In this paper, we consider the operator L generated in L2(R+)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$L_{2}(\mathbb{R}_{+})$\end{document} by the Sturm-Liouville equation −y″+q(x)y=λ2y\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$-y^{\prime\prime}+q(x)y=\lambda^{2}y$\end{document}, x∈R+=[0,∞)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$x\in \mathbb{R}_{+}= [ 0,\infty ) $\end{document}, and the boundary condition (α0+α1λ+α2λ2)y′(0)−(β0+β1λ+β2λ2)y(0)=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$( \alpha_{0}+\alpha_{1}\lambda+\alpha_{2}\lambda^{2} ) y^{\prime} ( 0 ) - ( \beta_{0}+\beta_{1}\lambda +\beta_{2}\lambda^{2} ) y ( 0 ) =0$\end{document}, where q is a complex-valued function, αi,βi∈C\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\alpha_{i},\beta_{i}\in\mathbb{C}$\end{document}, i=0,1,2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$i=0,1,2$\end{document}, and λ is an eigenparameter. Under the conditions q,q′∈AC(R+)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$q,q^{\prime}\in \operatorname{AC}(\mathbb{R}_{+})$\end{document}, limx→∞|q(x)|+|q′(x)|=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\lim_{x\rightarrow \infty} \vert q(x)\vert +\vert q^{\prime}(x) \vert =0$\end{document}, supx∈R+[eεx|q″(x)|]<∞\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\sup_{x\in \mathbb{R}_{+}} [ e^{\varepsilon\sqrt{x}} \vert q^{\prime\prime}(x)\vert ] <\infty$\end{document}, ε>0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\varepsilon>0$\end{document}, using the uniqueness theorems of analytic functions, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities.
