Eigenvalues spectrum

In quantum mechanics, the eigenvalues of an operator represent the only numerical values that can be observed if the physical property corresponding to that operator is measured. Operators for which the eigenvalue spectrum (i.e., the list of eigenvalues) is discrete thus possess discrete spectra when probed experimentally. 

Cycle-State Structure from Global Eigenvalue Spectrum... 

Figure 2. Evidence of quantum monodromy in the accurately computed eigenvalue spectrum of H2O [33]. The energy points are plotted for the Jjo rotational levels of the (0, V2, 0) progression, with Ka taken equal to J. Crosses indicate levels that have been reassigned with respect to V2. Open circles indicate reported, usually extra, states [20] that do not conform with the pattern. Taken from Ref. [1] with permission of Taylor and Francis, http //www.tandf.co.uk/journals.

The generalization to the case of a thermally averaged parent state describes an interesting modulation curve that reflects in position and width the rotational eigenvalue spectrum of the resonant intermediate [31]. This structure has been observed in studies of HI ionization in Ref. 33. A schematic cartoon depicting the excitation scheme and the form of the channel phase for the case of a thermally averaged initial state is shown in Fig. 5g. 

The reason for pursuing the reverse program is simply to condense the observed properties into some manageable format consistent with quantum theory. In favourable cases, the model Hamiltonian and wave functions can be used to reliably predict related properties which were not observed. For spectroscopic experiments, the properties that are available are the energies of many different wave functions. One is not so interested in the wave functions themselves, but in the eigenvalue spectrum of the fitted model Hamiltonian. On the other hand, diffraction experiments offer information about the density of a particular property in some coordinate space for one single wave function. In this case, the interest is not so much in the model Hamiltonian, but in the fitted wave function itself. 

Fig. 25. The vibrational eigenvalue spectrum from the cluster model (from Ref. 68>)...

This Exercise demonstrates that some care is needed in utilizing (4.3) for finding n and x because pRtm(X) may have a singularity X = 0. Let us suppose in particular that the M-equation has a discrete eigenvalue spectrum, as in V.7. The lowest eigenvalue is zero, with eigenfunction psn = pn m(oo). Then the Laplace transform has a pole,... 

In order to understand the subsequent discussion it is helpful first to elucidate the eigenvalue spectrum of the triatomic compound [see also Nesbitt (1988a,b)]. The body-fixed theory for total angular momenta / 0 outlined in Section 11.1, especially the set of coupled equations... 

The summation over i includes all occupied levels of the energy eigenvalue spectrum. In the case of a closed shell system this number equals N/2, where Nis the number of electrons in the system. Using eqs. (2.8) and (2.35), the electronic charge density can be expressed as ... 

With the abovementioned structure of the eigenvalue spectrum, the term with l = 1 in Eq. (4.131), which is proportional to exp(—e Gt/xo), at ct > 1 is far more long-living than any other one. The dominating role of the decrement A had been proved by Brown, who derived [47] the following asymptotic expression for it ... 

However, the case of il being exactly zero is to some extent an exception. At any finite Q, one has to remember that the time X is exponential in a that is, it grows infinitely with cooling the system. Therefore, at low temperatures the interwell transition is completely frozen, and the situation is governed by intrawell relaxation. The latter is sensitive to the details of the potential near the bottom of the well, and for the system in question is determined by the infinite eigenvalue spectrum A of the kinetic equation (4.225) for k > 3. 

Let us recall that the LRC potential, which is fitted on a panel of atomic IP, leads to good IP for most (if not any) atomic IPs. This is unfortunately not true anymore for the molecular systems for which a shift between experimental IPs and the eigenvalues spectrum remains [76]. This shift is definitely smaller than those of the LDA or other GGAs, but it is significantly larger than for atoms. Indeed, it corresponds to an overcorrection with respect to the LDA, and this could be related to a proportionnally smaller importance of low density domains in molecules than in atoms. This overcorrection was also noticed by Casida with the VLB potential [77]. 

The unitary group invariance of the hamiltonian assures that its exact eigenvalue spectrum is invariant to a unitary transformation of the basis. It means that a full Cl calculation must provide the same eigenvalues (many-electron energy states) as a full VB calculation. However, at intermediate levels of approximation, this equivalence is not straightforward. In this section we will sketch out the main points of contact between MO and VB related wave functions. 

If neither k — q = 0, nor k + q = 0 for any q, we obtain infinite dimensional irreps with unbounded J3 eigenvalue spectra, since from Eq. (17) it follows that J kq 0. These conditions are equivalent to k q0 integer. The parameter q0 is continuous and can be uniquely chosen so that its real part satisfies — j < Re(q0) < Each such q0 gives a different J3 eigenvalue spectrum. The other irrep series — k — 1 mentioned above gives the same J3 eigenvalue spectra. Thus, the infinite dimensional irreps of this type are labelled by Q and q0 and denoted by D(Q, q0). 

For unirreps of so(2, 1) the situation is quite different. The fundamental inequalities cannot be satisfied if the J3 eigenvalue spectrum is bounded unless k = 0, corresponding to the trivial one-dimensional unirrep. Thus, there are... 

Fig. 1. Regions in the (k, q) plane where the -eigenvalue spectrum corresponds to unitary irreducible representations of so(3) (horizontal shading) and so(2, 1) (vertical shading). See text for details. 

For our applications we need the unirreps of so(2, 1) from Section III which have the T3 eigenvalue spectrum bounded from below. Thus we shall consider unirreps of type D+( — k — 1) defined by [cf. Eq. (28), y = — 1 and... 

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