One Radius Theorem For A Radial Eigenfunction Of A Hyperbolic Laplacian

TL;DR

This paper establishes a lower bound for the interval where two radial eigenfunctions of a hyperbolic Laplacian with the same initial value differ, revealing insights into their uniqueness and the eigenvalues' influence.

Contribution

It provides a new estimate for the interval where radial eigenfunctions with the same initial value must differ, linking this to their eigenvalues and demonstrating the abundance of eigenfunctions with the same initial value.

Findings

Derived a lower bound for the interval where eigenfunctions differ

Showed the dependence of the bound on eigenvalues

Proved the existence of infinitely many eigenfunctions with the same initial value

Abstract

Let us fix two different radial eigenfunctions of a hyperbolic Laplacian and assume that both of them have the same value at the origin. Both eigenvalues can be complex numbers. The main goal of this paper is to estimate the lower bound for the interval (0,T], where these two eigenfunctions must assume different values at every point. We shall see that T is a function of two different eigenvalues corresponding to the given pair of radial eigenfunctions. On the other hand, we shall see that at every fixed point and for the value already assumed by a radial eigenfunction at the fixed point, there are infinitely many other radial eigenfunctions, assuming the same value at the fixed point and satisfying the same initial condition.