Eigenfunctions mapping operators

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

The scattering wave function Hpj = Etp can be mapped onto the interior space of the billiard by the projection operator ipB = PbV where Pb = J2n n)(n Then this truncated scattering wave function can be expanded in the eigenfunctions of the closed billiard ipn(x,y) (A.F. Sadreev et.al., 2003)... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

We now turn to the effective operator definitions generated by the non-norm-preserving mappings K, L). As mentioned in Section II.B, expression (2.16), in which is replaced by ), applies if the a )o are used, while expression (2.14), with (/) ) replaced by ), applies if the are incorporated into new model eigenfunctions. We first discuss the effective operator expressions that may be obtained from (2.16). 

Mappings (K, L) conserve the norms of the right eigenfunctions. In particular, they relate by (A.2) the unity normalized eigenvectors la )o and ). This, however, does not imply that they conserve the norm of an arbitrary vector because angles between basis vectors differ in these two sets of eigenfunctions. Consider an arbitrary vector -y )q of and its true counterpart ly) — )o- Left-operating on the former with the first... 

But he was not at all certain that the principles considered so far in atomic theory could, in fact, be used for the realization of such a program. This was because the characteristic interaction of the chemical forces deviated completely from other familiar forces These forces seemed to "awake" after a previous "activation," and they suddenly vanished after the "exhaustion" of the available "valences." By making use of elementary symmetry considerations, it was known that the mode of operation of the homopolar valence forces could be mapped onto the symmetry properties of the Schrodinger eigenfunction of the atoms of the periodic system and could be interpreted as quantum mechanical resonance effects. This interpretation was formally equivalent to its chemical model, that is, it produced the same valence numbers and... 

Q is the normal-ordered wave operator, mapping the eigenfunctions of the effective Hamiltonian onto the exact ones, = 1, It satisfies intermediate normalization,... 

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