A new expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Comparison to Bethe ansatz clarifies the relation between the two approaches.
Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function
本文提出了一种柱坐标系下亥姆霍兹格林函数的傅里叶分解新方法，它等价于求解一般环源的亥姆霍兹方程。将亥姆霍兹格林函数的傅里叶系数分为半超前+半滞后分量和半超前-半滞后分量。封闭形式的解决方案，给出了这些组件的霍恩函数和Kampe de Feriet函数，分别。与这些二维超几何函数相关联的偏微分方程系统被用来构造两个分量都满足的四阶常微分方程。本文从广义傅里叶系数的积分表示出发，导出了广义傅里叶系数的第二个四阶常微分方程，并证明了这两个微分方程是等价的。各种傅立叶系数的级数解也给出了，主要是在勒让德函数和贝塞尔/汉克尔函数。这些都是来自封闭形式的超几何解决方案或积分表示，或两者兼而有之。数值计算比较不同的方法计算傅立叶系数。
The exponential decay of lattice Green functions is one of the main technical ingredients of the Bałaban's approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The main sources of exponential decay are the Combes-Thomas method and the analyticity of the Fourier transforms. They are combined using a renormalization group equation and the method of images.
We compute a Green's function giving rise to the solution of the Cauchy problem for the source-free Maxwell's equations on a causal domain $\mathcal{D}$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$, where $AdS^5$ denotes the simply connected $5$-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on $\mathcal{D}$ and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on $\…
Fourier transform and related integral transforms in superspace
本文研究了经典Fourier变换、分数阶Fourier变换和Radon变换在超空间上的推广。以前，超空间中的傅里叶变换已经被研究过了，但是具有不同的内核。在这项工作中，费米子部分的傅立叶内核有一个自然的辛结构，使用克利福德分析方法。研究了这三种变换的几个基本性质。使用适当的推广的埃尔米特多项式的超空间（见[H。De Bie，F. Sommen，Hermite and Gegenbauer polynomials in superspace using Clifford analysis，J.Phys.A 40（2007）10441-10456]），构造了用于傅里叶变换的本征函数基。
In this paper, we prove that given a cut-and-project scheme $(G, H, \mathcal{L})$ and a compact window $W \subseteq H$, the natural projection gives a bijection between the Fourier transformable measures on $G \times H$ supported inside the strip $\cL \cap (G \times W)$ and the Fourier transformable measures supported inside $\oplam(W)$, and relate their Fourier transforms. We use this formula to relate the Fourier transforms of the measures, and explain how one can use this relation to re-derive some known results about Fourier analysis of measures with Meyer set support.
We propose an alternative definition for a Tsallis entropy composition-inspired Fourier transform, which we call "$τ_q$-Fourier transform". We comment about the underlying "covariance" on the set of algebraic fields that motivates its introduction. We see that the definition of the $τ_q$-Fourier transform is automatically invertible in the proper context. Based on recent results in Fourier analysis, it turns that the $τ_q$-Fourier transform is essentially unique under the assumption of the exchange of the point-wise product of functions with their convolution.
