Matching Tag: quantum-unique-ergodicity
Let Γ ⊆ SL 2 ( Z ) \Gamma\subseteq \operatorname{SL}_2(\operatorname{\mathbb{Z}}) Γ ⊆ SL 2 ( Z ) f ∈ S k ( Γ ) f\in S_k(\Gamma) f ∈ S k ( Γ ) μ f \mu_f μ f
Let ψ : R + → R \psi\colon \mathbb{R}^+ \to \mathbb{R} ψ : R + → R f ( z ) f(z) f ( z ) L 2 L^2 L 2 k k k
Holomorphic QUE conjecture. Fix a bounded continuous function ϕ \phi ϕ Sp 2 n ( Z ) \ H n \operatorname{Sp}_{2n}(\mathbb Z)\backslash\mathbb H_n Sp 2 n ( Z ) \ H n F ∈ S k ( Sp 2 n ( Z ) ) F\in S_k(\operatorname{Sp}_{2n}(\mathbb Z)) F ∈ S k ( Sp 2 n ( Z ))
Let X : = H / S L ( 2 , Z ) \mathfrak{X}:=\mathcal{H}/SL(2,\mathbb{Z}) X := H / S L ( 2 , Z ) Δ \Delta Δ ϕ i \phi_i ϕ i L 2 L^2 L 2
Holomorphic mass equidistribution conjecture. For every such Ψ \Psi Ψ
Let u j u_j u j 1 / 4 + T j 2 1/4+T_j^2 1/4 + T j 2 λ u j ( m ) \lambda_{u_j}(m) λ u j ( m ) u j ( − x + i y ) = δ j u j ( x + i y ) u_j(-x+iy)=\delta_j u_j(x+iy) u j ( − x + i y ) = δ j u j ( x + i y ) δ j ∈ { − 1 , 1 } \delta_j\in\{-1,1\} δ j ∈ { − 1 , 1 }
Fix n ∈ Z ⩾ 0 n\in\mathbb{Z}_{\geqslant 0} n ∈ Z ⩾ 0 N ⩾ n N\geqslant n N ⩾ n Y − N . . N \mathbf{Y}_{-N..N} Y − N .. N F − N . . N \mathcal{F}_{-N..N} F − N .. N
Let f f f k k k q q q d ν f = ∣ f ( z ) ∣ 2 y k − 2 d x d y d\nu_f=|f(z)|^2y^{k-2}\,dx\,dy d ν f = ∣ f ( z ) ∣ 2 y k − 2 d x d y Y 0 ( q ) = Γ 0 ( q ) \ H Y_0(q)=\Gamma_0(q)\backslash\mathbb H Y 0 ( q ) = Γ 0 ( q ) \ H
Let X X X ρ m \rho_m ρ m X × K ρ m X\times_K\rho_m X × K ρ m { s n } \{s_n\} { s n } L 2 L^2 L 2
Let ( M , g ) (M,g) ( M , g ) ( ψ j ) j ≥ 1 (\psi_j)_{j\geq1} ( ψ j ) j ≥ 1
Let Φ H t \Phi^t_H Φ H t E E \mathcal{E}_E E E H ^ ℏ \hat H_\hbar H ^ ℏ ψ ℏ , j \psi_{\hbar,j} ψ ℏ , j
Let H \mathbb{H} H d μ z = y − 2 d x d y d\mu z=y^{-2}dxdy d μ z = y − 2 d x d y Γ = SL 2 ( Z ) \Gamma=\operatorname{SL}_2(\mathbb{Z}) Γ = SL 2 ( Z ) X = Γ \ H X=\Gamma\backslash\mathbb{H} X = Γ\ H
