Poisson Summation Formulas and the Theta Function Summation Method for Porous Structures and Irregular Lattices in One Dimension.
Ewald's theta function method (TFM) is a powerful technique that is used to investigate the properties of lattice sums. An integral part of this method is the application of the Poisson summation formula (PSF) to theta functions (TFs), which are special types of lattice sums. The PSF provides an alternate expression (a sum that is also a TF) that is rapidly convergent in domains where the original TF is not. Limitations on the sums to which the PSF may be applied impose limitations on the sums tractable via the TFM. There are two ways to characterize these restrictions. First, almost all TFs to which one can apply the PSF are derived from TFs defined on periodic lattices. Second, even if the TF can be derived from a TF defined on a periodic lattice, it may not be possible to find an alternate expression because of the complexity of the theta function's summand. To enlarge the class whose members are lattice sums to which the TFM can be applied, it is necessary to extend the class whose members are TFs for which one can obtain alternate expressions or at least secure accurate approximations of alternate expressions. Such an extension for one-dimensional lattice sums is accomplished in two ways. First, Bochner's theorem, which in its original form has no practical applications, is considered. By elaborating and modifying this theorem, it is possible to obtain PSFs for some one-dimensional lattice sums (some of these sums being TFs) that were not previously known. These sums are defined on both periodic and non-periodic (irregular) lattices. Next, a fairly systematic approach for accurately approximating alternate expressions for TFs defined on non-periodic lattices (that cannot be effectively handled by the modified Bochner theorem) is presented. This technique relies upon the construction of a sum defined on a periodic lattice whose alternate expression accurately approximates the TF in the domain where the TF is slowly convergent.
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- Physics: Condensed Matter