Growth and nodal current of complexified horocycle eigenfunctions
Mikhail Dubashinskiy
TL;DR
This work extends semiclassical analysis of eigenfunctions from geodesic to horocycle flows on the hyperbolic plane by studying complexified horocycle eigenfunctions $u^{\mathbb C}$ on horocycle Grauert tubes $\mathcal G_1$. It constructs a Fay-type automorphic kernel to analytically continue horocycle eigenfunctions and shows that, under quantum ergodicity with $\hbar=1/\tau$, their complex growth obeys a precise exponential law governed by the gauge factor $B_0$, yielding a weak-* limit for $|u^{\mathbb C}|^2$ and a Lelong–Poincaré formula for nodal currents. A key innovation is the reduction to a semiclassical pseudodifferential operator on the energy surface $\{H_1=1/2\}$, enabling slice-wise (on $\Sigma_t$) and tube-wide asymptotics and a robust logarithmic growth description. These results parallel Zelditch’s geodesic-case theory but adapt to horocycle flow, providing microlocal quantum ergodicity implications for complexified horocycle eigenfunctions and their nodal geometry on hyperbolic spaces.
Abstract
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 $\left(-y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+ 2iτy\frac{\partial}{\partial x}\right)u(x+iy)=s^2 u(x+iy)$, $x+iy\in\mathbb C^+$, with $τ,s\in\mathbb R$, $τ$ large and $s/τ$ small. In other words, we study eigenfunctions of 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 $\hbar=1/τ$) 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 $τ\to\infty$ turns to be governed by the growth of complexified gauge factor occurring in $τ$-automorphic kernels for functions on $\mathbb H$.