Dense sets

Each time step thus involves a calculation of the effect of the Hamilton operator acting on the wave function. In fully quantum methods the wave function is often represented on a grid of points, these being the equivalent of basis functions for an electronic wave function. The effect of the potential energy operator is easy to evaluate, as it just involves a multiplication of the potential at each point with the value of the wave function. The kinetic energy operator, however, involves the derivative of the wave function, and a direct evaluation would require a very dense set of grid points for an accurate representation. 

Since this is true for every n, we must have ( (O >/)—0. But as such / are known to form a dense set in 2, we obtain =0 as we wished to show. 

The discrete level is coupled to the dense set of vibrational levels, and the vibrational levels are coupled to the continuum, but the discrete level is not directly coupled to the translational continuum (see Fig. 25). 

Discrete State Jra in the Dense Set of Exact States Corresponding to the States... 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

Figure 1.13 The ground (lower solid line) and excited (dashed line) potential energy curves of the molecular ion H2+.The upper potential curve represents the ground electronic potential curve shifted by the energy ha> of one photon of the electromagnetic radiation. The ground vibrational wavefunction in the ground electronic state is coupled to the continuum of scattering states of the excited electronic potential depicted here by dense set of energy levels.

FIGURE 6.15 Compound state associated with the conversion from zero-order states to the sparse set i//E (associated with a finite number of discrete energy levels) and cf>e and the dense set (consistent with an infinite or effectively infinite number of degrees of freedom). cf>0 is the ground state. 

Dense set 157 Deploye, see Split Derivation 83 Derived group 73 Descent data 131 Diagonalizable group scheme 14 Differential field 77 Differential operator 99 Differentials of an algebra 84 Dimension of an algebraic G 88 Direct limit 151... 

In an effort to understand the intramolecular dynamics in unimolecular dissociation. Remade and Levine [87] used an effective Hamiltonian approach that can account for different time scales associated with unimolecular reaction. In doing so, they assumed that a dense set of energy levels lies above the dissociation barrier and that the barrier is sufficiently high that the number of states from which dissociation occurs is small compared to the number of bound states. 

In [STW] it is shown that the assumption that Q has no points of accumulation can be dropped from the hypotheses of Theorem 4.4 with no change in the conclusions. One can also replace the assertion that almost all initial data, in the sense of Lebesgue measure, belong to orbits converging to a rest point in O by the assertion that this holds for an open and dense set of initial data. 

All of the open problems for the standard gradostat system of Chapter 6 are open problems for the unstirred chemostat model discussed in Chapter 10. It can be shown [HSW] that the dynamics of the unstirred chemostat system mirror those of the gradostat in the sense that there is an order interval, bounded by two (possibly identical) positive rest points, that attracts all solutions. Furthermore, an open and dense set of initial data generates solutions that converge to a stable rest point. The question of the uniqueness of the interior rest point is a major open problem. Another is how to handle the case where the diffusion coefficients of the competitors and nutrient are distinct. Although there must still be conservation of total nutrient, it is no longer a pointwise conservation relation and the reduction to two equations is not clear. Even if accomplished, it may be difficult to exploit. If one is forced to analyze the full... 

Normally, a narrow resonance is coupled with the dense set of background resonance states. Because the background states look alike, the coupling between the narrow state and the background is not specific and the variation of a parameter, for example J, does not appreciably change... 

If m > 2d, a set of embedding maps F from IR into IR"1 consists of an open and dense set. Here a condition that a set of the maps is open and dense (sometimes the properties of an open and dense set is referred to as generic) results in that, given an embedding map F, F will persist as an embedding under any arbitrarily small... 

For sufficiently smooth functions

boundary conditions, the energy functional H(formal operator H to find the self-adjoint operator in L2( 2). As a result, the Hamiltonian H may be defined for the set Du, the domain of definition of H, being a dense set in 2.2(D). Different boundary conditions generate different self-adjoint operators H with different domains. 

It would appear that this procedure can usually be carried out with confidence for a really dense set k), which one would have in a large molecule. Therefore, a broad absorption might be taken as a signature of such a large system. 

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