Mapping operators Theorem

The derivations of Theorems V and VI for the preservation of [ A, B] consist of (1) the replacement by P of the product KK that is present between A and B in [A, B], (2) the commutation of A with P, and (3) the incorporation of P into K and K. Step (1) uses Eq. (4.6), which is valid only for norm-preserving mappings. With other definitions, however, the products analogous to KK may be replaced by P if (criterion 1) they satisfy the fundamental relation (2.13). Step (2) clearly applies to any effective operator definitions. So does step (3) since P can be combined with any mapping operators using Eqs. (2.3) and (2.6). Hence, all effective operator definitions that fulfill criterion 1 conserve [A, B] if [A, B] =0. When these definitions, like A, produce the associated effec-... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

For the GS, the HK theorems" guarantee thatEq. (10) of different exact theories all deliver the same GS density in spite of distinct mathematical structures of Oeff (r [p]) within different theoretical approaches " (i.e. local vs. nonlocal operators). The reason is simple the density is one-to-one mapped on to the GS wavefimction, regardless of how the exact wavefimction and the exact density are calculated. 

The operator SR maps A on A, whereas the operator RS maps B on B. The "mirror theorem" says that the two operators SR and RS have the same non-vanishing eigenvalues and the same type of classical canonical structure. The matrix representations are defined by the relations... 

If g and h are unipotent and commute, their product is unipotent, since gh — 1 is the sum of commuting nilpotents g(h — 1) and g — 1 and hence is nilpotent. In particular the tensor product of unipotent operators is unipotent. The direct sum is so also, and clearly a unipotent map induces unipotent actions on invariant subspaces, quotients, and duals. As in (7.1) this gives us a persistence theorem ... 

Definition 71 On the basis of Theorem 70 we can determine the operator A which maps an element y G T into the proper element x G X, according to formula (D.2) ... 

The key step in deriving (4.7) is the commutation of P with H. Clearly, a similar reasoning applies when replacing H with any operator that commutes with H because such an operator also commutes with P. Therefore, this leads to Theorem V as follows state-independent effective operators produced by norm-preserving mappings conserve the commutation relations between H and an arbitrary operator B and between B and any operator that commutes with H. Given particular choices of P,... 

Hence, the commutation relation between A and B is conserved iff the right hand sides of Eqs. (4.8) and (4.9) are equal to each other, thereby leading to Theorem VII as follows the commutation relation between two operators A and B is preserved upon transformation to state-independent effective operators obtained with norm-preserving mappings iff A and B satisfy... 

Another important application of Theorem V is that (Corollary V.2) the dipole length and dipole velocity transition moments are equivalent when computed with state-independent effective operators obtained with norm-preserving mappings. According to definition A (see Table I), these computations evaluate o( p /8)o, and (a r )3)oWith... 

Section III is proven to classify this group of mappings, and then the effective operator definitions for the subset of Theorem II.a are derived. 

So the FC integral is added to the very few physical systems [18] which are realizations of this particular algebra. Using the Taylor theorem for shift operators due to Sack [19], and the Cauchy relation mentioned above, we can apply this very general idea to the specific case of the harmonic oscillator to obtain the closed formula (5). Recurrence relations can also be obtained by noticing that O is in reality a superoperator which maps normal ladder operators by the canonical transformation ... 

In conclusion, we emphasise the following points (i) we have re-derived a previously obtained operator array formulation, which in its complex symmetric form permits a viable map of gravitational interactions within a combined quantum-classical structure (ii) the choice of representation allows the implementation of a global superposition principle valid both in the classical as well as the quantum domain (iii) the scope of the presentation has focused on obtaining well-known results of Einstein s theory of general relativity particularly in connection with the correct determination of the perihelion motion of the planet Mercury (iv) finally, we have obtained a surprising relation with Godel s celebrated incompleteness theorem. 

The previous discussion subtly shifted between molecular similarity and molecular properties. It is important to elucidate the relationship between the two. If each of the molecular properties can be treated as a separate dimension in a Euclidean property space, and dissimilarity can be equated with distance between property vectors, similarity/diversity problems can be solved using analytical geometry. A set of vectors (chemical structures) in property space can be converted to a matrix of pairwise dissimilarities simply by applying the Pythagorean theorem. This operation is like measuring the distances between all pairs of cities from their coordinates on a map. 

Consider the map ip P Cdescending closure operator, whose image is C[Pg.242]

Finally, we would like to elaborate the proposed protocol of the high-friction map, eqn (13.17). Its construction is based purely on the thermodynamic consideration, eqn (13.15), validated by the central limit theorem. Therefore it may offer a general rule to obtain the Smoluchowski limit to any phase-space dynamics under study. The protocol proposed in this chapter is based on the fact that the map is universal at formal level and is therefore obtainable with thermodynamic consideration. It means the Smoluchowski dynamics can be taken care of by the related Fokker-Planck equation, upon the universal map being carried out. It is worth pointing out that the resultant diffusion operator in eqn (13.18) clearly originates from only the Hamiltonian part of the... 

It follows from a basic theorem on linear operators that, since H is selfadjoint, the evolution operator U t) is unitary (Jordan [93]). The evolution operator maps the state vector for time zero (that is, ip)) onto the corresponding vector for time t. 

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