The estimation of the ratio of two entire functions with the same zeros in the ball

TL;DR

This paper establishes bounds on the ratio of two entire functions with the same zeros, providing insights into their stability, which is relevant for inverse resonance problems in quantum mechanics.

Contribution

It introduces a new estimate for the ratio of entire functions with shared zeros, advancing understanding of their stability in inverse resonance problems.

Findings

The ratio of two such functions can be tightly bounded within a specific domain.

The bounds depend on the zeros' coincidence within a certain radius.

Results have implications for stability analysis in inverse resonance problems.

Abstract

The paper studies entire functions of finite order of growth for which a representation of the form $\psi(z) = 1+ O(|z|^{-\mu}), \mu >0,$ as $z\to \infty$, is valid on a fixed ray of the complex plane. The main result is the following. Assume that the zeros of two functions $\psi_1, \psi_2$ of this class coincide in the circle of radius $R$ with the center in zero. Then given arbitrary small $\delta\in (0,1)$ and $\varepsilon >0$ the relation of these functions admits the estimate $|\psi_1(z)/\psi_2(z) -1| \leqslant \varepsilon R^{-\mu(1-\delta)}$ for all $|z|\leqslant R^{1-\delta}$, provided that $R\geqslant R_0$ and $R_0 =R_0(\varepsilon, \delta)$ is sufficiently large. This result is of considerable interest in the analysis of the stability in the inverse resonance problem for the Schroedinger equation.