Estimate of the Ratio of Two Entire Functions whose Zeros Coincide in the Disk

V. L. Geynts ; A. A. Shkalikov

Matematičeskie zametki, Tome 99 (2016) no. 6, pp. 887-896.

Résumé

We study entire functions of finite growth order that admit the representation $\psi(z) = 1+ O(|z|^{-\mu})$, $\mu >0$, on a ray in the complex plane. We obtain the following result: if the zeros of two functions $\psi_1$, $\psi_2$ of such class coincide in the disk of radius $R$ centered at zero, then, for any arbitrarily small $\delta\in (0,1)$, $\varepsilon >0$, the ratio of these functions in the disk of radius $R^{1-\delta}$ admits the estimate $|\psi_1(z)/\psi_2(z) -1| \le \varepsilon R^{-\mu(1-\delta)}$ if $R\ge R_0(\varepsilon, \delta)$. The obtained results are important for stability analysis in the problem of the recovery of the potential in the Schrödinger equation on the semiaxis from the resonances of the operator.

Détail

Keywords: entire function of finite order, Hadamard theorem, Schrödinger operator, resonances of the Schrödinger operator, Jost function.

Bibliographie

[1] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR | Zbl

[2] M. Marletta, R. Shterenberg, R. Weikard, “On the inverse resonance problem for Schrödinger operators”, Comm. Math. Phys., 295:2 (2010), 465–484 | DOI | MR

[3] B. Ya. Levin, Lectures on Entire Functions, Trans. Math. Monogr., 150, Amer. Math. Soc., Providence, RI, 1996 | MR | Zbl