论文标题
复杂的震环素本征的生长和淋巴结电流
Growth and nodal current of complexified horocycle eigenfunctions
论文作者
论文摘要
我们研究了Lobachevsky平面的肌醇本征函数。这些是函数$ u \ colon \ mathbb h = \ mathbb c^+= \ {z \ in \ mathbb c \ colon \ colon \ im z> 0 \} \ to \ mathbb c $ x^2}+\ frac {\ partial^2} {\ partial y^2} \ right）+2iτy\ frac {\ partial} {\ partial x} \ partial x} \ right） r $，$τ$大，$ s/τ$小。换句话说，我们研究\ emph {磁}量子哈密顿量在双曲平面上的特征。根据半经典的对应原理，这种功能的渐近行为与$ t \ mathbb h $上的Horocycle流有关。如果一系列的烟叶函数在可允许的能级（$ \ hbar = 1/τ$）上具有微关量量子独特的牙齿性，那么我们可能会发现$ u $的除数的渐近分布在分析上继续延续到复杂的lobachevsky plane $ \ mathbb h^\ mathbb h^\ mathbb c $。这是通过建立$ | u | $ in $ \ mathbb h^\ mathbb c $的渐近估计来完成的。功能$ u $的增长为$τ\ to \ infty $转弯，由$τ$ -automorphic beans in $ \ mathbb h $上的$τ$ -automorphic beannels中的复杂化\ emph {gauge因子}的增长所支配。
We study horocycle eigenfunctions at Lobachevsky plane. These are functions $u\colon \mathbb H=\mathbb C^+=\{z\in\mathbb C\colon \Im z>0\}\to\mathbb C$ such that $\left(-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+ 2iτy\frac{\partial}{\partial x}\right)u(x+iy)=s^2 u(x+iy)$, $x+iy\in\mathbb C^+$, with $τ,s\in\mathbb R$, $τ$ large and $s/τ$ small. In other words, we study eigenfunctions of \emph{magnetic} quantum Hamiltonian on hyperbolic plane. By semiclassical correspondence principle, asymptotic behavior of such functions is related to horocycle flow on $T\mathbb H$. If a sequence of horocycle functions possesses microlocal quantum unique ergodicity at the admissible energy level (with $\hbar=1/τ$) then we may find asymptotic distribution of divisor of $u$ analytically continued to the complexified Lobachevsky plane $\mathbb H^\mathbb C$. This is done by establishing the asymptotic estimates on $|u|$ in $\mathbb H^\mathbb C$. The growth of functions $u$ as $τ\to\infty$ turns to be governed by the growth of complexified \emph{gauge factor} occurring in $τ$-automorphic kernels for functions on $\mathbb H$.
