On simple and interlaced property of the zeros of two entire functions
On simple and interlaced property of the zeros of two entire functions
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DOI：
10.1112/blms.12768
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发表时间：
2022-12
影响因子：
0.9
通讯作者：
Tao Liu;Guangsheng Wei
中科院分区：
文献类型：
--
作者：
Tao Liu;Guangsheng Wei
This paper is concerned with two pairs (Aj,Bj)$(A_j,B_j)$ , j=0,1$j=0,1$ of entire functions of m$m$ ‐type. We give conditions under which the zeros of A0+tA1$A_0+tA_1$ and B0+tB1$B_0+tB_1$ are real and interlaced for each t∈R$t\in \mathbb {R}$ . This result is used to deal with the inverse problems for the special transmission eigenvalue problem: −u′′+qu=λu$-u^{\prime \prime }+qu=\lambda u$ with u(0)=0=u′(1)sinλλ−u(1)cosλ.$$\begin{equation*}\hspace*{3.4pc} u(0)=0= u^{\prime }(1)\frac{\sin \sqrt {\lambda }}{\sqrt {\lambda }}-u(1)\cos \sqrt {\lambda }. \end{equation*}$$We prove that, if its characteristic function Δ(λ,q)$\Delta (\lambda ,q)$ has only nonreal zeros, then for each t∈R$t\in \mathbb {R}$ , there exists a unique real‐valued function q(·,t)∈L2(0,1)$q(\cdot ,t)\in L^2(0,1)$ such that the corresponding characteristic function of q(·,t)$q(\cdot , t)$ is tΔ(λ,q)$t\Delta (\lambda ,q)$ .
This paper is concerned with two pairs (Aj,Bj)$(A_j,B_j)$ , j=0,1$j=0,1$ of entire functions of m$m$ ‐type. We give conditions under which the zeros of A0+tA1$A_0+tA_1$ and B0+tB1$B_0+tB_1$ are real and interlaced for each t∈R$t\in \mathbb {R}$ . This result is used to deal with the inverse problems for the special transmission eigenvalue problem: −u′′+qu=λu$-u^{\prime \prime }+qu=\lambda u$ with u(0)=0=u′(1)sinλλ−u(1)cosλ.$$\begin{equation*}\hspace*{3.4pc} u(0)=0= u^{\prime }(1)\frac{\sin \sqrt {\lambda }}{\sqrt {\lambda }}-u(1)\cos \sqrt {\lambda }. \end{equation*}$$We prove that, if its characteristic function Δ(λ,q)$\Delta (\lambda ,q)$ has only nonreal zeros, then for each t∈R$t\in \mathbb {R}$ , there exists a unique real‐valued function q(·,t)∈L2(0,1)$q(\cdot ,t)\in L^2(0,1)$ such that the corresponding characteristic function of q(·,t)$q(\cdot , t)$ is tΔ(λ,q)$t\Delta (\lambda ,q)$ .