Full Hilbert space

Next, the full-Hilbert space is broken up into two parts—a finite part, designated as the P space, with dimension M, and the complementai y part, the Q space (which is allowed to he of an infinite dimension). The breakup is done according to the following criteria [8-10] ... 

The question to be asked is Under what conditions (if at all) do the components of X fulfill Eq. (B.8) In [34] it is proved that this relation holds for any full Hilbert space. Here, we shall show that this relation holds also for the P sub-Hilbert space of dimension M, as defined by Eq. (10). To show that we employ, again, the Feshbach projection operator foraialism [79] [see Eqs. (11)]. 

In the two-electronic-state Born-Huang expansion, the full-Hilbert space of adiabatic electronic states is approximated by the lowest two states and furnishes for the corresponding electronic wave functions the approximate closure relation... 

Notice that, the physical variables are described by non-tilde operators, while the tilde operators, up to now, play a role of ancillary variables only. However, as we will see in Section 3, the full Hilbert space has the original set of non-tilde operators associated to dynamical observables, whilst the tilde operators are connected with the generators of symmetries. 

The formulas of Appendix A are easily transformed to time-dependent forms via the Fourier-Laplace transform. For the purpose of upcoming extensions to the nonself adjoint case, we introduce the self-adjoint Hamiltonian H = FT = Htt. Furthermore the function 0 considered above is said to belong to the domain of the operator H, i.e.,

self-adjoint problems D(W) = D(H) and for bounded operators, the range R(H) and the domain D(H) are identical with the full Hilbert space however, in general applications they will differ as we will see later. 

This is not the full Hilbert space of the Hamiltonian which would also include eigenfunctions corresponding to the continuous eigenvalues. 

The renormalized theory of the effective Hamiltonian implied by the restriction to some subspace S of the full Hilbert space also imposes a requirement for renonnalisation of expectation values of other operators (Freed ). Suppose that we have some operator B and we require its expectation value in a state 0 of the full Schrddinger Eq. (2-2) in complete analogy with the effective Hamiltonian theory described above we define an effective operator B by the requirement that its expectation value in a state A ) in the subspace should equal the exact expectation value (c.f. Eq. (2-4)),... 

Left-multiplying the full Hilbert space Schrodinger equation,... 

These eigenfunctions define a basis in the N-electron Hilbert space. To account for the effects of the rest interaction V perturbatively, the full Hilbert space H of Hg is separated into two parts the model space M and its complement, TL M, which includes all the remaining dimensions. To these subspaces, we assign the projection operators... 

In the full Hilbert space the many-body Hamiltonian H is of the form... 

Similar to what is done in QDPT, a model space So of dimension N is defined as a subspace of the full Hilbert space S of dimension M. Remember that QDPT is used to determine accurate wave functions and energies starting from a limited description of the system based on the model space. However, in the present case, the accurate energies and wave functions are already known and the action goes in the opposite direction the lengthy wave function of length M is mapped on the smaller subspace So ensuring a minimum loss of the information contained in the full solution. 

Select the N eigenfunctions of the full Hilbert space (e.g. obtained in an ab initio calculation) with the largest projection onto the model space. (Bi-)orthonormalize the projections of these vectors and take the total energy of one of the roots as zero of energy. 

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