Spectra discrete

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

In (10-124) and (10-125), n and m refer to the eigenvalues of a complete set of commuting observables so Snm stands for a delta function m those observables in the set that have a continuous spectrum, and a Kronecker 8 in those that have a discrete spectrum. 

Regardless of how we may choose to represent the GF, we should note that it is for a finite chain, so that there is a discrete spectrum of m states, whose energies are given by the poles of GijTO(1, 1). 

As noted earlier, for each particle i, there is a discrete spectrum of positive energy bound states and positive and negative energy continuum states. Let us consider a product wave function of the form = i//i(l)i//2(2), a normalizable stationary bound-state eigenfunction of... 

It turns out there are increasing roots, y, of the parameter y that lead to successive (shell after shell) smoothing of the discrete spectrum g(E). In this investigation, we have chosen the first roots y= y, that fulfill the successive smearing of the different main shells and for which a well pronounced plateau is observed. Fig. (la) displays the smoothed energy level distribution (E) for atom Zn (Z = 30), which has four shells (Is 2s 2p 3s 3p 3d 4s ), for the plateau values of y y (continuous line) and Yo (dotted line). We see that the root y" " corresponds to complete smoothing (in energy space) of the two outer-shell electrons 4s but that the spectrum Ihfr(E, y ") preserves a clear information on the existence of the first two shells. For the root y " such information is still kept only about the deeply bonded Is electrons. Fig. (lb) displays the function hfr(E, yS ). which shows that fory=y the one-particle spectrum is completely smoothed out. 

In accordance with these observations one can completely smooth an atomic spectrum with some optimal value of y, yS . The two-shell, second-row atoms (3 Z S 10) have only one stationary value y yS" (the first that smoothes the whole spectrum). The three-shell, third-row atoms (11SZ<18) have two such values of y the first one, y ", corresponds to complete smoothing of the two-shell upper part of the discrete spectrum, and the second one, y, leads to total smoothing of the whole spectrum. In the case of four-shell, fourth-row atoms, as it was illustrated on Zinc, there exist three such roots y, y , yS The use of the second one (y ) results in complete smoothing of gHFR(E) beyond the K-shell region. The same role was played by the first root in the case of three-shell atoms (that is why we denote these two different roots with the same symbol). 

The Fourier series, which has a discrete spectrum but periodic spatial function, is actually a special case of the Fourier transform. (Note that an equally spaced discrete spectrum necessarily implies a periodic function having a finite period given by the wavelength of the lowest frequency.) See Bracewell (1978) to see how the explicit form of the Fourier series may be obtained from the Fourier transform. Taking discrete, equally spaced... 

Let the discrete spectrum, which consists of the coefficients of u(k) and v(k), be denoted by U(n) and V(n), respectively. The low-frequency spectral components U(n) are most often given by the most noise-free Fourier spectral components that have undergone inverse filtering. For these cases V(n) would then be the restored spectrum. However, for Fourier transform spectroscopy data, U(n) would be the finite number of samples that make up the interferogram. For these cases V(n) would then represent the interferogram extension. 

However, unless special care is exercised, generally the discrete spectrum does not estimate very well the sampled continuous spectrum. The problems we face are as follows. 

Interpolation and smoothing by addition of zeros. We may need to add zeros to the sample simply in order to obtain 2 1 points. The addition of zeros, however, also increases the length of the observation interval [0,T], and hence the number of frequences in the discrete spectrum. Smoothing the spectrum by an appropriate window and applying the inverse transformation then results in an... 

In the classical limit h - 0, the spectrum of the Landau-von Neumann superoperator tends to the spectrum of the classical Liouvillian operator. If the classical system is mixing, the classical Liouvillian spectrum is always continuous so that we may envisage an analytic continuation to define a discrete spectrum of classical resonances. It has been shown that such classical resonances are given by the zeros of the classical zeta function (2.44) and are called the Pollicott-Ruelle resonances sn(E) [63], These classical Liouvillian resonances characterize exponential decay and relaxation processes in the statistical description of classical systems. The leading Pollicott-Ruelle resonance defines the so-called escape rate of the system,... 

Gaseous emission of Infrared radiation differs in character from solid emission in that the tenner consists of discrete spectrum lines or bands, with significant discontinuities, while the latter shows a continuous distribution of energy throughout the spectrum. The predominant source or molecular radiation in the infrared is the result of vibration of the molecules in characteristic modes. Energy transitions between various stales of molecular rotation also produce infrared radiation. Complex molecular gases radiate intricate spectra, which may be analyzed to give information of the nature of the molecules or of the composition of the gas. 

The concept of probability density may also be utilized to describe the discrete spectrum or, particularly, to solve the inverse problem of the approximate reconstruction of the spectrum from its envelope line using a certain number of its lower moments. The density function of the energy levels of the discrete spectrum may have the form... 

The discussion of the operator - ( ) should have the preceding as a starting point, but for our purposes it is sufficient to have found an unperturbed operator with a partially discrete spectrum, with known solutions and therefore tractable by methods of the partitioning technique-... 

In the previous section we discussed the Hermitean operator -a (t) It is just a one particle operator with a complete, discrete spectrum, and the following relations hold, for <.0 i... 

The distinction between a truly continuous absorption spectrum and a banded absorption spectrum for diatomic molecules may be made by instruments of relatively low resolving power. Even though individual rotational lines are not resolved, a discrete spectrum will have sharp band heads and the appearance will in no way resemble the appearance of a continuum. 

As we have seen in discussing the behavior of diatomic molecules, there is often great difficulty in identifying a predissociation spectrum since the appearance may at one extreme be essentially that of a truly discrete spectrum and at the other that of a true continuum. The facts of photochemistry may be of great use to the spectroscopist in distinguishing between predissociation and truly discrete spectra. 

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