Growth and nodal current of complexified horocycle eigenfunctions

We study horocycle eigenfunctions at Lobachevsky plane. They are functions $u\colon \mathbb H=\mathbb C^+=\{z\in\mathbb C\colon \Im z>0\}\to\mathbb C$ such that $\left(-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+ 2i\tau y\frac{\partial}{\partial x}\right)u(x+iy)=s^2 u(x+iy)$, $x+iy\in\mathbb C^+$, with $\tau,s\in\mathbb R$, $\tau$ large and $s/\tau$ small. In other words, we study eigenfunctions of magnetic quantum Hamiltonian on hyperbolic plane. By Bohr semiclassical correspondence principle, the asymptotic behavior of such functions is related to horocycle flow on $T\mathbb H$. Let $u^{\mathbb C}$ be analytic continuation of function $u$ to Grauert tube; the latter is an open neighbourhood of $\mathbb H$ in the complexified Lobachevsky plane $\mathbb H^{\mathbb C}$. If a sequence of horocycle functions possesses microlocal quantum ergodicity at the admissible energy level (with $\hbar=1/\tau$) then we may find asymptotic distribution of divisor of $u^{\mathbb C}$. This is done by establishing the asymptotic estimates on $|u^{\mathbb C}|$ in $\mathbb H^{\mathbb C}$. Under imaginary-time horocycle flow, microlocalization of $u$ in $T^*\mathbb H$ is taken to localization of $u^{\mathbb C}$ on $\mathbb H^{\mathbb C}$. The growth of functions $u^{\mathbb C}$ as $\tau\to\infty$ turns to be governed by the growth of complexified gauge factor occurring in $\tau$-automorphic kernels for functions on $\mathbb H$.