On the spectrum of the hierarchical Laplacian

Abstract

Let $(X,d)$ be a locally compact separable ultrametric space. We assume that $(X,d)$ is proper, that is, any closed ball $B$ in $X$ is a compact set. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of balls (the choice function), we define the hierarchical Laplacian $L_C$ which is closely related to the concept of the hierarchical lattice of F.J. Dyson. $L_C$ is a non-negative definite, self-adjoint operator in $L^2(X,m)$. We address in this paper to the following question: How general can be the spectrum $\mathsf{Spec}(L_C)$ as a subset of the non-negative reals? When $(X,d)$ is compact, $\mathsf{Spec}(L_C)$ is an increasing sequence of eigenvalues of finite multiplicity which contains $0$. Assuming that $(X,d)$ is not compact we show that, under some natural conditions concerning the structure of the hierarchical lattice (= the tree of $d$-balls), any given closed subset $S$ of $[0,\infty )$, which contains $0$ as an accumulation point and is unbounded if $X$ is non-discrete, may appear as $\mathsf{Spec}(L_C)$ for some appropriately chosen function $C(B)$. The operator $-L_C$ extends to $L^q(X,m)$, $0<q<\infty$, as Markov generator and its spectrum does not depend on $q$. As an example, we consider the operator ${\mathfrak{D}}^{\alpha }$ of fractional derivative defined on the field ${\mathbb{Q}}_p$ of $p$-adic numbers.

27 pages

References in corpus (1)

• Poisson statistics of eigenvalues in the hierarchical Anderson model

Cited by in corpus (3)

• Oscillating heat kernels on ultrametric spaces
• Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis
• Hierarchical Schrödinger-type operators: the case of potentials with local singularities