Spectral function derivation

Fig. 3.2. The (smoothed) spectral functions derived from the measurements, Fig. 3.1, of the normalized binary absorption coefficients of helium-argon, neon-argon and argon-krypton mixtures at room temperature in a semi-logarithmic grid, Eq. 3.2.

Acetylacetonates and Related Complexes (Table 6). Metal /3-diketonates and related functional derivatives have been extensively investigated in UV-P.E. spectroscopy. Their spectra are similar to those of the free j3-diketone ligands, whose spectral patterns can always more or less directly be traced back in the spectra of the metal complexes, of which they often represent the major and more characteristic part, d-... 

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. 

In an attempt to model the spectral functions of rare gas mixtures, Fig. 3.2, it was noted that a Gaussian function with exponential tails approximates the measurements reasonably well [75], about as well as the Lorentzian core with exponential tails. Two free parameters were chosen such that at the mending point a continuous function and a continuous derivative resulted the negative frequency wing was again chosen as that same curve, multiplied by the Boltzmann factor, to satisfy Eq. 3.18. Subsequent work retained the combination of a Lorentzian with an exponential wing and made use of a desymmetrization function [320],... 

Using the so-called planar libration-regular precession (PL-RP) approximation, it is possible to reduce the double integral for the spectral function to a simple integral. The interval of integration is divided in the latter by two intervals, and in each one the integrands are substantially simplified. This simplification is shown to hold, if a qualitative absorption frequency dependence should be obtained. Useful simple formulas are derived for a few statistical parameters of the model expressed in terms of the cone angle (5 and of the lifetime x. A small (3 approximation is also considered, which presents a basis for the hybrid model. The latter is employed in Sections IV and VIII, as well as in other publications (VIG). 

We derive an analytical expression for the spectral function in terms of a double integral, which differs from the formulas given in Section III by account of finiteness of the well depth. Two important approximations are also given, in which the spectral function is represented by simple integrals. These... 

A brief list of basic assumptions used in the ACF method precedes the detailed analysis of the results of calculations. The derivation of the formula for the spectral function is given at the end of the section. The calculations demonstrate a substantial progress as compared with the hat-flat model but also reveal two drawbacks related to disagreement with experiment of (i) the form of the FIR absorption spectrum and (ii) the complex-permittivity spectrum in the submillimeter wavelength region. We try to overcome these drawbacks in the next two sections, to which Fig. 2c refers. 

At the end of the section, derivation of a modified spectral function is given with account of the p(f)-part of a dipole moment. 

Third, the expression for the spectral function pertinent to the HO model is derived in detail using the ACF method. Some general results given in GT and VIG (and also in Section II) are confirmed by calculations, in which an undamped harmonic law of motion of the bounded charged particles is used explicitly. The complex susceptibility, depending on a type of a collision model,... 

The spectral function (22a) is a main object of further calculations given in Sections III-VI and VII-IX. In the next subsection we shall derive a more convenient expression for L(z). 

Finally, we derive the formula for the spectral function convenient for calculations ... 

Substituting this relation into Eq. (31), we finally derive unambiguous relation between the complex susceptibility and spectral function relevant to the Gross collision model12 ... 

We shall get from (70a) and (70b) another formula for the spectral function L(z). At first, we express the series S(k) through elementary functions (the derivation is given in Section III.E) ... 

The paper [50] was written, when our general linear response-theory (ACF method) was still in progress, so the derivation of the spectral function described in Ref. 50 is more specialized than that given, for example, in VIG and GT. 

A general approach (VIG, GT) to a linear-response analytical theory, which is used in our work, is viewed briefly in Section V.B. In Section V.C we consider the main features of the hat-curved model and present the formulae for its dipolar autocorrelator—that is, for the spectral function (SF) L(z). (Until Section V.E we avoid details of the derivation of this spectral function L). Being combined with the formulas, given in Section V.B, this correlator enables us to calculate the wideband spectra in liquids of interest. In Section V.D our theory is applied to polar fluids and the results obtained will be summarized and discussed. 

The corresponding spectral functions, denoted L(z) and L(z), are derived, as well as the SF L for the rotators, in Section V.E in the form of simple integrals from elementary functions over a full energy of a dipole (or over some function of this energy). The total spectral function is thus represented as... 

The multiplier 2 here accounts for existence of two potential wells with oppositely directed symmetry axes (the case of mirror symmetry is assumed). The derivation of the spectral function of the rotators, namely the last term in Eq. (170), will be given in Section V.E.6. 

Substituting all variables into integral (A2) and taking into account Eq. (197d), we finally derive Eqs. (174)-(178) for the spectral function of the precessors. 

The main purpose of this section is consideration of the FIR spectra due to the second dipole-moment component, p(f). However, for comparison with the experimental spectra [17, 42, 51] we should also calculate the effect of a total dipole moment ptot. In Refs. 6 and 8 the modified hybrid model44 was used, where reorientation of the dipoles in the rectangular potential well was considered. In this section the effect of the p(f) electric moment will be found for the hat-curved, potential, which is more adequate than the rectangular potential pertinent to the hybrid model. In Section VI.B we present the formula for the spectral function of the hat-curved model modified by taking into account the p(f) term (derivation of the relevant formula is given in Section VI.E). The results of the calculations and discussion are presented, respectively, in Sections VI.C and VI.D. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

In summary, we have derived formulas for the Weyl-Titchmarsh m-function, where the imaginary part serves as a spectral function of the differential equation in question. Before we look at the full m-function, we will see how it works in connection with the spectral resolution of the associated Green s function... 

In this addendum, we will derive the spectral function from Weyl s theory and in particular demonstrate the relationship between the imaginary part of the Weyl-Titchmarsh m-function, mi, and the concept of spectral concentration. For simplicity we will restrict the discussion to the spherical symmetric case with the radial coordinate defined on the real half-line. Remember that m could be defined via the Sturm-Liouville problem on the radial interval [0,b] (if zero is a singular point, the interval [a,b], b > a > 0), and the boundary condition at the left boundary is given by [commensurate with Eq. (5)]... 

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