数論的量子カオスと量子エルゴード性

Transcription

1 $\lambda$ (Shin-ya Koyama) ( (Toyo University))* $\lambdaarrow\infty$ $u_{\lambda}$ 2 ( ) $($ 1900, $)$ $*$ $ $ 2100

2 2 (1915 ) (1956 ) ( $)$ (1980 ) 3 $\lambda$ (1) : $GOE$ - (2) $u_{\lambda}$ If- $L^{\infty}-$ $\lambdaarrow\infty$ ( $L^{\infty}$- ) 2,3 $L^{\infty}-$ (3) $u_{\lambda}(z)$ $ u_{\lambda}(z) $ $z$ $\lambdaarrow\infty$ ) (

3 3 (3 ) (3) ( ) $\infty$ $L$ $L$ $L$ $L$ 2. $M$ ( $)$ $\lambda$, $u_{\lambda}$ $1A,$ $B$ $\lim_{\lambdaarrow\infty}\frac{\int_{a} u_{\lambda}(z) ^{2}\frac{dxdy}{y^{2}}}{\int_{B} u_{\lambda}(z) ^{2}\frac{dxdy}{y^{2}}}=\frac{vo1(A)}{vo1(B)}$ (1) $M$ ( ) $\frac{dxdy}{y^{2}}$ vol(a) $= \int_{a}\frac{dxdy}{y^{2}}$ 1

4 4 (1) $ u_{\lambda}(z) ^{2} \frac{dxdy}{y^{2}}$ $\frac{dxdy}{y^{2}}$ 2 $\lambdaarrow\infty$ $\lambda$ $\lambda$ $\lambda$ $u_{\lambda}$ ( $1O$ ) $\lambda=\frac{1}{4}+r^{2}$ $r$ (1) $\int_{a} u_{\lambda}(z) ^{2}\frac{dxdy}{y^{2}}\sim Cvo1(A)\log r$ $(rarrow\infty, C r )$. (2) (2) $L$- ( 14 ) $H$ $z=x+iy\in H,$ ${\rm Re}(s)>1,$ $\Gamma=SL(2, \mathbb{z})$, $\Gamma_{\infty}=\{\pm(\begin{array}{ll}1 b0 1\end{array}) b\in \mathbb{z}\}\subset\gamma$ $E(z, s)= \sum_{\gamma\in\gamma_{\infty}\backslash \Gamma}{\rm Im}(\gamma z)^{s}$ (3) $E(z, s)$

5 5 $E(z, s)=y^{s}+^{\hat{\zeta}(s-1)}y^{1-s} \hat{\zeta}(s)+\frac{2}{\hat{\zeta}(2s)}\sum_{n=1}^{\infty} n ^{s-\frac{1}{2}}\sigma_{1-2s}(n)e^{2\pi inx}k_{s-\frac{1}{2}}(2\pi n y)\sqrt{y}.$ (4) $\sigma_{s}(n)=\sum_{d n}d^{s}$ $\int_{a} E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}\sim\frac{48}{\pi}vol(A)\log r (rarrow\infty)$ $A$ $M=\Gamma\backslash H$ $A$ $f_{a}(z)$ $\int_{a} E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}=\int_{M}f_{A}(z) E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}$ $f_{a}\in L^{2}(M)$ $\lambdaarrow\infty$ $L^{2}(M)$ ( ) 1 $M=SL(2, \mathbb{z})\backslash H$ ) ( $L^{2}(M)$ $\lim_{rarrow\infty}\int_{m^{u}}j(z) E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}=0$ $J_{j}(r)= \int_{m}uj(z)e(z, \frac{1}{2}+ir)e(z, \frac{1}{2}-ir)\frac{dxdy}{y^{2}}$ (5)

6 6 $I_{j}(s)= \int_{m}u_{j}(z)e(z, \frac{1}{2}+ir)e(z, s)\frac{dxdy}{y^{2}}$. (6) $u_{j}$ (6) $E(z, s)$ (3): $E(z, s)= \sum_{\gamma\in\gamma_{\infty}\backslash \Gamma}{\rm Im}(\gamma z)^{s}$ $M=\Gamma\backslash H$ $H$ $I_{j}(s)= \int_{\gamma_{\infty}\backslash H}u_{j}(z)E(z, \frac{1}{2}+ir)y^{s}\frac{dxdy}{y^{2}}$ $= \int_{0}^{\infty}\int_{0}^{1}u_{j}(z)e(z, \frac{1}{2}+ir)y^{s}\frac{dxdy}{y^{2}}$ (7) $u_{j}(-\overline{z})=u_{j}(z)$ $u_{j}(-\overline{z})=-u_{j}(z)$ $E(z, s)=$ $E(1-\overline{z}, s)$ $u_{j}$ $u_{j}$ $I_{j}(s)\equiv 0$ $e^{2\pi inx}+e^{-2\pi inx}=2\cos(2\pi nx)$ $n$ $-n$ 8 $u_{j}(z)= \sqrt{y}\sum_{n=1}^{\infty}a_{j}(n)k_{ir_{j}}(2\pi n y)\cos(2\pi nx) (a_{j}(1)=1)$ (8) $\frac{1}{4}+r_{j}^{2}=\lambda_{j}$ $L$- $a_{j}(n)$ : $L(s, u_{j})= \sum_{n=1}^{\infty}\frac{a_{j}(n)}{n^{s}}$ $= \prod_{p}(1-\frac{a_{j}(p)}{p^{s}}+\frac{1}{p^{2s}})^{-1}$ (9)

7 7 2 (4) (8) (7) $I_{j}(s)= \int_{0}^{\infty}\int_{0}^{1}(y\sum_{n=1}^{\infty}a_{j}(n)k_{ir_{j}}(2\pi n y)\cos(2\pi nx))$ $(y^{\frac{1}{2}+ir}+y^{\frac{1}{2}-ir} \frac{\hat{\zeta}(ir)}{\hat{\zeta}(1+2ir)}+\frac{2\sqrt{y}}{\hat{\zeta}(1+2ir)}\sum_{m=1}^{\infty}\frac{\sigma_{-2ir}(m)}{m^{-ir}}e^{2\pi imx}k_{ir}(2\pi my))$ $y^{s} \frac{dxdy}{y^{2}}.$ $\int_{0}^{1}\cos(2\pi nx)dz=\{\begin{array}{ll}0 (n\neq 0)1 (n=0),\end{array}$ $\cos\alpha\cos\beta=\frac{1}{2}(\cos(\alpha+\beta)+\cos(\alpha-\beta))$ $n=m$ $ny\mapsto y$ $I_{j}(s)= \frac{2}{\hat{\zeta}(1+2ir)}(\sum_{n=1}^{\infty}\frac{\sigma_{-2ir}(n)a_{j}(n)}{n^{s-ir}})\int_{0}^{\infty}k_{ir}(2\pi y)k_{ir_{j}}(2\pi y)y^{s}\frac{dy}{y}$ $\int_{0}^{\infty}k_{ir}(2\pi y)k_{ir_{j}}(2\pi y)y^{s}\frac{dy}{y}=\frac{\gamma(\frac{s+ir_{j}+ir}{2})\gamma(\frac{s+ir_{j}-ir}{2})\gamma(\frac{s-ir_{j}+ir}{2})\gamma(\frac{s-ir_{j}-ir}{2})}{\pi^{s}\gamma(s)}$ $R(s)= \sum_{n=1}^{\infty}\frac{\sigma_{-2ir}(n)a_{j}(n)}{n^{s-ir}}$ $I_{j}(s)= \frac{2\pi^{-s}}{\hat{\zeta}(1+2ir)}\cross\frac{\gamma(\frac{s+ir_{j}+ir}{2})\gamma(\frac{s+ir_{j}-ir}{2})\gamma(\frac{s-ir_{j}+ir}{2})\gamma(\frac{s-ir_{j}-ir}{2})}{\gamma(s)}r(s)$

8 8 $R(s)$ : $J_{j}(r)=I_{j}( \frac{1}{2}-ir)$ $= \frac{2\pi^{-\frac{1}{2}+ir}\gamma(\frac{\frac{1}{2}+ir_{j}}{2})\gamma(\frac{\frac{1}{2}+ir_{j}-2ir}{2})\gamma(\frac{\frac{1}{2}-ir_{j}}{2})\gamma(\frac{\frac{1}{2}-ir_{j}-2ir}{2})l(\frac{1-2ir}{2},u_{j})l(\frac{1}{2},u_{j})}{\hat{\zeta}(1+2ir)\gamma(\frac{1}{2}-ir)\zeta(1-2ir)}.$ (11) $ \Gamma(\sigma+ir) \sim e^{-\pi r/2} r ^{\sigma-\frac{1}{2}} (rarrow\infty)$ (11) $=O( r ^{-1/2})$ (12) $\frac{1}{\zeta(1+2ir)}=o(\log r)$ (13)

9 9 (11) $L( \frac{1}{2}+ir, uj)$ (12), (13) $J_{j}(r)=O( \frac{l(\frac{1}{2}+ir,u_{j})}{ r ^{\frac{1}{2}}}) (rarrow\pm\infty)$ (14) $L$ $L( \frac{1}{2}+\dot{\iota}r, u_{j})=o( r ^{\frac{1}{2}}) (rarrow\pm\infty)$ $L( \frac{1}{2}+\dot{\iota}r, uj)=o( r ^{\frac{1}{2}-\delta}) (rarrow\pm\infty)$ $\delta>0$ (14) $J_{j}(r)=O( r ^{-\delta}) (rarrow\pm\infty)$ $\lim_{rarrow\pm\infty}j_{j}(r)=0$ $L( \frac{1}{2}+\dot{\iota}r, u_{j})=o( r ^{\frac{1}{3}+\epsilon}) (\forall\epsilon>0)$ (15) ( ) $h(y)$ $\infty$ $0$ $y$ $0$ $\infty$ $h(y)=o_{n}(y^{n})$ $(N\in \mathbb{z})$ ( $O_{N}$ $O$ $N$ ) $h(y)$ $y$ $N$ $H(s)= \int_{0}^{\infty}h(y)y^{-s}\frac{dy}{y}$ $h(y)$ $H(s)$ $s$ $r$

10 10 $\sigma+ir$ $\sigma\in \mathbb{r}$ $h(y)= \frac{1}{2\pi i}\int_{(\sigma)}h(s)y^{s}ds$ $\int_{(\sigma)}$ ${\rm Re}(s)=\sigma$ $h$ $F_{h}(z)= \sum_{\gamma\in\gamma_{\infty}\backslash \Gamma}h({\rm Im}(\gamma z))$ $h(y)=y^{s}$ (3) $1^{\lambda}$ $h(y)=o_{n}(y^{n})$. 2 $L^{2}(M)$ 2 $yarrow 0,$ $h(y)$ $\infty$ $yarrow 0,$ $\infty$ $E(z, s)$ $F_{h}(z)$ $E(z, s)$ $(\sigma)arrow(2)$ $F_{h}(z)= \frac{1}{2\pi i}\int_{(2)}h(s)e(z, s)ds$ 2 $M=SL(2, \mathbb{z})\backslash H$ $F(z)$ $rarrow\infty$ $\int_{m}f(z) E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}\sim\frac{48}{\pi}(\int_{M}F(z)\frac{dxdy}{y^{2}})\log r$ $\infty$ $C^{\infty}(M)$

11 11 $\int_{m}f_{h}(z) E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}$ $= \frac{1}{2\pi i}\int_{m}\int_{(2)}h(s)e(z, s)ds E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}$ $= \frac{1}{2\pi i}\int_{0}^{\infty}\int_{(2)}h(s)y^{s}ds\int_{0}^{1} E(z, \frac{1}{2}+ir) ^{2}\frac{dxdy}{y^{2}}$ $= \frac{1}{2\pi i}\int_{0}^{\infty}\int_{(2)}h(s)y^{s}ds( y^{\frac{1}{2}+ir}+y^{\frac{1}{2}-ir}\frac{\hat{\zeta}(2ir)}{\hat{\zeta}(1+2ir)} ^{2}$ $+ \frac{2y}{\hat{\zeta}(1+2ir)} ^{2}\sum_{n=1}^{\infty} \sigma_{-2ir}(n)k_{ir}(2\pi ny) ^{2})\frac{dy}{y^{2}}$ $=F_{1}(r)$ $F_{2}(r)$. $F_{1}(r)= \frac{1}{2\pi i}\int_{0}^{\infty}$ (2) $H(s)y^{s}ds y^{\frac{1}{2}+ir}+y^{\frac{1}{2}-ir} \frac{\hat{\zeta}(2ir)}{\hat{\zeta}(1+2ir)} ^{2}\frac{dy}{y^{2}}$ $ \frac{\hat{\zeta}(2ir)}{\hat{\zeta}(1+2ir)} =1$ $F_{1}(r)=2 \int_{0}^{\infty}h(y)\frac{dy}{y}+$ ( ) (16) $r$ $F_{2}(r)= \frac{2}{\pi i \hat{\zeta}(1+2ir) ^{2}}$ (2) $H(s) \sum_{n=1}^{\infty}\frac{ \sigma_{-2ir}(n) ^{2}}{n^{s}}\int_{0}^{\infty} K_{ir}(2\pi y) ^{2}y^{s}\frac{dy}{y}ds.$ (17)

12 12 : $\sum_{n=1}^{\infty}\frac{ \sigma_{a}(n) ^{2}}{n^{s}}=\prod_{p}\sum_{k=0}^{\infty}\frac{\sigma_{a}(p^{k})\sigma_{-a}(p^{k})}{p^{ks}}$ $= \prod_{p}\sum_{k=0}^{\infty}\frac{1}{p^{ks}}(\frac{1-p^{a(k+1)}}{1-p^{a}})(\frac{1-p^{-a(k+1)}}{1-p^{-a}})^{2}$ $= \prod_{p}\frac{1}{(1-p^{a})(1-p^{-a})}\sum_{k=0}^{\infty}(2p^{-ks}-p^{(a-s)k+a}+p^{(-a-s)k-a})$ $= \prod_{p}\frac{1}{(1-p^{a})(1-p^{-a})}(\frac{2}{1-p^{-s}}-\frac{p^{a}}{1-p^{a-s}}-\frac{p^{-a}}{1-p^{-a-s}})$ $= \prod_{p}\frac{1+p^{-s}}{(1-p^{-s})(1-p^{-(s-a)})(1-p^{-(s+a)})}$ $= \prod_{p}\frac{1-p^{-2s}}{(1-p^{-s})^{2}(1-p^{-(s-a)})(1-p^{-(s+a)})}$ $= \frac{\zeta(s)^{2}\zeta(s-a)\zeta(s+a)}{\zeta(2s)}$. (18) $\Gamma$ (17) $y\}$ $F_{2}(r)= \frac{2}{\pi i \hat{\zeta}(1+2ir) ^{2}}$ (2) $H(s) \sum_{n=1}^{\infty}\frac{ \sigma_{-2ir}(n) ^{2}}{n^{s}}\int_{0}^{\infty} K_{ir}(2\pi y) ^{2}y^{s}\frac{dy}{y}ds$ $= \frac{2}{\pi i \hat{\zeta}(1+2ir) ^{2}}$ $\cross\int_{(2)}\frac{h(s)\zeta(s)^{2}\zeta(s+2ir)\zeta(s-2ir)\gamma(\frac{s}{2}+ir)\gamma(\frac{s}{2}-ir)\gamma(\frac{s}{2})^{2}}{\pi^{s}\zeta(2s)\gamma(s)}ds$ $= \frac{2}{\pi i \hat{\zeta}(1+2ir) ^{2}}\int_{(2)}B(s)ds$ (19) $B(s)= \frac{h(s)\zeta(s)^{2}\zeta(s+2ir)\zeta(s-2ir)\gamma(\frac{s}{2}+ir)\gamma(\frac{s}{2}-ir)\gamma(\frac{s}{2})^{2}}{\pi^{s}\zeta(2s)\gamma(s)}$ (20) $\Gamma$ ${\rm Re}(s)=1/2$ $H(\sigma+ir)$ $r$ (19) $s=1$

13 13 $F_{2}(r)= \frac{4{\rm Res}_{s=1}B(s)}{ \hat{\zeta}(1+2ir) ^{2}}+\frac{2}{\pi i \hat{\zeta}(1+2ir) ^{2}}\int_{(1/2)}B(s)ds+O(r^{-1})$. (21) $O(r^{-1})$ $s=1\pm 2ir$ $B(s)$ $tarrow\infty$ (21) $\zeta(\frac{1}{2}+ir)=o(r^{\frac{1}{6}+\epsilon})$ $,$ $B(s)$ $\zeta(s+2ir)\zeta(s-2ir)$ $\frac{2}{\pi i \hat{\zeta}(1+2ir) ^{2}}\int_{(1/2)}B(s)ds=O((r^{\frac{1}{3}+\epsilon})^{2}r^{-1/2})=O(r^{-\frac{1}{6}+\epsilon})$ $\epsilon$ ( ). (21) $s=1$ $G(s)= \frac{h(s)\zeta(s+2ir)\zeta(s-2ir)\gamma(\frac{s}{2}+ir)\gamma(\frac{s}{2}-ir)\gamma(\frac{s}{2})^{2}}{\pi^{s}\zeta(2s)\gamma(s)}$ $B(s)=\zeta(s)^{2}G(s)$ $2_{\gamma}$ $\zeta(s)$ $sarrow 1$ $\zeta(s)=\frac{1}{s-1}+\gamma+o(s-1) (sarrow 1)$. $B(s)$ $B(s)=( \frac{1}{s-1}+\gamma+o(s-1))^{2}(g(1)+g (1)(s-1)+O(s-1)^{3})$ $(s-1)^{-1}$ ${\rm Res}_{s=1}B(s)=2G(1)\gamma+G (1)$ $G$ ${\rm Res}_{s=1}B(s)=G(1)(2 \gamma+\frac{g }{G}(1))$ (22) 2 $\gamma=\lim_{narrow\infty}(1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n)= \cdots$

14 14 $\frac{g }{G}(1)=\frac{H }{H}(1)+\frac{\zeta (1+2ir)}{\zeta(1+2ir)}+\frac{\zeta (1-2ir)}{\zeta(1-2ir)}$ $+ \frac{\gamma (\frac{1}{2}+ir)}{\gamma(\frac{1}{2}+ir)}+\frac{\gamma (\frac{1}{2}-ir)}{\gamma(\frac{1}{2}-ir)}+c.$ $C$ $r$ - $\triangleright\grave{}$ $\frac{\zeta (1+2ir)}{\zeta(1+2ir)}=O(\frac{\log r}{\log\log r})$ $\frac{\gamma }{\Gamma}(\frac{1}{2}+\dot{\iota}r)\sim\log r$ (22) $2\log r$ $G(1)= \frac{h(1) \zeta(1+2ir)\gamma(\frac{1}{2}+ir) ^{2}\Gamma(\frac{1}{2})^{2}}{\pi\zeta(2)}$ $= \frac{h(1)\pi \hat{\zeta}(1+2ir) ^{2}}{\zeta(2)}$ $= \frac{6}{\pi}h(1) \hat{\zeta}(1+2ir) ^{2}$ ${\rm Res}_{s=1}B(s)= \frac{6}{\pi}h(1) \hat{\zeta}(1+2ir) ^{2}(2\log r+o(\frac{\log r}{\log\log r}))$ (21) $\frac{4{\rm Res}_{s=1}B(s)}{ \hat{\zeta}(1+2ir) ^{2}}=\frac{48H(1)}{\pi}\log r+o(1)$. $H(1)= \int_{0}^{\infty}h(y)\frac{dy}{y^{2}}=\int_{m}f_{h}(z)\frac{dxdy}{y^{2}}$ ( )

15 15 3 $F$ $M$ $\int_{m}f(z)d\mu_{r}(z)\sim\frac{48}{\pi}(\int_{m}f(z)\frac{dxdy}{y^{2}})\log r (rarrow\infty)$. $O$ $F$ $\epsilon>0$ $\Vert G-F\Vert_{\infty}<\epsilon$ $G$ $G=G_{1}+G_{2}$ $G_{1}$ $G_{2}$ $G_{1}$ 1 $rarrow\infty$ $G_{2}$ 2 $H=G-F$ $rarrow\infty$ ( ) 1 $SL(2, \mathbb{z})$ $A$ $f_{a}$ $F(z)$ ( ) 1995 W. Luo and P. Sarnak: Quantum ergodicity of Eigenfunctions on $PSL_{2}(\mathbb{Z})/H_{2}$ Publications Mathematiques de L IHES 81 (1995) ( 14 ) 1

16 16 3. $\Gamma_{j}(j=1,2,3, \ldots)$ $SL(2, \mathbb{r})$ $H=\{x+iy y>0\}$ $M_{j}=\Gamma_{j}\backslash H$ $M_{j}$ $\varphi_{j}:m_{j}arrow M_{j+1}$ $f_{j}:m_{j}arrow \mathbb{c}$ $M_{j}$ $d\mu j$ $d \mu_{j}:= f_{j}(z) ^{2}dz, dz=\frac{dxdy}{y^{2}}$ 1( ) $f_{j}:m_{j}arrow \mathbb{c}$ (equidistributed) $A_{1},$ $B_{1}\subset M_{1}$ $\lim_{jarrow\infty}\frac{\int_{a_{j}}d\mu_{j}}{\int_{b_{j}}d\mu_{j}}=\frac{\int_{a_{1}}dz}{\int_{b_{1}}dz}$ $A_{j}=\varphi_{j-1}0\varphi_{j-2}\circ\cdots\circ\varphi_{1}(A_{1})$ 1

17 17 1( (Luo-Sarnak[4] 1995) ) $M_{j}=SL(2, \mathbb{z})\backslash H(\forall j=1,2,3, \ldots),$ $\varphi j$ $E(z, s)$ $SL(2, \mathbb{z})$ $\in \mathbb{r}$ $f_{j}(z)=e(z, \frac{1}{2}+it_{j})$ Koyama[1] 3 Truelsen [6] 2 ( (Lindenstrauss[3], Soundararajan[5]) ) 1 $M_{j},$ $M_{j}$ $\varphi J$ $0=\lambda_{0}<\lambda_{1}\leq$ $\lambda_{2}\leq\cdots$ $\lambda_{j}$ $f_{j}(z)(\vert f_{j}\vert_{2}=1)$ $f_{j}(z)$ $M$ ( Lindenstrauss Soundararajan 1 ) 3 ( (Koyama[2] 2009) $q_{1}=1$ $qj(j=2,3, \ldots)$ $M_{j}=\Gamma_{0}(qj)\backslash H$ $\pi J$ : $M_{j}arrow M_{1}$ $\psi_{j}$ : $M_{1}arrow M_{j+1}$ $\varphi_{j}:m_{j}arrow^{\pi_{j}}m_{1}arrow^{\psi_{j}}m_{j+1}$ $t\in \mathbb{r}$ $E_{q_{j},\nu_{j}}(z, s)$ $f_{j}(z)=e_{q_{j},\nu_{j}}(z, \frac{1}{2}+it)$ $\Gamma_{0}(qj)$ $vj$ $v_{j}$ 3 2 (S. Koyama and S. Nak jima) $q_{j}=j(j=2,3, \ldots)$ $M_{j}=\Gamma_{0}(qj)\backslash H$ $\psi_{j}$ : $\pi j$ : $M_{1}arrow M_{j+1}$ $M_{j}arrow M_{1}$ $\varphi_{j}:m_{j}arrow^{\pi_{j}}m_{1}arrow^{\psi_{j}}m_{j+1}$ $t\in \mathbb{r}$ $E_{q_{j},\nu_{j}}(z, s)$ $f_{j}(z)=e_{q_{j},\nu_{j}}(z, \frac{1}{2}+it)$ $\Gamma_{0}(qj)$ $\nu j$ $vj$

18 ( ) [1] S. Koyama: Quantum ergodicity of Eisenstein series for arithmetic 3-manifolds. Communications in Mathematical Physics 215 (2000), no. 2, [2] S. Koyama: Equidistribution of Eisenstein series in the level aspect. Commumications in Mathematical Physics 289 (2009), no. 3, [3] E. Lindenstrauss: Invariant measures and arithmetic quantum unique ergodicity. Annals of Mathematics 163 (2006) no. 1, [4] L. Wen Zhi and P. Sarnak: Quantum ergodicity $PSL_{2}(\mathbb{Z})\backslash H^{2}$ of eigenfunctions on. Inst. Hautes Etudes Sci. Publ. Math. 81 (1995) [5] K. Soundararajan: Quantum unique ergodicity $SL2(\mathbb{Z})\backslash H$ for. Annals of Mathematics 172 (2010) no. 2, [6] J.L. Truelsen: Quantum unique ergodicity of Eisenstein series on the Hilbert modular group over a totally real field. Forum Math. 23 (2011), no. 5.