Spectral functions

We seek the poles of the spectral function g(E) given by (3.7). In the WKB approximation the path integral in (3.7) is dominated by the classical trajectories which give an extremum to the action functional... 

Here we shall describe how the periodic-orbit theory of section 3.4, relating the energy levels with the poles of the spectral function g E), can be extended to two dimensions. For simplicity we shall exemplify this extension by the simplest model in which the total PES is constructed of two paraboloids crossing at some dividing line. Each paraboloid is characterized by two eigenfrequen-cies, o + and [Pg.72]

If the resolving capacity of the instruments is ideal then vibrational-rotational absorption and Raman spectra make it possible in principle to divide and study separately vibrational and orientational relaxation of molecules in gases and liquids. First one transforms the observed spectrum of infrared absorption FIR and that of Raman scattering FR into spectral functions... 

In quantum theory as well as in classical theory, linear absorption of light at frequency co is described by a spectral function... 

The modulation index is related to the transmission spectra of the gas-filled reference cell, TRef( ), the measurement cell, with its unknown gas concentration, Ty Ca,(/.), and the optical filter, F(/.), all of which are shown in Equation 1. If required, the source spectra can also be taken into account, by using additional spectral functions (although this will usually have little spectral variation over the linewidth of the gas band) ... 

Transitions between superbarrier states (a > N, see Fig. 4.4) involve nondiagonal in cr elements of the density matrix which give, at AU > T, a small contribution to the spectral function for frequencies of the order a>h + 0cr - 0[Pg.98]

For a single subbarrier level we have A0(f) = 1, B0(4) = C0(4) = 0, and the halfwidth of the spectral function is determined solely by the reorientation rate, which is equal to the transition rate to the first excited state of deformation vibrations 164... 

The quantity (A 1.56) at a>> 0 is associated with the so-called spectral function... 

We demonstrate that the spectral function of valence harmonic vibrations of a diatomic group that effects rotational reorientations is broadened by w. The vector of atom C displacements relative to the atom B (see Fig. A2.1) may be represented as x(t)e(t), where x(t) is the change in the length of the valence bond oriented at the time t along the unit vector e(/). Characteristic periods of valence vibrations are much shorter than periods of changes in unit vector orientations. As a consequence, the GF of the displacements defined by Eq. (4.2.1) can be expressed approximately as ... 

The spectral function thus has a Lorentz shape with a halfwidth at the half distribution height equal to the average reorientation frequency w. If expressed in spectroscopic units (cm 1), the halfwidth Avv2 amounts to cd2nca (c0 designates the velocity of light in vacuo). 

The spectral function of local vibration (low-temperature approximation involving collectivized high-frequency and low-frequency modes)... 

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

Now as t -> 0 the spectral function L reduces to the area under the distribution. This is the steady state compliance Je ... 

Once the Green functions have been obtained, together with the self-energies, they allow calculation of the quantities of interest. In particular, the spectral function is given as... 

The diagonal elements of the spectral function yield the local DOS at the corresponding site, while their sum, the trace, yields the DOS ... 

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