Hyperbolic plane wave

Definition

Let $P_b(z)$ be the hyperbolic Poisson kernel of the Poincaré disk, with $b\in S^1$ and $z\in\mathbb D$ . For $r\in\mathbb R$ , the hyperbolic plane wave based at $b$ is

\psi_b^r(z)=P_b(z)^{1/2+ir}.

With the nonnegative Laplacian convention it satisfies

-\Delta\psi_b^r=(r^2+1/4)\psi_b^r.

Incoming and outgoing conventions

For the displayed phase convention, $r>0$ is called outgoing and $r<0$ incoming. Reversing the time or Fourier sign convention swaps these labels.

Boundary synthesis

Generalized eigenfunctions can be synthesized by integrating $\psi_b^r(z)$ against a boundary distribution in $b$ . Incoming and outgoing boundary data are related by an oscillatory integral operator; at high frequency this relation is the Fourier-like transform to which a fractal uncertainty principle applies.

References

Sigurdur Helgason, Groups and Geometric Analysis, AMS, 2000. Publisher record.
Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: §1.6.