Abstract
Let (X, d) be a locally compact separable ultrametric space. We assume that (X, d) is proper, that is, any closed ball B⊂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). In this paper we address the following question: How general can be the spectrum \(\mathsf {Spec}(L_{C})\subseteq \mathbb {R}_{+}?\) When (X, d) is compact, S p e c(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 ⊆ ℝ+, which contains 0 as an accumulation point and is unbounded if X is non-discrete, may appear as S p e c(L C ) for some appropriately chosen function C(B). The operator −L C extends to L q(X, m), 1 ≦ q < ∞, as Markov generator and its spectrum does not depend on q. As an example, we consider the operator 𝔇α of fractional derivative defined on the field ℚ p of p-adic numbers.
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The research of the first author was supported by the Polish Government Scientific Research Fund, Grant 2012/05/B/ST1/00613
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Bendikov, A., Krupski, P. On the spectrum of the hierarchical Laplacian. Potential Anal 41, 1247–1266 (2014). https://doi.org/10.1007/s11118-014-9409-6
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DOI: https://doi.org/10.1007/s11118-014-9409-6