[Paper Review] Patterson--Sullivan distributions for rank one symmetric spaces of the noncompact type

Joachim Hilgert, Michael Schröder|ArXiv.org|Sep 11, 2009

Advanced Algebra and Geometry15 references3 citations

TL;DR

This paper generalizes the asymptotic equivalence between Wigner and Patterson–Sullivan distributions from hyperbolic surfaces to all rank one symmetric spaces of noncompact type. Using non-Euclidean pseudodifferential calculus and the method of stationary phase, it establishes that off-diagonal Wigner distributions and normalized off-diagonal Patterson–Sullivan distributions are asymptotically equivalent as eigenvalues grow, with an error term of order $O(1/ ho)$, linking quantum ergodicity to boundary distributions on symmetric spaces.

ABSTRACT

There is a remarkable relation between two kinds of phase space distributions associated to eigenfunctions of the Laplacian of a compact hyperbolic manifold: It was observed in \cite{AZ} that for compact hyperbolic surfaces $X_Γ=Γ\backslash\mathbb{H}$ Wigner distributions $\int_{S^* X_Γ} a dW_{ir_j} = < \mathrm{Op}(a)ϕ_{ir_j},ϕ_{ir_j} >_{L^2(X_Γ)}$ and Patterson--Sullivan distributions $PS_{ir_j}$ are asymptotically equivalent as $r_j o\infty$. We generalize the definitions of these distributions to all rank one symmetric spaces of noncompact type and introduce off-diagonal elements $PS_{λ_j,λ_k}$. Further, we give explicit relations between off-diagonal Patterson--Sullivan distributions and off-diagonal Wigner distributions and describe the asymptotic relation between these distributions.

Motivation & Objective

• To extend the observed asymptotic equivalence between Wigner and Patterson–Sullivan distributions—previously known only for compact hyperbolic surfaces—to all rank one symmetric spaces of noncompact type.
• To define and study off-diagonal Patterson–Sullivan distributions $PS_{ ho_j, ho_k}$ for eigenfunctions with different spectral parameters.
• To establish a precise asymptotic relation between off-diagonal Wigner distributions and off-diagonal Patterson–Sullivan distributions in the high-energy limit.
• To use the method of stationary phase and $G$-equivariant pseudodifferential calculus to derive explicit asymptotic expansions for spectral distributions on symmetric spaces.
• To provide a framework linking quantum ergodicity on symmetric spaces to boundary-invariant measures via Patterson–Sullivan theory.

Proposed method

• Adopt a $G$-equivariant calculus of non-Euclidean pseudodifferential operators on symmetric spaces $X = G/K$, where $G$ is a semisimple Lie group with finite center and $K$ a maximal compact subgroup.
• Define Helgason boundary values and use the Iwasawa decomposition $G = KAN$ to parametrize the geodesic boundary $B = K/M$.
• Construct off-diagonal Patterson–Sullivan distributions $PS_{ ho_j, ho_k}$ as distributional kernels associated to eigenfunctions of the Laplacian on $X_{ ho} = \Gamma\backslash X$.
• Apply the method of stationary phase (MSP) to the integral representation of the spectral projection $L_{ ho_k}(\chi a)$, identifying a unique non-degenerate critical point at the identity in $\overline{N}$.
• Derive an asymptotic expansion $L_{ ho_k}(\chi a)(g) = C(2\pi/\rho_k)^{s/2} \sum_{n=0}^N \rho_k^{-n} R_{2n}(\chi a)(g) + O(\rho_k^{-N-1+2K})$, where $s = \dim(N)$.
• Invert the leading-order operator $L_{\rho_k}^{(N)}$ via a formal inverse $M_{\rho_k}^{(N)}$ to relate $\langle \mathrm{Op}(a)\varphi_{\rho_j}, \varphi_{\rho_k} \rangle$ to $\langle a, \widehat{PS}_{\rho_j,\rho_k} \rangle$ up to $O(1/\rho_k)$.

Experimental results

Research questions

• RQ1 Is the asymptotic equivalence between Wigner and Patterson–Sullivan distributions valid beyond hyperbolic surfaces, in the broader class of rank one symmetric spaces of noncompact type?
• RQ2 How can off-diagonal Patterson–Sullivan distributions be defined and characterized for eigenfunctions with distinct spectral parameters $\rho_j, \rho_k$?
• RQ3 What is the precise asymptotic relation between off-diagonal Wigner distributions $\langle \mathrm{Op}(a)\varphi_{\rho_j}, \varphi_{\rho_k} \rangle$ and the normalized off-diagonal Patterson–Sullivan distributions $\widehat{PS}_{\rho_j,\rho_k}$?
• RQ4 Can the method of stationary phase be adapted to non-Euclidean pseudodifferential operators on symmetric spaces to derive spectral asymptotics?
• RQ5 To what extent do Patterson–Sullivan distributions capture the same quantum ergodicity information as Wigner distributions in symmetric space settings?

Key findings

• The off-diagonal Wigner distribution $\langle \mathrm{Op}(a)\varphi_{\rho_j}, \varphi_{\rho_k} \rangle$ and the normalized off-diagonal Patterson–Sullivan distribution $\widehat{PS}_{\rho_j,\rho_k}$ are asymptotically equivalent as $\rho_k \to \infty$, with an error term of order $O(1/\rho_k)$.
• The leading-order asymptotic expansion of the spectral projection $L_{\rho_k}(\chi a)$ is derived via the method of stationary phase, yielding $C(2\pi/\rho_k)^{s/2} \sum_{n=0}^N \rho_k^{-n} R_{2n}(\chi a) + O(\rho_k^{-N-1+2K})$, where $s = \dim(N)$.
• The normalization factor $\langle 1, PS_{\rho_k,\rho_k} \rangle$ satisfies $C(2\pi/\rho_k)^{s/2} \cdot \langle 1, PS_{\rho_k,\rho_k} \rangle = 1 + O(1/\rho_k)$, ensuring consistency in the normalization of $\widehat{PS}_{\rho_j,\rho_k}$.
• The asymptotic equivalence is expressed as $\left(1 + O(1/\rho_k)\right) \cdot \langle a, \widehat{PS}_{\rho_j,\rho_k} \rangle = \langle a, W_{\rho_j,\rho_k} \rangle + O(1/\rho_k)$, showing that the two distributions are asymptotically indistinguishable.
• The $G$-equivariance of the pseudodifferential calculus ensures that $\Gamma$-invariant symbols on $SX$ induce well-defined operators on the quotient $X_\Gamma = \Gamma\backslash X$, enabling the study of automorphic eigenfunctions.
• The Wigner and Patterson–Sullivan distributions are uniformly bounded, and their asymptotic equivalence implies that the study of one can substitute for the other in quantum ergodicity questions on symmetric spaces.