# At Most Two Radii Theorem For A Real Eigenvalue Of The Hyperbolic Laplacian

## TL;DR

This paper investigates the relationship between eigenvalues of the hyperbolic Laplacian and the average values of their eigenfunctions over spheres, providing conditions for unique identification of eigenvalues based on these averages.

## Contribution

It introduces a method to uniquely determine real eigenvalues of the hyperbolic Laplacian using sphere averages of eigenfunctions, with specific radius conditions.

## Key findings

- Eigenvalues can be identified from sphere averages when the average is large enough.
- Additional average computations are necessary when the initial average is zero.
- The results apply to hyperbolic spaces of any dimension with constant negative curvature.

## Abstract

We study a $(k+1)$-dimensional hyperbolic space of a negative constant sectional curvature $\kappa=-1/\rho^2$. Let $\lambda$ be a real eigenvalue and $f_{\lambda} (x)$ be an eigenfunction of the hyperbolic Laplacian assuming a non-zero value at $x_0$. Then the average value of $f_{\lambda}(x)$ over any sphere centered at $x_0$ allows to identify the corresponding eigenvalue $\lambda$ uniquely as long as that average value is large enough. Otherwise, to identify the corresponding eigenvalue uniquely, we need to make sure that the computed average value is not zero and then we need to compute an additional average value of $f_{\lambda}(x)$ over a small enough sphere centered at the same point $x_0$.