## Eigenvalues of a Class of Second-Order Differential Operator with Eigenparameters Dependent Internal Point Conditions

Liu Wei,, Xu Meizhen,*

College of Sciences, Inner Mongolia University of Technology, Hohhot 010051

 基金资助: 国家自然科学基金(12261066)内蒙古自然科学基金(2021MS01020)内蒙古自然科学基金(2023LHMS01015)内蒙古自治区直属高校基本科研业务费(JY20240043)

 Fund supported: NSFC(12261066)NSF of Inner Mongolia(2021MS01020)NSF of Inner Mongolia(2023LHMS01015)Basic Science Research Fund of the Universities Directly Under the lnner Mongolia Autonomous Region(JY20240043)

Abstract

This paper mainly discusses the self-adjointness and eigenvalue dependence of a class of second-order differential operator with internal point conditions containing an eigenparameter. First, a problem-related linear operator $T$ is defined in an appropriate Hilbert space, and the study of the problem to be transformed into the research of the operator $T$ in this space, and the operator $T$ is proved to be self-adjoint according to the definition of self-adjoint operator. In addition, on the basis of self-adjoint, it is proved that the eigenvalues are not only continuously dependent but also differentiable on each parameter of the problem, and the corresponding differential expressions are given. Meanwhile, the monotonicity of the eigenvalues with respect to the part parameters of the problem is also discussed.

Keywords： Internal point conditions; Eigenparameters; Self-adjointness; Dependence of eigenvalue; Monotonicity of eigenvalues

Liu Wei, Xu Meizhen. Eigenvalues of a Class of Second-Order Differential Operator with Eigenparameters Dependent Internal Point Conditions[J]. Acta Mathematica Scientia, 2024, 44(4): 815-828

## 1 引言

$$$ly:=-y ''+q(x)y=\lambda y, x\in J'=(M',a)\cup (a,N').$$$

\begin{aligned} &U(y):=y'(M)-hy(M)=0, V(y):=y'(N)+Hy(N)=0, \end{aligned}

\begin{aligned} &y(a+)=cy(a-), y'(a+)=c^{-1}y'(a-)+(b-\lambda m)y(a-). \end{aligned}

$$$b,c,m,h,H\in \mathbb{R},c>0,m>0,q(x)\in L^{1}(J,\mathbb{R}).$$$

## 2 算子的自伴性

$$$\langle f,g \rangle_{1}=\int^{a}_{M}f(x)\overline{g}(x){\rm d}x+\int^{N}_{a}f(x)\overline{g}(x){\rm d}x, \forall f,g\in L^{2}(J).$$$

$$$(F,G)=\langle f,g \rangle_{1}+cmf_{1}\overline{g}_{1},$$$

$$$TF=\left( \begin{array}{c} -f''+qf \\ \frac{1}{m}[c^{-1}f'(a-)-f'(a+)+bf(a-)] \\ \end{array} \right)=\left( \begin{array}{c} \lambda f \\ \lambda f_{1} \\ \end{array} \right)=\lambda F,$$$

$\forall F=(f,f_{1})^{T}\in \mathcal{H}$. 其中, 算子 $T$ 的定义域为

$\begin{array}{c} D(T)=\left\{F=\left(f, f_{1}\right)^{T} \in \mathcal{H}: f \in D_{l}, f(a+)-c f(a-)=0, f_{1}=f(a-)\right\}, \\ D_{l}=\left\{f \in L^{2}(J): f, f^{\prime} \in A C(J), l f \in H_{1}, U(f)=V(f)=0\right\}.\end{array}$

$\langle lx,y\rangle_{1}-\langle x,ly\rangle_{1}=[x,y]_{M}^{a-}+[x,y]_{a+}^{N},$

$$$[x,y]:=x\overline{y}'-x'\overline{y}.$$$

$$$\begin{split} &f(M)=\alpha_{1},f(N)=\beta_{1},f(a+)=\gamma_{1},f(a-)=\delta_{1},\\ &f'(M)=\alpha_{2},f'(N)=\beta_{2},f'(a+)=\gamma_{2},f'(a-)=\delta_{2}. \end{split}$$$

(1) $g,g'\in AC(J), lg\in H_{1};$

(2) $w_{1}=\frac{1}{m}[c^{-1}g'(a-)-g'(a+)+bg(a-)];$

(3) $w=-g''+qg;$

(4) $g'(M)-hg(M)=0, g'(N)+Hg(N)=0;$

(5) $g(a+)=cg(a-).$

$\widetilde{ C_{0}^{\infty}}$ 表示

$\phi(x)=\left\{ \begin{array}{cc} \varphi_{1}(x),&x\in[M,a), \\ \varphi_{2}(x),& x\in(a,N] \\ \end{array} \right.$

$\begin{split} (TF,G)=\int^{a}_{M}l(f)\overline{g}{\rm d}x+\int^{N}_{a}l(f)\overline{g}{\rm d}x =\int^{a}_{M}f\overline{w}{\rm d}x+\int^{N}_{a}f\overline{w}{\rm d}x =(F,W), \end{split}$

$$$\langle lf,g \rangle_{1}=\langle f,w\rangle_{1},$$$

$\begin{split} &f(M)\overline{g}'(M)-f'(M)\overline{g}(M)=0, f(N)\overline{g}'(N)-f'(N)\overline{g}(N)=0. \end{split}$

$( TF, G)=( F, W)$ 及 (2.1)-(2.2) 式可知

$$$\langle lf,g \rangle_{1}+cm\cdot \frac{1}{m}[c^{-1}f'(a-)-f'(a+)+bf(a-)]\cdot \overline{g}_{1}=\langle f,w\rangle_{1}+cmf_{1}\overline{w}_{1}.$$$

$\begin{matrix} cmf_{1}\overline{w}_{1}&=[f,g](a-)-[f,g](a+)+f'(a-)\overline{g}(a-)-cf'(a+)\overline{g}(a-)+cbf(a-)\overline{g}(a-) \\ &=f(a-)\overline{g}'(a-)-f(a+)\overline{g}'(a+)+f'(a+)\overline{g}(a+)-cf'(a+)\overline{g}(a-)+cbf(a-)\overline{g}(a-), \end{matrix}$

$\overline{w}_{1} =\frac{1}{cm}\overline{g}'(a-)-\frac{1}{cmf_{1}}f(a+)\overline{g}'(a+)+\frac{1}{cmf_{1}}f'(a+)\overline{g}(a+)-\frac{1}{mf_{1}}f'(a+)\overline{g}'(a-)+\frac{1}{m}b\overline{g}(a-).$

$-cf(a-)\overline{g}'(a+)=-f(a+)\overline{g}'(a+)+f'(a+)\overline{g}(a+)-cf'(a+)\overline{g}(a-),$

$\begin{equation*} \int^{a}_{M}f(x)\overline{g}(x){\rm d}x+\int^{N}_{a}f(x)\overline{g}(x){\rm d}x+cmf_{1}\overline{g}_{1}=0.\end{equation*}$

## 3 特征值和特征函数的连续性

$CY(M)+DY(N)=0,$

$AY(a+)=B_\lambda Y(a-),$

$\varphi_{11}, \varphi_{12}$ 是方程(1.1) 在区间 $[M,a)$ 上满足如下初始条件的线性无关解

$$$(\Psi_{11}(M, \lambda), \Psi_{12}(M, \lambda))=E,$$$

$\left\{ \begin{array}{ll} -y''+q(x)y=\lambda y, \\ y(t_{0})=d, y'(t_{0})=k, \\ \end{array} \right.$

$|t-t_0|+| d-d_0|+|k-k_0|+\int^{a}_{M}(|q-q_0|){\rm d}x+\int^{N}_{a}(|q-q_0|){\rm d}x< \delta.$

$\begin{split} &|y(x,t,d,k,q)-y(x,t_0,d_0,k_0,q_0)|<\varepsilon,\\ &|y'(x,t,d,k,q)-y'(x,t_0,d_0,k_0,q_0)|< \varepsilon.\\ \end{split}$

$\lambda(w_0)$ 是单重特征值, 则 $\Delta' (\lambda(w_0))\neq0$. 因为 $\Delta (\lambda)$$\lambda 的整函数, 由定理 3.2 可知结论成立. 定义 3.1u 满足问题(1.1)-(1.3), u_1=u(a-), 且有 $$\int^{a}_{M}u\overline{u}{\rm d}x+\int^{N}_{a}u\overline{u}{\rm d}x+cmu_{1}\overline{u}_{1}=1$$ 成立, 则称 (u,u_1)^T 为正规化特征向量. 定理 3.3 假设记号同定理 3.2. 设特征值 \lambda(w_{0})$$w_{0}\in \Omega$ 内是单重的, 令 $(u(x,w_{0}),$$u_{1}(w_{0}))^T \in \mathcal{H}$$\lambda(w_{0})$ 的一个正规化特征向量, 则存在一个对应于特征值 $\lambda(w)$ 的正规化特征向量 $(u(x,w), u_{1}(w))^T\in \mathcal{H}$, 使得在 $\Omega$ 中当 $w\rightarrow w_{0}$ 时, 有

$$$u(x,w)\rightarrow u(x,w_0), u'(x,w)\rightarrow u'(x,w_0), u_{1}(w)\rightarrow u_{1}(w_{0})$$$

$\lambda(w_{0})$ 是算子 $T$ 的单重特征值, $(y,y_1)^T\in\mathcal{H}$ 是其对应的特征向量, 且满足

$\parallel y(x,w_{0})\parallel^{2}=\int^{a}_{M}|y(x,w_{0})|^2{\rm d}x+\int^{N}_{a}|y(x,w_{0})|^2{\rm d}x=1.$

$(A,B_{\lambda})_{2\times 4}(w)\rightarrow (A,B_{\lambda})_{2\times 4}(w_{0}).$

$$$\begin{split} & \parallel y(x,w)\parallel^{2}=\int^{a}_{M}|y(x,w)|^2{\rm d}x+\int^{N}_{a}|y(x,w)|^2{\rm d}x=1. \\ & y(x,w)\rightarrow y(x,w_{0}), y'(x,w)\rightarrow y'(x,w_{0}). \end{split}$$$

$$$y_{1}(w)\rightarrow y_{1}(w_{0}).$$$

$\lambda(w_{0})$ 的正规化特征向量 $(u(x,w_{0}),u_{1}(w_{0}))^T$ 和第一个分量导数 $u'(x,w_{0})$ 分别为

$$$\begin{split} &(u(x,w_{0}),u_{1}(w_{0}))^T=\frac{(y(x,w_{0}),y_{1}(w_{0}))^T}{\parallel (y(x,w_{0}),y_{1}(w_{0}))^T \parallel},\\ &u'(x,w_{0})=\frac{y'(x,w_{0})}{\parallel (y(x,w_{0}),y_{1}(w_{0}))^T \parallel}. \end{split}$$$

## 4 特征值的可微性和单调性

$$$\begin{split} & (\nu-\mu)\bigg[\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x+cmu_{1}\overline{v}_{1}\bigg] =-[u,v]^{a-}_{M}-[u,v]^{N}_{a+}+(\nu-\mu)cmu_{1}\overline{v}_{1}. \end{split}$$$

$\lim_{l\rightarrow 0}\frac{1}{l}\int^{x+l}_{x}f=f(x) \quad {\rm {\rm a.e.}}\quad (a,b).$

$$$d \lambda_{q}(\tau)=\int^{a}_{M}|u|^2 \tau {\rm d}x+\int^{N}_{a}|u|^2 \tau {\rm d}x, \tau\in L^{1}(J,\mathbb{R}).$$$

(2) 固定 $w$ 中除 $h$ 以外的所有变量, 令 $\lambda=\lambda(h)$, 则 $\lambda$ 是可微的且有

$\lambda'(h)=|u(M)|^{2}.$

(3) 固定 $w$ 中除 $H$ 以外的所有变量, 令 $\lambda=\lambda(H)$, 则 $\lambda$ 是可微的且有

$\lambda'(H)=|u(N)|^{2}.$

(4) 固定 $w$ 中除 $b$ 以外的所有变量, 令 $\lambda=\lambda(b)$, 则 $\lambda$ 是可微的且有

$\lambda'(b)=c|u(a-)|^{2}.$

(5) 固定 $w$ 中除 $c$ 以外的所有变量, 令 $\lambda=\lambda(c)$, 则 $\lambda$ 是可微的且有

$\lambda'(c)=-\frac{1}{c}u(a-)\overline{u}'(a-).$

(6) 固定 $w$ 中除 $m$ 以外的所有变量, 令 $\lambda=\lambda(m)$, 则 $\lambda$ 是可微的且有

$\lambda'(m)=-c\lambda|u(a-)|^{2}.$

(1) 取$\tau$ 充分小, 令特征值 $\mu=\lambda(q), \nu=\lambda(q+\tau)$ 对应的正规化特征向量分别为 $(u,u_1)^T, (v,v_1)^T$, 其中 $u=u(x,q), v=u(x,q+\tau)$, 由 (4.1) 式可知

$\begin{matrix} &[\lambda(q+\tau)-\lambda(q)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x] \\ = & \int^{a}_{M}\lambda(q+\tau)u\overline{v}{\rm d}x-\int^{a}_{M}\lambda(q)u\overline{v}{\rm d}x+\int^{N}_{a}\lambda(q+\tau)u\overline{v}{\rm d}x-\int^{N}_{a}\lambda(q)u\overline{v}{\rm d}x \\ =& -[u,v](a-)+[u,v](M)-[u,v](N)+[u,v](a+) \\ & +\int^{a}_{M}(q+\tau) u\overline{v}{\rm d}x-\int^{a}_{M}qu\overline{v}{\rm d}x+\int^{N}_{a}(q+\tau) u\overline{v}{\rm d}x-\int^{N}_{a}qu\overline{v}{\rm d}x, \end{matrix}$
$\begin{split} & \lambda(q)cmu_{1}=u'_{1}+cbu_{1}-cu'(a+), \lambda(q+\tau)cm\overline{v}_{1}=\overline{v}'_{1}+cb\overline{v}_{1}-c\overline{v}'(a+). \end{split}$

$[u,v](a+)=[u,v](a-).$

$[u,v](M)=0,$
$[u,v](N)=0.$

$[\lambda(q+\tau)-\lambda(q)]cmu_{1}\overline{v}_{1}=0.$

$\begin{matrix} & [\lambda(q+\tau)-\lambda(q)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x+cmu_{1}\overline{v}_{1}] \\ =&\int^{a}_{M}(q+\tau) u\overline{v}{\rm d}x-\int^{a}_{M}qu\overline{v}{\rm d}x+\int^{N}_{a}(q+\tau) u\overline{v}{\rm d}x-\int^{N}_{a}qu\overline{v}{\rm d}x \\ =&\int^{a}_{M}\tau u\overline{v}{\rm d}x+\int^{N}_{a}\tau u\overline{v}{\rm d}x \end{matrix}$

$\tau \rightarrow 0$, 由定义 3.1, 定理 3.3 和定义 4.1 即可得结果(4.2).

(2) 取 $\varepsilon$ 充分小, 令特征值 $\mu=\lambda(h), \nu=\lambda(h+\varepsilon)$ 对应的正规化特征向量分别为 $(u,u_1)^T, (v,v_1)^T,$其中 $u=u(x,h), v=u(x,h+\varepsilon)$, 由(1.1) 式我们有

$-u''+q(x)u=\lambda(h) u,$
$-\overline{v}''+q(x)\overline{v}=\lambda(h+\varepsilon)\overline{v},$

$[\lambda(h+\varepsilon)-\lambda(h)]u\overline{v}=-u\overline{v}''+u''\overline{v},$

$[\lambda(b+\varepsilon)-\lambda(b)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x]=0,$
$[\lambda(b+\varepsilon)-\lambda(b)]cmu_{1}\overline{v}_{1}=c(b+\varepsilon)u_{1}\overline{v}_{1}-cbu_{1}\overline{v}_{1}=\varepsilon cu_{1}\overline{v}_{1},$

$$$[\lambda(b+\varepsilon)-\lambda(b)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x+cmu_{1}\overline{v}_{1}]=\varepsilon cu_{1}\overline{v}_{1}.$$$

(4) 取 $\varepsilon$ 充分小, 令特征值 $\mu=\lambda(c), \nu=\lambda(c+\varepsilon)$ 对应的正规化特征向量分别为 (u,u_1)^T, (v,v_1)^T, 其中 u=u(x,c), v=u(x,c+\varepsilon), 根据转移条件 (1.3), (4.1) 及 (4.10)-(4.12) 式计算可得 [\lambda(c+\varepsilon)-\lambda(c)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x]=0 \begin{align*} & [\lambda(c+\varepsilon)-\lambda(c)]cmu_{1}\overline{v}_{1} \\ ={}&cmu(a-)\lambda(c+\varepsilon)\overline{v}(a-)-cm\overline{v}(a-)\lambda(c)u(a-)\\ ={}&u(a-)cm\cdot\frac{1}{m}[(c+\varepsilon)^{-1}\overline{v}'(a-)-\overline{v}'(a+)+b\overline{v}(a-)]\\ &-\overline{v}(a-)cm\cdot\frac{1}{m}[c^{-1}u'(a-)-u'(a+)+bu(a-)]\\ ={}&\frac{c}{c+\varepsilon}u(a-)\overline{v}'(a-)-\overline{v}(a-)u'(a-)-cu(a-)\overline{v}'(a+)+c\overline{v}(a-)u'(a+)\\ ={}&-\frac{\varepsilon}{c+\varepsilon}u(a-)\overline{v}'(a-). \end{align*} 综上所述 \begin{equation*} [\lambda(c+\varepsilon)-\lambda(c)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x+cmu_{1}\overline{v}_{1}]=-\frac{\varepsilon}{c+\varepsilon}u(a-)\overline{v}'(a-). \end{equation*} 两边同时除以 \varepsilon 并令 \varepsilon\rightarrow 0, 由定义 3.1 和定理 3.3 即可得结果 (4.6) 式. (5) 取 \varepsilon 充分小, 令特征值 \mu=\lambda(m), \nu=\lambda(m+\varepsilon) 对应的正规化特征向量分别为 (u,u_1)^T, (v,v_1)^T, 其中 u=u(x,m), v=u(x,m+\varepsilon), 根据转移条件 (1.3), (4.1) 及 (4.10)-(4.12) 式计算可得 [\lambda(m+\varepsilon)-\lambda(m)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x] =0 \begin{align*} [\lambda(m+\varepsilon)-\lambda(m)]cmu_{1}\overline{v}_{1} ={}&cmu(a-)\lambda(m+\varepsilon)\overline{v}(a-)-cm\overline{v}(a-)\lambda(m)u(a-)\\ ={}&u(a-)cm\cdot\frac{1}{m+\varepsilon}[c^{-1}\overline{v}'(a-)-\overline{v}'(a+)+b\overline{v}(a-)]\\ &-\overline{v}(a-)cm\cdot\frac{1}{m}[c^{-1}u'(a-)-u'(a+)+bu(a-)]\\ ={}&-\frac{\varepsilon}{m+\varepsilon}u(a-)c[c^{-1}\overline{v}'(a-)-\overline{v}'(a+)+b\overline{v}(a-)]\\ ={}& -\varepsilon u(a-)c\lambda\overline{v}(a-). \end{align*} 所以 [\lambda(m+\varepsilon)-\lambda(m)][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x+cmu_{1}\overline{v}_{1}] =-\varepsilon cu(a-)\lambda \overline{v}(a-). 两边同时除以 \varepsilon 并令 \varepsilon\rightarrow 0, 由定义 3.1 和定理 3.3 即可得结果 (4.7) 式. 下面, 我们给出 \lambda 关于 a+, a- 的微分表达式, 令 a_{1}=a-, a_{2}=a+, 然后有 定理 4.2 w=(q,b,h,H,c,m,a+,a-,M,N)\in \Omega, \lambda=\lambda(w) 是算子 T 的特征值, (u,u_{1})^{T} 是相应的正规化特征向量. 若 \lambda(w)w 的某邻域 $\mathcal{M}\subset\Omega$ 内的几何重数不变, 则 $\lambda$ 关于内部不连续点 $a$ 左右两侧是可微的且有如下表达式

(1) 固定 $w$ 中除 $a_{1}$ 以外的所有变量, 令 $\lambda=\lambda(a_{1})$, 则 $\lambda$ 是可微的且有

$$$\lambda'(a_{1})=-|u(a_{1},a_{1})|^{2}[\lambda(a_{1})-q(a_{1})]-|u'(a_{1},a_{1})|^{2}, {\rm a.e.}\quad a_{1}\in[M,a).$$$

(2) 固定 $w$ 中除 $a_{2}$ 以外的所有变量, 令 $\lambda=\lambda(a_{2})$, 则 $\lambda$ 是可微的且有

$$$\lambda'(a_{2})=|u(a_{2},a_{2})|^{2}[\lambda(a_{2})-q(a_{2})]+|u'(a_{2},a_{2})|^{2}, {\rm a.e.}\quad a_{2}\in(a,N].$$$

$[\lambda(a_{1}+l)-\lambda(a_{1})][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x]=-[u,v](a_{1})+[u,v](a_{2})$

\begin{align*} & \ [\lambda(a_{1}+l)-\lambda(a_{1})]cmu_{1}\overline{v}_{1}\\ &=cmu_{1}\lambda(a_{1}+l)\overline{v}_{1}-cm\overline{v}_{1}\lambda(a_{1})u_{1}\\ &=u_{1}cm\cdot\frac{1}{m}[c^{-1}\overline{v}'(a_{1})-\overline{v}'(a_{2})+b\overline{v}(a_{1})]+\overline{v}_{1}cm\cdot\frac{1}{m}[c^{-1}u'(a_{1})-u'(a_{2})+bu(a_{1})]\\ &=u(a_{1})\overline{v}'(a_{1})-c\overline{v}'(a_{2})u(a_{1})-\overline{v}(a_{1})u'(a_{1})+c\overline{v}(a_{1})u'(a_{2})\\ & =[u,v](a_{1})-cu(a_{1})\overline{v}'(a_{2})+cu'(a_{2})\overline{v}(a_{1}), \end{align*}

$\begin{split} -cu(a_{1})\overline{v}'(a_{2})&=-cu(a_{1})[c^{-1}\overline{v}'(a_{1})+(b-\lambda m)\overline{v}(a_{1})],\\ cu'(a_{2})\overline{v}(a_{1})& =c\overline{v}(a_{1})[c^{-1}u'(a_{1})+(b-\lambda m)u(a_{1})]. \end{split}$

$\begin{matrix} & [\lambda(a_{1}+l)-\lambda(a_{1})][\int^{a}_{M}u\overline{v}{\rm d}x+\int^{N}_{a}u\overline{v}{\rm d}x+cmu_{1}\overline{v}_{1}]\notag\\ ={}&-u(a_{1})\overline{v}'(a_{1})+u'(a_{1})\overline{v}(a_{1})+u(a_{2})\overline{v}'(a_{2})-u'(a_{2})\overline{v}(a_{2})\notag\\ ={}& -u(a_{1},a_{1})\overline{u}'(a_{1},a_{1}+l)+u'(a_{1},a_{1})\overline{u}(a_{1},a_{1}+l)\\ & +u(a_{2},a_{1})\overline{u}'(a_{2},a_{1}+l)-u'(a_{2},a_{1})\overline{u}(a_{2},a_{1}+l)\notag\\ ={}&u'(a_{1},a_{1})[\overline{u}(a_{1},a_{1}+l)-\overline{u}(a_{1}+l,a_{1}+l)]-{u(a_{1},a_{1})[\overline{u}'(a_{1},a_{1}+l)-\overline{u}'(a_{1}+l,a_{1}+l)]}\notag, \end{matrix}$

$\begin{matrix} & \overline{u}'(a_{1},a_{1}+l)-\overline{u}'(a_{1}+l,a_{1}+l) =-\int^{a_{1}+l}_{a_{1}}\overline{u}''(s,a_{1}+l){\rm d}s \notag \\ ={}&\int^{a_{1}+l}_{a_{1}}[\lambda(a_{1}+l)\overline{u}(s,a_{1}+l)-q(s)\overline{u}(s,a_{1}+l)]{\rm d}s\notag \\ ={}& \lambda(a_{1}+l)\int^{a_{1}+l}_{a_{1}}\overline{u}(s,a_{1}){\rm d}s-\lambda(a_{1}+l)\int^{a_{1}+l}_{a_{1}}[\overline{u}(s,a_{1})-\overline{u}(s,a_{1}+l)]{\rm d}s\notag \\ & -\int^{a_{1}+l}_{a_{1}}q(s)\overline{u}(s,a_{1}){\rm d}s+\int^{a_{1}+l}_{a_{1}}q(s)[\overline{u}(s,a_{1})-\overline{u}(s,a_{1}+l)]{\rm d}s, \end{matrix}$

$\begin{matrix} & \overline{u}(a_{1},a_{1}+l)-\overline{u}(a_{1}+l,a_{1}+l)=-\int^{a_{1}+l}_{a_{1}}\overline{u}'(s,a_{1}+l){\rm d}s \\ =&-\int^{a_{1}+l}_{a_{1}}\overline{u}'(s,a_{1}){\rm d}s+\int^{a_{1}+l}_{a_{1}}[\overline{u}'(s,a_{1})-\overline{u}'(s,a_{1}+l)]{\rm d}s.\end{matrix}$

$l\rightarrow 0$ 时, $\overline{u}(s,a_{1})-\overline{u}(s,a_{1}+l)\rightarrow 0$, $\overline{u}'(s,a_{1})-\overline{u}'(s,a_{1}+l)\rightarrow 0$, 由引理 4.2 可知

$\begin{split} & \lim_{l\rightarrow 0}\frac{\overline{u}'(a_{1},a_{1}+l)-\overline{u}'(a_{1}+l,a_{1}+l)}{l}=\overline{u}(a_{1},a_{1})[\lambda(a_{1})-q(a_{1})], \\ & \lim_{l\rightarrow 0}\frac{\overline{u}(a_{1},a_{1}+l)-\overline{u}(a_{1}+l,a_{1}+l)}{l}=-\overline{u}'(a_{1},a_{1}). \end{split}$

$\lambda(\widetilde{h})>\lambda(h).$

(3) 固定 $w$ 中除 $H$ 之外的所有参数, 特征值 $\lambda=\lambda(H)$ 是单调递增的, 即若存在一个实数 $\widetilde{H}$$\widetilde{H}>H, 则有 \lambda(\widetilde{H})>\lambda(H). (4) 固定 w 中除 b 之外的所有参数, 特征值 \lambda=\lambda(b) 是单调递增的, 即若存在一个实数 \widetilde{b}$$\widetilde{b}>b$, 则有

$\lambda(\widetilde{b})>\lambda(b).$

$f(s(t))=\lambda(s(t)),s(t)=q+t(Q-q), t\in[0,1].$

$s(t)$ 在区间 $[0,1]$ 是勒贝格可积的. 由链式求导法则和 (4.2) 式可知

$f'(t)=\lambda'(s(t))s'(t)=\int^{a}_{M}|u(r,s(t))|^2(Q-q){\rm d}r+\int^{N}_{a}|u(r,s(t))|^2(Q-q){\rm d}r>0,$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Pöeschel J, Trubowitz E. Inverse Spectral Theory. New York: Academic Press, 1987

Dauge M, Hellfer B.

Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators

J Differ Equ, 1993, 104(2): 243-262

Dauge M, Hellfer B.

Eigenvalues variation. II. Multidimensional Problems

J Differ Equ, 1993, 104(2): 263-297

Kong Q K, Zettl A.

Eigenvalues of regular Strum-Liouville problems

J Differ Equ, 1996, 131: 1-19

Kong Q K, Zettl A.

Dependence of eigenvalues of Strum-Liouville problems on the boundary

J Differ Equ, 1996, 126: 389-407

Suo J Q, Wang W Y.

Eigenvalues of a class of regular fourth-order Sturm-Liouville problems

Appl Math Comput, 2012, 218(19): 9716-9729

Ge S Q, Wang W Y, Suo J Q.

Dependence of eigenvalues of a class of fourth-order Sturm-Liouville problemson the boundary

Appl Math Comput, 2013, 220: 268-276

Kong Q K, Wu H Y, Zettl A.

Dependence of eigenvalues on the problems

Math Nachr, 1997, 188: 173-201

Zheng Z W, Ma Y J.

Dependence of eigenvalues of 2$m$th-order spectral problems

Bound Value Probl, 2017, 2017(1): Article 126

Yang Q X, Wang W Y, Gao X C, Rajendran S.

Dependence of eigenvalues of a class of higher-order Sturm-Liouville problems on the boundary

Math Problems Eng, 2015, 2015: Article 686102

Zhang M Z, Wang Y C.

Dependence of eigenvalues of Sturm-Liouville problems with interface conditions

Appl Math Comput, 2015, 265: 31-39

Wang A P, Zettl A.

Eigenvalues of Sturm-Liouville problems with discontinuous boundary conditions

Electron J Differ Equ, 2017, 2017(127): 1-27

Zhang M Z, Li K.

Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary conditions

Appl Math Comput, 2020, 378: 125214

Zhang H Y, Ao J J, Li M L.

Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and interface conditions

Mediterr J Math, 2022, 19(2): 1-17

Bai Y L, Wang W Y, Li K, Zheng Z W.

Eigenvalues of a class of eigenparameter dependent third-order differential operators

J Nonlinear Math Phy, 2022, 29(3): 447-492

Zhang H Y, Ao J J, Mu D.

Eigenvalues of discontinuous third-order boundary value problems with eigenparameter-dependent boundary conditions

J Math Anal Appl, 2022, 506(2): 125680

Qin J F, Li K, Zheng Z W, Cai J M.

Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary condition

AIMS Math, 2022, 7(5): 9247-9260

Yan W W, Xu M Z.

The self-adjointness and dependence of eigenvalues of fourth-order differential operator with eigenparameters in the boundary conditions

Acta Math Sci, 2022, 42A(3): 671-693

Carlson R.

Hearing point masses in a string

Siam J Math Anal, 1995, 26: 583-600

Akdogan Z, Demirci M, Mukhtarov Sh O.

Green function of discontinuous boundary-value problem with transmission conditions

Math Meth Appl Sci, 2007, 30: 1719-1738

Erdogan S.

A class of second-order differential operators with eigenparameter-dependent boundary and transmission conditions

Math Meth Appl Sci, 2014, 37(18): 2952-2961

Guo Y X.

Inverse spectral problem of regular Sturm-Liouville operator

Shanxi: Shanxi Normal University, 2015: 63-80

Wei Z, Wei G.

Inverse spectral problem for non-selfadjoint Dirac operator with boundary and jump conditions dependent on the spectral parameter

J Comput Appl Math, 2016, 308: 199-214

Aydemir K, Mukhtarov O.

A class of Sturm-Liouville problems with eigenparameter dependent transmission conditions

Numer Func Anal Opt, 2017, 38(10): 1260-1275

Zettl A. Sturm-Liouville Theory. New York: Amer Math Soc, 2005

Dieudonné J. Foundations of Modern Analysis. New York: Academic Press, 1969

/

 〈 〉