论文标题
半同质树上的拉普拉斯人的球形功能和光谱
Spherical functions and spectrum of the Laplacian on semi-homogeneous trees
论文作者
论文摘要
在半同质树上，我们研究了拉普拉斯操作员的$ \ ell^p $ -spectrum $μ_1$（各向同性最近的邻邻过渡算子）；在更简单的均匀树环境中，已知的结果是特定情况。频谱由球形函数的特征值（即，相对于参考顶点$ v_0 $ radial the radial and radial的特征功能，即在此归一化。我们表明，球形函数是广义泊松核的边界积分，与均匀的环境不同，它不是通常的泊松内核的复杂功能。我们通过马尔可夫链及其生成功能来计算这些广义的泊松仁，从此我们为球形函数发出明确的表达式。在半均匀的树上，球形功能事实证明具有$ \ ell^p $行为，在同质树上没有发生：其中之一，对于$ v_0 $的适当选择，属于$ \ ell^p $，对于某些$ p <2 $。 达到归一化，操作员$μ_1^2 $与步骤2 laplacian $μ_2$不同。另一方面，与半同质性$μ_2$相关的复发关系是多边形图的复发关系，类似于同质树中的拉普拉斯式的图形。 By this token, we compute the spectra of $μ_1^2$ on a semi-homogeneous tree, hence, by extracting square roots, the $\ell^p$-spectrum of $μ_1$ for $1\leqslant p <\infty$, and show that it is disconnected for $p$ in an interval containing 2 but it is connected at all other values of $p$.
On a semi-homogeneous tree, we study the $\ell^p$-spectrum of the Laplace operator $μ_1$ (the isotropic nearest-neighbor transition operator); the known results in the much simpler setting of homogeneous trees are obtained as particular cases. The spectrum is given by the eigenvalues of spherical functions, i.e., eigenfunctions of $mu_1$ that are radial with respect to a reference vertex $v_0$ and normalized there. We show that spherical functions are boundary integrals of generalized Poisson kernels that, unlike the homogeneous setting, are not complex powers of the usual Poisson kernel. We compute these generalized Poisson kernels via Markov chains and their generating functions, whence we work out explicit expressions for spherical functions. On semi-homogeneous trees, spherical functions turn out to have an $\ell^p$ behavior that does not occur on homogeneous trees: one of them, for an appropriate choice of $v_0$, belongs to $\ell^p$ for some $p<2$. Up to normalization, the operator $μ_1^2$ differs from the step-2 Laplacian $μ_2$ only by a shift. On the other hand, the recurrence relation associated to the semi-homogeneous $μ_2$ is that of a polygonal graph, akin to that of the Laplacian in a homogeneous tree. By this token, we compute the spectra of $μ_1^2$ on a semi-homogeneous tree, hence, by extracting square roots, the $\ell^p$-spectrum of $μ_1$ for $1\leqslant p <\infty$, and show that it is disconnected for $p$ in an interval containing 2 but it is connected at all other values of $p$.
