This work deals with the characterization of eigenfunctions of the Laplacian L on a homogeneous tree X, which satisfy certain growth conditions. More precisely, we shall prove that the Poisson transform on X provides an one-to-one correspondence between the subspace of all Hardy-type eigenfunctions of L on X and the Lebesgue spaces (possibly the set of all complex measures) on the boundary of X.

In this article we prove the restriction theorem for Helgason-Fourier transform on homogeneous tree. Our proof is based on the duality argument and the norm estimates of Poisson transform. We also characterize all eigenfunctions of the laplacian on homogeneous tree which are Poisson transform of L p functions defined on the boundary.

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Let Ψ be a non-constant complex-valued analytic function defined on a connected, open set containing the L p -spectrum of the Laplacian L on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup T (t ) = e t Ψ(L ) to be chaotic on L p -spaces. We also study the chaotic dynamics of the semigroup T (t ) = e t (aL +b) separately and obtain a sharp range of b for which T (t ) is chaotic on L p -spaces. It includes some of the important semigroups such as the heat semigroup and the Schrödinger semigroup.

Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .