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Stieltjes‐integral approximations to elementary dispersion relations
Author(s) -
Langhoff P. W.
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560080839
Subject(s) - riemann–stieltjes integral , dispersion relation , fredholm determinant , mathematics , hilbert space , dispersion (optics) , mathematical analysis , lattice (music) , quantum mechanics , integral equation , physics , acoustics
Variational calculations in Hilbert space and theorems from the Tchebycheff‐Stieltjes‐Markoff moment theory are employed in the construction of Stieltjes‐integral approximations to a number of elementary dispersion relations. Appropriate derivatives of the Stieltjes integrating functions provide convergent estimates for the absorptive and dispersive conjugate functions in their nonanalytic regions on the real axis, where the dispersion relations are of particular interest. Illustrative applications are given in the cases of the Kramers‐Krönig‐Heisenberg dispersion formula, the closely related wave number‐dependent dielectric response function, a dispersion relation that arises in connection with lattice vibrations and the thermodynamic properties of crystals, and the Schwinger‐Baker formula for the elastic‐scattering Fredholm determinant.