Annihilation condition conditions satisfied

The structure of the parametric UA for the 4-RDM satisfies the fourth-order fermion relation (the expectation value of the commutator of four annihilator and four creator operators [26]) for any value of the parameter which is a basic and necessary A-representability condition. Also, the 4-RDM constructed in this way is symmetric for any value of On the other hand, the other A-representability conditions will be affected by this value. Hence it seems reasonable to optimize this parameter in such a way that at least one of these conditions is satisfied. Alcoba s working hypothesis [48] was the determination of the parameter value by imposing the trace condition to the 4-RDM. In order to test this working hypothesis, he constructed the 4-RDM for two states of the BeHa molecule in its linear form Dqo/,. The calculations were carried out with a minimal basis set formed by 14 Hartree-Fock spin orbitals belonging to three different symmetries. Thus orbitals 1, 2, and 3 are cr orbitals 4 and 5 are cr and orbitals 6 and 7 are degenerate % orbitals. The two states considered are the ground state, where... 

Once concentrated in the center, neutralinos annihilate copiously. The annihilation rate is maximal when all captured neutralinos annihilate (a condition called equilibrium between capture and annihilation). Whether this condition is satisfied depends on the relative strength of the annihilation and scattering cross sections, and ultimately on the parameters of the particle and halo models. (See Jungman, Kamionkowski Griest (1996) and references therein for complete formulas.)... 

Let us further assume that the reference function satisfies the annihilation condition (1.65), i.e., that... 

In order to proceed, we observe that the two functions and obtained in this way are usually not orthogonal and further that the annihilation condition (2.38) is seldom exactly satisfied. Starting out from the approximate eigenoperator D = Bd, we will now introduce a refined normalized wave function for the final state through the relation... 

Combining (5) and (9) immediately yields (6) and (7). However, Herman and Freed have shown that the annihilation condition (8) is, in general, not satisfied for the excitation energy problem when ) is of the same symmetry as 0>. In fact, the equation C>J 0) = A>, which is usually taken to define Ol, does not hold for Oj s that are general solutions of (6). These conclusions result from the realization that the set of operators ( 0><0, A >, A"> are eigenstates of H) give only zero matrix elements when inserted for <7, or Oj in <0 [O,[//, C) ]] 0> and <0 [O, Therefore, the most general Oj that satisfies... 

Our discussion may readily be extended from 2-positivity to p-positivity. The class of Hamiltonians in Eq. (70) may be expanded by permitting the G, operators to be sums of products of p creation and/or annihilation operators for p > 2. If the p-RDM satisfies the p-positivity conditions, then expectation values of this expanded class of Hamiltonians with respect to the p-RDM will be nonnegative, and a variational RDM method for this class will yield exact energies. Geometrically, the convex set of 2-RDMs from p-positivity conditions for p > 2 is contained within the convex set of 2-RDMs from 2-positivity conditions. In general, the p-positivity conditions imply the (7-positivity conditions, where q < p. As a function of p, experience implies that, for Hamiltonians with two-body interactions, the positivity conditions converge rapidly to a computationally sufficient set of representability conditions [17]. 

Let us make clear now the correspondence between our treatment here and Erdahl s 1978 treatment [4, Sec. 8]. Erdahl works in general Fock space and his operators conserve only the parity of the number of nuclei. He exhibits two families of operators that are polynomials in the annihilation and creation operators containing a three-body and a one-body term. Generic instances of these operators are denoted y and w. The coefficients are real, and Erdahl stresses that this is essential for his treatment. The one-body term is otherwise unrestricted, but the three-body term must satisfy conditions to guarantee that y+y or H +w does not contain a six-body term. For the first family the conditions amount to the three-body term being even under taking the adjoint, and for... 

Here ak and a, are creation and annihilation operators, and hk(ri) are mode functions that satisfy (a) the normalization condition hkhk — h kh k = % (a prime indicates a derivative with respect to conformal time), and (b) the mode... 

In (1) a ic.b Kjaic and b f are boson creation and annihilation operators for the a and b Hartree-Fock particle states with momentum hn and kinetic energy K-h k /2m. pa and pb denote the densities of the component holon gases while pa and pb denote their respective chemical potentials. In equilibrium, pa=Pb-Pt where u is determined by the condition that the statistical average of A Eic (a icaic b Kbic) be equal to p=pa+pb=N/A, the total number of holons per unit area (N , A ). V is taken to satisfy V<pairing interaction. Finally, V is restricted to operate between holons with k[Pg.45]

Structure of the constituents the previous condition of saturation is mostly satisfied by producing or annihilating entropy-relevant molecular conformations and configurations Qcy(< )-... 

Until now we neglected possible interactions between the propagating waves (except for their annihilation after collisions). This is justified if the time interval between any two subsequent waves is much larger than the recovery time of the individual elements of the excitable medium. Since the rotation period of spiral waves goes to infinity in the limit Go 0 (i.e. for weakly excitable systems) and thus the spirals become very sparse there, this condition can always be satisfied close enough to the existence boundary dR of spiral waves in the parameter space defined in [4]. The kinematical approach which was formulated above allows us to find the rotation frequency of free sparse spirals and to investigate relaxation to the regime of steady rotation. 

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