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Michael Hardy
Michael Hardy
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Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} dx dy$$dV(z)=y^{-2} \,dx \,dy$ be the volume element of the upper half plane $H^2$. In analogy with quantum mechanics, Luo-Sarnak defined the measure $\mu_t=|E(z,1/2+it)|^2 dV(z)$$\mu_t=|E(z,1/2+it)|^2 \,dV(z)$ and showed that it fulfills \begin{equation} \lim_{t\to\infty} \frac{\mu_t(K_1) }{\mu_t(K_2)}=\frac{Vol(K_1)}{Vol(K_2)} \end{equation}\begin{equation} \lim_{t\to\infty} \frac{\mu_t(K_1) }{\mu_t(K_2)}=\frac{\operatorname{Vol}(K_1)}{\operatorname{Vol}(K_2)} \end{equation} for compact, Jordan-measurable subsets $K_1,K_2$ of $X$. In analogy with the case of compact manifolds they called it quantum ergodicity. Another piece of work by Koyama, Sarnak and Petridis showed that this is also the case for certain arithmetic 3-manifolds (e.g. $X=PSL_2(\mathcal{O}_K)\setminus H^3$, where $\mathcal{O}_K$ is the integer ring of an imaginary quadratic field $K$ of class number one).

Now I am wondering how much is known for general arithmetic quotients of n-dimensional hyperbolic space or if the knowledge about these kind of examples ends with dimension 3. I am definitely grateful for any information on the current state of the matter and further references!

Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} dx dy$ be the volume element of the upper half plane $H^2$. In analogy with quantum mechanics, Luo-Sarnak defined the measure $\mu_t=|E(z,1/2+it)|^2 dV(z)$ and showed that it fulfills \begin{equation} \lim_{t\to\infty} \frac{\mu_t(K_1) }{\mu_t(K_2)}=\frac{Vol(K_1)}{Vol(K_2)} \end{equation} for compact, Jordan-measurable subsets $K_1,K_2$ of $X$. In analogy with the case of compact manifolds they called it quantum ergodicity. Another piece of work by Koyama, Sarnak and Petridis showed that this is also the case for certain arithmetic 3-manifolds (e.g. $X=PSL_2(\mathcal{O}_K)\setminus H^3$, where $\mathcal{O}_K$ is the integer ring of an imaginary quadratic field $K$ of class number one).

Now I am wondering how much is known for general arithmetic quotients of n-dimensional hyperbolic space or if the knowledge about these kind of examples ends with dimension 3. I am definitely grateful for any information on the current state of the matter and further references!

Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} \,dx \,dy$ be the volume element of the upper half plane $H^2$. In analogy with quantum mechanics, Luo-Sarnak defined the measure $\mu_t=|E(z,1/2+it)|^2 \,dV(z)$ and showed that it fulfills \begin{equation} \lim_{t\to\infty} \frac{\mu_t(K_1) }{\mu_t(K_2)}=\frac{\operatorname{Vol}(K_1)}{\operatorname{Vol}(K_2)} \end{equation} for compact, Jordan-measurable subsets $K_1,K_2$ of $X$. In analogy with the case of compact manifolds they called it quantum ergodicity. Another piece of work by Koyama, Sarnak and Petridis showed that this is also the case for certain arithmetic 3-manifolds (e.g. $X=PSL_2(\mathcal{O}_K)\setminus H^3$, where $\mathcal{O}_K$ is the integer ring of an imaginary quadratic field $K$ of class number one).

Now I am wondering how much is known for general arithmetic quotients of n-dimensional hyperbolic space or if the knowledge about these kind of examples ends with dimension 3. I am definitely grateful for any information on the current state of the matter and further references!

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Claudius
Claudius
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Quantum ergodicity of Eisenstein series on arithmetic quotients of hyperbolic space

Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} dx dy$ be the volume element of the upper half plane $H^2$. In analogy with quantum mechanics, Luo-Sarnak defined the measure $\mu_t=|E(z,1/2+it)|^2 dV(z)$ and showed that it fulfills \begin{equation} \lim_{t\to\infty} \frac{\mu_t(K_1) }{\mu_t(K_2)}=\frac{Vol(K_1)}{Vol(K_2)} \end{equation} for compact, Jordan-measurable subsets $K_1,K_2$ of $X$. In analogy with the case of compact manifolds they called it quantum ergodicity. Another piece of work by Koyama, Sarnak and Petridis showed that this is also the case for certain arithmetic 3-manifolds (e.g. $X=PSL_2(\mathcal{O}_K)\setminus H^3$, where $\mathcal{O}_K$ is the integer ring of an imaginary quadratic field $K$ of class number one).

Now I am wondering how much is known for general arithmetic quotients of n-dimensional hyperbolic space or if the knowledge about these kind of examples ends with dimension 3. I am definitely grateful for any information on the current state of the matter and further references!