Feshbach projection operators

To continue, we define the following two relevant Feshbach projection operators [79], namely. Pm, the projection operator for the P space... 

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)]. 

Let us now suppose the state s) to be represented by the approximate Hamiltonian Ho of Equation Al. If, for example, the state s) and another state i>) of the manifold (Al) between the transitions are isolated from the remaining states of Ho, the problem is relatively simple. However, if the states are not isolated, the calculation of the transition probability is not simple. To reduce the complexity and to make clear the states of the system to be included in the calculation of the matrix elements of the Green s function, we use the Feshbach projection operator formalism. In this formalism, a projection operator associated with a model function or a set of states, the so-called dose coupled states, is introduced... 

The formal theory of resonances due to Feshbach begins with the decomposition of the Hamiltonian in terms of a projection operator Q [8]. He defines Q as the projection onto the closed-channel space, just like the example of H discussed around Eqs. (4) and (5). Then, QBSs described well by the eigenfunctions Q4> of Eq. (5) with his Q may be called Feshbach resonances." A simplified picture would be that eigenstates Q are supported by some attractive effective potential approaching asymptotically the threshold energy of a closed channel. If this is the case, then the energies EQ of... 

Using the projection-operator formalism of Feshbach [ 115,116], an implicit variational solution for the coefficients cIJiS in can be incorporated into an equivalent partitioned equation for the channel orbital functions. This is a multichannel variant of the logic used to derive the correlation potential operator vc in orbital-functional theory. Define a projection operator Q such that... 

Molecular optical potentials for non-reactive processes may be rigorously defined by means of partitioning techniques (see e.g. Feshbach, 1962), which are based on the classification of scattering channels in two groups the first one includes states which are asymptotically selected or detected, and is characterized by a projection operator P the second one includes all other states (in practice those to which flux is lost) and is characterized by the projector Q. An optical potential operator VH may then be constructed as... 

The so-called ISR [98, 99] provides a powerful, general argument to explain nonexponential decay for fhe survival probability (although not necessarily for other quantities, see Section 4.3). Let us define, following Feshbach [80], projection operators P = 4>o>( Fol and Q = 1 - P. Then... 

By means of the projection operators thus defined, we can focus our attention on the subsystem of special interest and derive a Schrodinger equation for that part of the state vector that governs the properties of the molecular excitations (Feshbach, 1973). The equation for the total system is written as... 

To solve the problem, we first define a projection operator P which specifies the open target eigenstates of the collision. By open channels we mean the states that are accessible by the collision energy. P is an At-body operator and, unlike the projection operator in the Feshbach formalism [130], is unique. With as a target eigenstate P is simply... 

In SRS-PT, the Feshbach-Ldwdin Hamiltonian Heft, which is obtained by partitioning the exact Hamiltonian with the help of the projection operators P and Q, is best known. 

To simplify the calculation of fft), we make use of the Feshbach projection formalism [80, 81] by introducing the operators P and Q satisfying the relations... 

The coupled Schrodinger equations can be projected onto the fa fa subspace by Feshbach partitioning, giving an equation for the coefficient function Xd(q) in the component faxdiq) of the total wave function. The effective Hamiltonian in this equation is tn + Vd(q) + Vopt, which contains an optical potential that is nonlocal in the <7-space. This operator is defined by its kernel in the fa - fa subspace,... 

The Lippmann—Schwinger equations (6.73) are written formally in terms of a discrete notation i) for the complete set of target states, which includes the ionisation continuum. For a numerical solution it is necessary to have a finite set of coupled integral equations. We formulate the coupled-channels-optical equations that describe reactions in a channel subspace, called P space. This is projected from the chaimel space by an operator P that includes only a finite set of target states. The entrance channel 0ko) is included in P space. The method was first discussed by Feshbach (1962). Its application to the momentum-space formulation of electron—atom scattering was introduced by McCarthy and Stelbovics... 

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