自相似拟连续线性链中的色散关系和波算子
Dispersion Relations and Wave Operators in Self-Similar Quasi-Continuous Linear Chains
摘要：
我们构造自相似函数和线性算子，推导出拉普拉斯算子和达朗伯波算子的自相似变体。自相似性作为一种对称性质的迫切性要求引入非局域粒子间相互作用。我们推导出一个自相似线性波算子，描述具有空间自相似分布的非局部粒子间弹簧的无限长的准连续线性链的动力学。非局域谐振子-粒子相互作用的自相似性导致了具有自相似性和分形特征的Weierstrass-Mandelbrot函数形式的色散关系。我们还推导出一个连续近似的自相似拉普拉斯算子的分数阶积分和收益率在低频制度的幂律频率依赖的振子密度。
英文摘要：
We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of non-local particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We also derive a continuum approximation which relates the self-similar Laplacian to fractional integrals and yields in the low-frequency regime a power law frequency-dependence of the oscillator density.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
