Lost eigenvalues

It is clear that if F bejongs to the domain D(U), then the function F = t/ P also belongs to L2 and E = E is a persistent discrete eigenvalue. On the other hand, if P belongs to the complement C(U), the function F = U V is outside L2 and it cannot be an eigenfunction to H hence E becomes a lost eigenvalue, one that does not necessarily belong to the spectrum of H. 

If anyone of the functions = (v/jJ is situated outside the domain of the operator u, the transformed determinant Cua = UCa is no longer an element of L2, and this may be taken as an indiaction of a lost eigenvalue. However, in such a case the operator pu = upu 1 defined by (3.20) no longer exists, and the entire Hartree-Fock scheme breaks down. 

However, since the projector p does not necessarily exist, the invariance theorem t = tu could very well break down, corresponding to the case of "lost eigenvalues" in the general theory. 

This boundary condition will be automatically fulfilled for all eigenfunctions associated with discrete or continuous eigenvalues to the Hamiltonian, except for those eigenfunctions P which accidentally lack projections in the reference space and then correspond to "lost eigenvalues". Since the matrix C = > has em inverse, it may now... 

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Even if the conditions for persistent, lost, and new eigenvalues are completely clear for the exact eigenvalue problems to the operators T and Tt, it is considerably more difficult to translate them to the approximate eigenvalue problems associated with the application of the bi-variational principle for the operators T and Tt to truncated basis sets. In this connection, the relations (A. 1.40-1.49) may turn out to be useful in formulating the problem. Some of the computational aspects, particularly the choice of the dual basis sets, are further discussed in reference A. 

When nuclei in molecules are treated classically, the concept of molecular geometry emerges in a natural way. To be more precise, the equilibrium geometry is defined as the set of internal coordinates r for which the ground-state eigenvalue q(r) of the electronic Hamiltonian attains a local minimum. Different minima in q(r) correspond to equilibrium geometries of isomeric species with identical compositions. Needless to say, this naive picture is inevitably lost in a fully quanturn-mechanical treatment. 

Thus in this case the dynamics is at all times adiabatic in the sense that it mainly follows the dressed eigenstate whose eigenvalue is continuously connected to the one associated to the initial dressed state. This adiabatic transport results at the end of the pulse in an (almost) complete return in the initially populated state. It is important to point out that the dynamics is affected by the resonance in the sense that the excited bare state 2) is highly populated during the pulse if ft is of the same order as A or larger at the peak laser amplitude. For two-level systems, the nonadiabatic small corrections lost to the other eigenstate have been extensively studied (see, for example, Ref. 59 and references therein). 

The INS technique can be simply summarised. The observed positions of the transitions (the eigenvalues) are a function of the molecule s structure and the intramolecular forces, as in optical spectroscopy, and correspond to the energies lost by the neutron. The strength of the observed transition is a function of the atomic displacement occurring during that vibration (the eigenvector) and the momentum lost by the neutron. The atomic displacements are again determined by the molecule s structure and the intramolecular forces but... 

This relation defines a p-fold fimction Zj = /(z), and one then looks for the "crossing" points with the straight line Zj = z, which gives the eigenvalues z = z = E. Substitution into the relation (4.3) gives then the exact wave functions. In comparison to the previous sections, this approach deals also with a secular equation of order p, but the wave operator now contains the energy E explicitly, and further all the degeneracies of the Hamiltonian H are removed. This means that the connection with the idea of the existence of a "model Hamiltonian" and a set of "model functions" is definitely lost. However, from the point-of-view of ab-initio applications this approach may offer other... 

It contains two p E, i) producing terms (either from different energy levels of A, term 1, or from p E, t), term 4) and two consuming terms, in which p E, t) is lost to other energy levels of A (term 2) or to B (term 3). For the population density of B an analogous ME exists. Both populations are coupled by the mass conservation requirement and therefore the set of coupled differential equations contains both species. We can define a new vector p E, t) which contains the populations of both isomers. This leads to the same eigenvalue equation as discussed earlier. 

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