Four Particular Cases of the Fourier Transform

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

Three-dimensional Korteweg-de Vries equation and traveling wave solutions

The three-dimensional power Korteweg-de Vries equation[ut+unux+uxxx]x+uyy+uzz=0, is considered. Solitary wave solutions for any positive integernand cnoidal wave solutions forn=1andn=2are obtained. The cnoidal wave solutions are shown to be represented as infinite sums of solitons by using Fourier series expansions and Poisson's summation formula.

Nearly conconcentric Korteweg-de Vries equation and periodic traveling wave solution

The generalized nearly concentric Korteweg-de Vries equation[un+u/(2η)+u2uζ+uζζζ]ζ+uθθ/η2=0is considered. The author converts the equation into the power Kadomtsev-Petviashvili equation[ut+unux+uxxx]x+uyy=0. Solitary wave solutions and cnoidal wave solutions are obtained. The cnoidal wave solutions are shown to be representable as infinite sums of solitons by using Fourier series expansions and Poisson's summation formula.

Distribution of quantum states in enclosures of finite size: II

By making use of the modified form of Poisson's summation formula, we calculate the expression for the number of eigenstates, N(K), with eigenvalues [Formula: see text] of a particle in spherical and cylindrical enclosures of finite size, and with its wave-function subject to Dirichlet boundary conditions and Neumann boundary conditions at the walls of the container. We also obtain the oscillatory terms in addition to the important nonoscillatory terms already known and compare our results with the actual number of such states computed from the tables of the zeros of the relevant special mathematical functions. The inclusion of these oscillatory terms improves the accuracy of the expressions in all cases, especially in the case of the cylinder, where these are quite significant. Some possible applications of the results obtained here are also indicated.

Distribution of quantum states in enclosures of finite size: I

We show that the expression for the density of states of a particle in a three-dimensional rectangular box of finite size can be obtained by using directly the Poisson's summation formula instead of using the Walfisz formula or the generalized Euler formula both of which can be derived from the former. We also derive the expression for the density of states in the case of an enclosure in the form of an infinite rectangular slab and apply it to the problem of the Bose–Einstein condensation of a Bose gas of noninteracting particles confined to a thin-film geometry.