Titchmarsh

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, UK, 1948, Chap. V. 

If we assume that the shape of each disc is circular, the CLD of an uncorrelated hard-disc fluid is the Mellin convolution of the intrinsic chord distribution, gc r 2), of an ideal disc of diameter 1 and the diameter distribution, ho (D) which characterizes the structure. The definition of the Mellin convolution (Titchmarsh [202], S. 53 Marichev [203] ... 

E. C. Titchmarsh, in the preface to his Theory of Fourier Integrals, remarks that, in doing problems on heat conduction, he has retained such terms as heat and temperature but that the reader need not know that such things exist. He was a pupil of G. H. Hardy, and shared his mentor s aversion to applied mathematics. By contrast, it is important in mathematical modeling to see as many links between the physical and mathematical as possible. 

Realizing that Eq. (13) gives an explicit solution of (1) with an appropriate V, in terms of logarithmic derivatives, it is possible to identify u with the well-known Jost solution denoted as/(r, 2), see more below and Ref. [44], which here must be proportional to the Weyl s solution x(f, )- With this identification, we obtain the generalized Titchmarsh formula (generalized since it applies to all asymptotically convergent exponential-type solutions commensurate with Weyl s limit point classification)... 

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)]... 

E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Qaredon, Oxford, Vol. 1,1946,1962 Vol. 2,1958. 

E. Brandas, M. Rittby, N. Elander, Titchmarsh-Weyl Theory and Its Relations to Scattering Theory Spectral Densities and Cross Sections Theory and Applications, J. Math. Phys. 26 (1985) 2648. 

M. Rittby, N. Elander, E. Brandas, Scattering in View of the Titchmarsh-Weyl Theory, Int. J. Quant. Chem. XXIII (1983) 865. 

Stribeck N (1989) Colloid Polym Sci 267 301 Station WO (1962) J Polym Sci 58 205 Station WO (1968) Z Kristallogr 127 229 Stribeck N (1999) J Polym Sci Part B Polym Phys 37 975 Cohen Y, Thomas EL (1987) J Polym Sci Part B Polym Phys B25 1607 Titchmarsh EC (1948) Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford... 

Titchmarsh, E. C., The Theory of Functions, Oxford Oxford University Press, 1988. 

The very roots of DPT can be traced back to Pauli s Handbuch article [38]. A perturbation expansion based on a change of the metric was first explicitly considered by Sewell [33], who only studied the leading order. After a not entirely fair criticism of this approach by Titchmarsh [66] it was forgotten for decades. Rutkowski [15] rediscovered this idea in a somewhat different form. He did not make a change of the metric, but instead he decomposed the Dirac equation, into what he called an equivalent of the Schrodinger equation and a remainder. 

I would like to thank my colleagues at Johnson Matthey and in particular Ken Griffin, Brian Harrison and Dave Grove for their ideas and critical review of the paper. In addition, Peter Wells (University of Hull) and John Titchmarsh (AEA Technology) for their contributions and Alison Neilson for preparing the manuscript. 

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