom refers to the chosen zeroth-order manifold 0)° and the position at the top refers to the states in the complementary subspace. Each line joining these two positions stands for a connecting matrix element. 

These equations show that if we ignore AH e) and diagonalize Hin the subspace 5 we are in effect diagonalizing the projected Hamiltonian P HP thus the complementary subspace 5 is treated implicitly through the energy-dependent operator AH s). A useful way of thinking about the relationship between Eqs. (2-2) and (2-3) follows from the observation that ifis completely determined by the requirement that it should reproduce the exact eigenvalues E ,... 

The set of functions, (3-1) which has some definite fixed value of Nn=Nj thus defines a subspace, say of the n-electron function space, and if we define the complementary subspace (see 2-1 b) through a set of orthogonal functions A, such that. 

In the review presented, the authors have considered the modern state of the theory of diatomic interactions in two comparative ways, one from the energetic and force-theoretical viewpoints, and the other from the mathematical and interpretative viewpoints. Undoubtedly, these points of view enhance each other. Also, these viewpoints have been studied in the framework of two complementary subspaces of the whole phase space of a given diatomic molecule, the position and the momentum subspaces. Probably it is of interest to study the theory of diatomic interactions based on the whole phase space. Of course, such an approach will be also useful in the study of the dynamical aspects of diatomic interactions. 

Resonant enhancements of scattering cross-sections in multichannel collision physics are often described in terms of the Feshbach theory of closed-channel resonance states [57], Feshbach s general formalism involves projecting the stationary Schrddinger equation onto complementary subspaces associated with the open and closed scattering channels. This theory has been applied in the context of the nearthreshold collision physics of ultracold gases consisting of alkali-metal atoms in a variety of different approaches (e.g.. Refs. [9,30,58]). 

The partitioning technique for solving secular equations is discussed. It is shown that the original secular equation may be transformed to a contracted secular equation referring to the specific subspace under consideration. The technique may be applied to hetero-atoms in a molecule, to central atoms in a crystal field, to chemical bonds in a molecular enviromnent, and to discuss the question of dressed and undressed particles. If the system under consideration is not complete, the addition of a complementary subspace leads to a new term in the Hamiltonian corresponding to a dressing of the system involved. 

The division into two sets of orthonormal spin-orbitals (here designated occ and unocc ) is of course quite arbitrary. We say that T = ips S occ) and V = ipu U unocc) are complementary subspaces, of the space spanned by a complete set of spin-orbitals, and that p and p project onto y and Y respectively. It follows that p and p have the properties ... 

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