Spectral function theorem

In the second period, which was ended by review GT after the average perturbation theorem was proved, it became possible to get the Kubo-like expression for the spectral function L(z) (GT, p. 150). This expression is applicable to any axially symmetric potential well. Several collision models were also considered, and the susceptibility was expressed through the same spectral function L(z) (GT, p. 188). The law of motion of the particles should now be determined only by the steady state. So, calculations became much simpler than in the period (1). The best achievements of the period (2) concern the cone-confined rotator model (GT, p. 231), in which the dipoles were assumed to librate in space in an infinitely deep rectangular well, and applications of the theory to nonassociated liquids (GT, p. 329). 

The spectral density theorem, using Eq. (5.6), can also be written in terms of the correlation function o( )... 

The use of the function theorem can be seen in conjunction with the representation theorem. We choose the spectral representation of the observable a, that is the representation in which the basis states are the eigenstates (corresponding to the eigenvalue spectrum) of a. 

We first replace the resolvent by a number by introducing its spectral representation and using the function theorem. At the same time we introduce the coordinate representation. 

Using the impact approximation presented in Chapter 6, they may easily be found for any rotational band even if rotational-vibrational interaction is nonlinear in J. In 1954 R W. Anderson proved as a theorem [104] that expansion of the spectral wings in inverse powers of frequency is controlled by successive odd derivatives of the correlation function at the origin. In impact approximation the lowest non-zero derivative of this type is the third and therefore asymptotics G/(co) is described by the power expansion [20]... 

The fundamental quantity for interferometry is the source s visibility function. The spatial coherence properties of the source is connected with the two-dimensional Fourier transform of the spatial intensity distribution on the ce-setial sphere by virtue of the van Cittert - Zemike theorem. The measured fringe contrast is given by the source s visibility at a spatial frequency B/X, measured in units line pairs per radian. The temporal coherence properties is determined by the spectral distribution of the detected radiation. The measured fringe contrast therefore also depends on the spectral properties of the source and the instrument. 

The central-limit theorem (Section III.B) suggests that when a measurement is subject to many simultaneous error processes, the composite error is often additive and Gaussian distributed with zero mean. In this case, the least-squares criterion is an appropriate measure of goodness of fit. The least-squares criterion is even appropriate in many cases where the error is not Gaussian distributed (Kendall and Stuart, 1961). We may thus construct an objective function that can be minimized to obtain a best estimate. Suppose that our data i(x) represent the measurements of a spectral segment containing spectral-line components that are specified by the N parameters... 

A proof of this relation may be found in Bracewell (1978). Note that the spectral variable used in this and the next chapter is the same as that defined in Eqs. (7) and (8). Now consider a spatial distribution /(x) and its Fourier spectrum F(w) that come close to satisfying the equality in Eq. (4). We may take Ax and Aw as measures of the width, and hence the resolution, of the respective functions. To see how this relates to more realistic data, such as infrared spectral lines, consider shifting the peak function /(x) by various amounts and then superimposing all these shifted functions. This will give a reasonable approximation to a set of infrared lines. To discuss quantitatively what is occurring in the frequency domain, note that the Fourier spectrum of each shifted function by the shift theorem is given simply by the spectrum of the unshifted function multiplied by a constant phase factor. The superimposed spectrum would then be... 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

We are most interested in integrating products of characters of representations. In this case, we can use the Spectral Theorem (Proposition 4.4) to simplify the expression of the integral. The proposition implies that for any function f invariant under conjugation, we have... 

The results of this section, even with their limitations, are the punch line of our story, the particularly beautiful goal promised in the preface. Now is a perfect time for the reader to take a few moments to reflect on the journey. We have studied a significant amount of mathematics, including approximations in vector spaces of functions, representations, invariance, isomorphism, irreducibility and tensor products. We have used some big theorems, such as the Stone-Weierstrass Theorem, Fubini s Theorem and the Spectral Theorem. Was it worth it And, putting aside any aesthetic pleasure the reader may have experienced, was it worth it from the experimental point of view In other words, are the predictions of this section worth the effort of building the mathematical machinery ... 

Wiener-Khintchine theorem). The right-hand side of this equation is often called the power spectrum. It is given by the autocorrelation function, Eq. 2.55. The Fourier transform of the autocorrelation function is related to the spectral moments,... 

The spectral theorem can also be used to express many functions of A, by recognizing that all powers of A have the same eigenvectors as A and the associated eigenvalues are equivalent functions of the a . 

To describe the effect of the environment one usually needs to determine the bath correlation function C(t). Let us start discussing this function for a bosonic bath where the subscript Ph indicates a bath of phonons. Using the numerical decomposition of the spectral density Eq. (2) together with the theorem of residues one obtains the complex bath correlation functions... 

According to the fluctuational dissipation theorem, the spectral density function in terms of % reads [96,97] as follows ... 

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