Spectral concentration

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)]... 

E. Brandas, Time Evolution and Spectral Concentration in Quantum Systems, Phys-ica 82A (1976) 97 E. Brandas, P. Froelich, A. Remark on Time Evolution and Spectral Concentration, Int. J. Quant. Chem. S9 (1975) 457. 

M.S.P. Eastham, On the Location of Spectral Concentration for Sturm-liouville Problems with Rapidly Decaying Potential, Mathematika 45 (1998) 25. 

D.J. Gilbert, B.J. Harris, Bounds for the Points of Spectral Concentration of Sturm-liouville Problems, Mathematika 47 (2000) 327. 

In all cases, the cations display spectral changes consistent with pseudobase formation at electronic spectral concentrations, with the disproportionation reaction becoming important at much higher concentrations of the heterocycle. This concentration dependence indicates the bimolecular nature of these reactions, and the 1 1 ratio of oxidized and reduced products, in those cases which have been carefully investigated, is indicative of a true disproportionation reaction. A particularly interesting case is the report217 of equimolar amounts of dihydropyridines (a mixture of 125 — 127) and pyridinones (a mixture of 128 and 129) from the reaction of the l-methyl-3-cyanopyridinium cation in aqueous base. The 1 1 ratio of oxidized and reduced products is sometimes obscured by subsequent air oxidation of the reduced product to the oxidized product. 

From Eq. (9) it is seen that at E E, it is the energy-dependence of r(E) fhat affects, in principle, the spectral concentration. The standard assumption is that for narrow resonances F(E) is a constant, whose value at the exact resonance energy is T(E,) s r =, where r is the mean lifetime, Eq. (3b). The distribution is then an exact Lorentzian. The dependence on energy of r in fhe neighborhood E< E, fhus follows the Lorentzian distribution, 1 ( E-Er + r /4 ( = 1)- other hand, the energy-dependence of... 

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