微分算子是线性算子中有着非常深刻应用背景的一类无界线性算子.数学物理及其它应用科学中许多问题都可归结为确定微分算子的特征值和特征函数以及将任意函数按特征函数展开成级数(或积分)的问题.其中很多实际问题, 例如具有叠层的热传导问题, 带有结点的弦振动问题, 势函数是广义函数的微分算子等, 都可以转化为内部具有不连续性的微分算子问题.广为被关注的“弹子动力系统”也可以从微分算子谱理论的角度来观察和研究, 即:考虑一类与其相关的微分算子(带有无穷多个不连续点的微分算子), 在不连续点附加转移条件来刻画质点的碰撞运动.因此, 近些年来, 伴随应用领域提出的众多问题, 本领域数学工作者对这类内部具有不连续性Sturm-Liuouville问题(内部具有有限多个或者无穷多个不连续点)的研究给予了特别的关注^[1-10].

本文研究了一类内部具有无穷多个不连续点Sturm-Liouville算子的亏指数问题.考虑Sturm-Liouville表达式

$ \begin{equation}\label{eq1.1} Mf(t)=-f''(t)+q(t)f(t), \ t\in I=\bigcup\limits_{i=0}^{\infty}(t_i, t_{i+1}) \end{equation} $

(1.1)

和不连续条件

$ \begin{array}{l}\label{eq1.3} \left\{ \begin{array}{l} f(t_i+0)=\alpha_{1i}f(t_i-0)+\beta_{1i}f'(t_i-0) \\[5pt] f'(t_{i}+0)=\alpha_{2i}f(t_i-0)+\beta_{2i}f'(t_i-0) \end{array} \right., \quad i\in {\Bbb N}\setminus \{0\}, \end{array} $

(1.2)

其中$t_0=0, \ \lim_{i\to \infty}t_i=\infty, \ q\in L_{loc}(I, {\Bbb R}), $且$A_i=\left( \begin{array}{cc} \alpha_{1i} &\beta_{1i} \\[5pt] \alpha_{2i} &\beta_{2i} \end{array} \right)\in M_2({\Bbb R})$满足

$ \rho_i=\det (A_i)>0 ; $

$f^{(k)}(t_i+0), \ f^{(k)}(t_i-0)\ (k=0, 1)$的定义如下

$ f^{(k)}(t_i+0)=\lim\limits_{t\rightarrow t_i+0}f^{(k)}(t), f^{(k)}(t_i-0)=\lim\limits_{t\rightarrow t_i-0}f^{(k)}(t), i\in {\Bbb N}\setminus \{0\}. $

注1.1  我们假设存在常数$\varepsilon_0>0$使得

$ \begin{array}{l}%\label{assumption} \inf\limits_{i\geq 1}(t_i-t_{i-1}) \geq 4\varepsilon_0. \end{array} $

本节, 我们引进与不连续条件相关的新内积, 构造新的Hilbert空间, 并其上定义与不连续条件有关的最小算子和最大算子.

定义微分算式列$\{M_i|i\in {\Bbb N}\}$如下

$ \begin{array}{l} D(M_i)=\{f_i:(t_i, t_{i+1})\rightarrow {\mathbb{C}}|f_i, f_i^{\prime }\in AC_{loc}(t_i, t_{i+1})\}, \end{array} $

$ \begin{equation} \label{e.q.1.5} M_if_i=-f''_i+q_if_i\, \, , t\in (t_i, t_{i+1}), \ f_i\in D(M_i). \end{equation} $

(2.1)

设$H_i$表示$L^2(t_i, t_{i+1})$赋予内积

$ \begin{array}{l} (f_i, g_i)_i=h_i\int ^{t_{i+1}}_{t_i}f_i\overline{g}_i \end{array} $

的Hilbert空间.新内积引导的范数为

$ \begin{array}{l} \|f_i\|_i=\left(h_i\int ^{t_{i+1}}_{t_i}|f_i|^2\right)^{\frac{1}{2}}, \end{array} $

其中

$ \begin{array}{l} h_i=\left\{ \begin{array}{ll} 1, &i=0\hbox{;} \\[2mm] \frac{1}{\rho_1\rho_2\cdot\cdot\cdot\rho_i}, &i\in {\Bbb N}\setminus \{0\}\hbox{.} \end{array} \right. \end{array} $

对任意的$i\in {\Bbb N}, $最大算子$S_{\max, i}:D(S_{\max, i})\subset H_i\to H_i$的定义为

$ D(S_{\max, i})=\{f_i\in D(M_i)|M_if_i\in H_i\}, \ i\in {\Bbb N}, $

$ S_{\max, i}f_i=M_if_i, \quad f_i\in D(S_{\max, i}); $

最小算子$S_{\min, i}:D(S_{\min, i})\subset H_i\to H_i$的定义为

$ D(S_{\min, i})=\{f_i\in D(S_{\max, i})| f_i(t_{i})=f'_i(t_{i})=f_i(t_{i+1})=f'_i(t_{i+1})=0 , \ \forall g_i\in D(S_{\max, i})\}, $

$ S_{\min, i}f_i=M_if_i, \ f_i\in D(S_{\min, i}). $

设$H$为由所有形如$f=\{f_i|i\in {\Bbb N}\}$且满足

$ \begin{array}{l} f_i\in H_i \quad (i\in {\Bbb N}) \ \mbox{和} \ \sum\limits_{i=0}^{\infty}\|f_i\|^2_i < \infty \end{array} $

的元素所构成的线性空间.其上定义如下内积及其诱导的范数

$ (f, g)_H=\langle f, g\rangle =\sum\limits_{i=0}^{\infty}(f_i, g_i)_i, $

$ \|f\|_H=\left(\sum\limits_{i=0}^{\infty}\|f_i\|_i^2\right)^{\frac{1}{2}}. $

显然, 内积空间$(H, \langle \cdot, \cdot \rangle)$是Hilbert空间, 仍记为$H$.

为了简便起见, 令

$ D(M)=\{f=\{f_i|i\in {\Bbb N}\}|f_i\in D(S_{\max, i}), \sum\limits_{i\in {\Bbb N}}\|f_i\|^2_i < \infty\}, $

$ D(M, A)=\{f=\{f_i|i\in {\Bbb N}\}\in D(M)| f \mbox{满足不连续条件} \}, $

其中$A=\{A_i\in M_2({\Bbb R})|i\in {\Bbb N}\setminus \{0\}\}$.

定义2.1   $D(M, A)$的定义如上.与不连续条件(1.2)有关的最大算子$C_{\max}$和最小算子$C_{\min}$定义如下$:$

$ \begin{equation} \label{gmo} \begin{array}{ll} D(C_{\max})=\{f\in D(M, A)|f, Mf\in H\}, \\ C_{\max}f=Mf, f\in D(C_{\max}). \end{array} \end{equation} $

(2.2)

$ \begin{equation} \label{gmio} \begin{array}{ll} D(C_{\min})=\{f\in D(C_{\max})|f(t_0)=f'(t_0)=0, \lim\limits_{i\to\infty}[f_i, g_i]_i(c_i)=0, \ \forall g\in D(C_{\max})\}, \\ C_{\min}f=Mf, f\in D(C_{\min}), \end{array} \end{equation} $

(2.3)

其中$c_i=\frac{t_i+t_{i+1}}{2}, \ [f_i, g_i]_i(c_i)=h_i\left(f_i(c_i)\overline{g^{'}_i(c_i)}-f^{'}_i(c_i)\overline{g_{i}(c_i)}\right), \ i\in {\Bbb N}.$

注2.1  如上定义的算子$C_{\min }$和$C_{\max }$不仅依赖于微分算式的系数, 而且还依赖于不连续条件(1.2).

引理2.1  对任意的$f\in D(C_{\max}), $极限

$ f^{(k)}(t_i+0)=\lim\limits_{t\rightarrow t_i+0}f^{(k)}(t), f^{(k)}(t_i-0)=\lim\limits_{t\rightarrow t_i-0}f^{(k)}(t), i\in {\Bbb N}\setminus \{0\}, k=0, 1, $

存在且有限的.

证   见文献[11-12].

引理2.2  设算子$C_{\max}$和$C_{\min}$的定义如(2.2)和(2.3)式, 则$C_{\max}$和$C_{\min}$在空间$H$中稠定的, $C_{\min}$为$H$中的闭对称算子, 且$C_{\min}$和$C_{\max}$是相互共轭的, 即

$ \begin{array}{l} C_{\min}^*=C_{\max}, \quad C_{\max}^*=C_{\min}. \end{array} $

证  见文献[10].

这一节, 我们讨论与不连续点附加的不连续条件(1.2)相关联的最小算子$C_{\min}$的亏指数.首先给出亏指数的定义.

定义3.1   设$H$是一个Hilbert空间. $T$是$H$中的闭对称线性算子, 记

$ d_{+}=\dim {\cal N}(T^{*}-iI), \ d_{-}=\dim {\cal N}(T^{*}+iI). $

我们称数对$(n_{+}, n_{-})$为$T$的亏指数.

定义3.2   设$f, f'$是区间$I$上的绝对连续函数, 并满足Sturm-Liouville方程

$ \begin{equation}\label{SLeq} Mf=zf, \ z\in {\mathbb{C}} \end{equation} $

(3.1)

和不连续条件(1.2).则称$f$为带有不连续条件(1.2)的Sturm-Liouville方程(3.1)的解.

由上述定义可知, 对任何$z\in {\mathbb{C}}, $内部具有无穷多个不连续条件(1.2)的Sturm-Liouville方程(3.1)均有两个线性无关的解.因此最小算子$C_{\min}$的亏指数$d^{\pm}\in \{0, 1, 2\}.$另外, 我们所考虑的方程系数函数$q$和不连续条件的系数矩阵$\{A_i|\ i\in {\Bbb N}\setminus \{0\}\}$都是实的, 这表明了$d^{+}=d^{-}=d.$

引理3.1  算子$C_{\min}$的定义如(2.3)式.则算子$C_{\min}$有形如$(m, m)$的亏指数, 其中$1\leq m\leq 2.$

证  因为$d^{\pm}\in \{0, 1, 2\}, $且$d^{+}=d^{-}=d$, 所以仅需证明不等式$1\leq m.$为此, 利用熟知的直接和的展开式

$ D(C_{\max})=D(C_{\min})+{\cal N}_{\lambda}+{\cal N}_{\overline{\lambda}}, $

只需要证明在$D(C_{\max})$中存在$2$个按模$D(C_{\min})$线性无关的元素就够了.上述的2个元素可以这样作出:取$D(C_{\max})$中的两个函数$y_1, y_2$, 使它们满足如下条件

$ \begin{array}{l} \left | \begin{array}{cc} y_1(0) &y_1'(0) \\ y_2(0) &y_2'(0) \end{array} \right |\neq0. \end{array} $

显然, 满足上述条件的$y_1, y_2$是存在的且是$D(C_{\max})$中按模$D(C_{\min})$线性无关元素.

这引理说明此类具有无穷多个不连续点Sturm-Liouville问题的亏指数或为$(1, 1), $或为$(2, 2).$类似于经典奇异Sturm-Liouville问题, 第一种称为极限点的情形, 第二种称为极限圆的情形.下面的定理, 对于鉴别$C_{\min}$的极限点, 极限圆属性是很重要的.

定理3.1  若存在$\lambda_0\in {\mathbb{C}}$ (可为实数), 使得带有不连续条件(1.2)的Sturm-Liouville方程

$ \begin{array}{l}\label{eq3.2} Mf=\lambda_{0}f, \quad t\in I \end{array} $

(3.2)

的解均属于$H, $则对于任何复数或实数$\lambda, $带有不连续条件的Sturm-Liouville方程

$ \begin{array}{l}\label{eq3.3} Mf=\lambda f, \quad t \in I \end{array} $

(3.3)

的解均属于$H.$

证  设$\varphi_0(t)=\varphi(t, \lambda_0), \ \psi_0(t)=\psi(t, \lambda_0)$为方程满足不连续条件的两个线性无关解, 满足

$ W(\varphi_0, \psi_0)(t_0)=\left| \begin{array}{cc} \varphi_0(t_0) &\psi_0(t_0) \\[5pt] \varphi'_0(t_0) &\psi'_0(t_0) \end{array}\right|=1. $

经简单计算, 得

$ W(\varphi_0, \psi_0)(t)=\frac{1}{h_i}, \quad t\in (t_i, t_{i+1}), i\in {\Bbb N}. $

令

$ \| \varphi_{0} \| ^{t}_{{t_i}, H}=\left\{\sum\limits_{k=i}^{n-1}\int_{t_k}^{t_{k+1}}h_k\mid\varphi_{0}(s, \lambda)\mid^{2}{\rm d}s+\int^{t}_{t_n}h_n\mid\varphi_{0}(s, \lambda)\mid^{2}{\rm d}s\right\}^{\frac{1}{2}}, t\in (t_n, t_{n+1}), n\geq i, $

$ \| \psi_{0} \| ^{t}_{{t_i}, H}=\left\{\sum\limits_{k=i}^{n-1}\int_{t_k}^{t_{k+1}}h_k\mid\psi_{0}(s, \lambda)\mid^{2}{\rm d}s+\int^{t}_{t_n}h_n\mid\psi_{0}(s, \lambda)\mid^{2}{\rm d}s\right\}^{\frac{1}{2}}, t\in (t_n, t_{n+1}), n\geq i $

和

$ r_{i}=\mbox{max}\{\| \varphi_{0} \| ^{\infty}_{t_i, H}, \| \psi_{0} \| ^{\infty}_{t_i, H}\}. $

根据$\varphi_0, \psi_0 \in H$的假设知, 当$i\rightarrow \infty$时$r_i \rightarrow 0.$于是可选择充分大的$i$使得$| \lambda -\lambda_0 |r^{2}_i $充分小, 即存在充分大的$i_0$使得当$i\geq i_0$时有

$ |\lambda -\lambda_0|r^{2}_i \leq \frac{1}{4}. $

设$\chi(t, \lambda)$是方程(3.3)满足不连续条件(1.2)的任意解.显然$\chi(t, \lambda)$满足下列方程

$ Mf-\lambda_{0}f=(\lambda-\lambda_{0})\chi , \ \ t\in I $

和不连续条件(1.2).由熟知的常数变易公式, 可将上述方程转化成积分方程

$ \chi_i(t, \lambda)=c_{i1}\varphi_0(t)+c_{i2}\psi_0(t)+(\lambda-\lambda_{0})\int^{t}_{t_i}\frac{\psi_0(t)\varphi_0(s)-\varphi_0(t)\psi_0(s)}{W(\varphi_0, \psi_0)(s)}\chi(s, \lambda){\rm d}s, t\in(t_i, t_{i+1}), $

其中$c_{i1}, c_{i2}$为常数.因为$\chi(t, \lambda)$满足不连续条件(1.2), 所以对任意的$t\in\bigcup\limits_{i=i_0}^{\infty}(t_i, t_{i+1})$有

$ \begin{array}{l}\label{eq3.4} \chi(t, \lambda)=c_{i_01}\varphi_0(t)+c_{i_02}\psi_0(t)+(\lambda-\lambda_{0})\int^{t}_{t_{i_0}}\frac{\psi_0(t)\varphi_0(s)-\varphi_0(t)\psi_0(s)}{W(\varphi_0, \psi_0)(s)}\chi(s, \lambda){\rm d}s. \end{array} $

(3.4)

由Schwarz不等式, 当$t\in(t_n, t_{n+1}), \ n\geq i_0$时有

$ \begin{array}{l} \left|\int^{t}_{t_{i_0}}\frac{\psi_0(t)\varphi_0(s)-\varphi_0(t)\psi_0(s)}{W(\varphi_0, \psi_0)(s)}\chi(s, \lambda){\rm d}s\right|\\ =\left|\sum\limits_{i=i_0}^{n-1}\int^{t_{i+1}}_{t_{i}}h_i\{\psi_0(t)\varphi_0(s)-\varphi_0(t)\psi_0(s)\}\chi(s, \lambda){\rm d}s\right.\\ \left.+\int^{t}_{t_{n}}h_n\{\psi_0(t)\varphi_0(s)-\varphi_0(t)\psi_0(s)\}\chi(s, \lambda){\rm d}s\right|\\ \leq \sum\limits_{i=i_0}^{n-1}\int^{t_{i+1}}_{t_{i}}h_i\{|\psi_0(t)||\varphi_0(s)|+|\varphi_0(t)||\psi_0(s)|\}|\chi(s, \lambda)|{\rm d}s\\ +\int^{t}_{t_{n}}h_n\{|\psi_0(t)||\varphi_0(s)|+|\varphi_0(t)||\psi_0(s)|\}|\chi(s, \lambda)|{\rm d}s\\ \leq |\varphi_0(t)|\|\psi_0\|^{t}_{t_{i_0}}\|\chi\|^{t}_{t_{i_0}}+|\psi_0(t)|\|\varphi_0\|^{t}_{t_{i_0}}\|\chi\|^{t}_{t_{i_0}} \\ \leq r_{i_0}(|\varphi_0(t)|+|\psi_0(t)|)\|\chi\|^{t}_{t_{i_0}}. \end{array} $

再对(3.4)式运用三角不等式, 并注意不等式

$ \|\chi\|^{\tau}_{t_{i_0}}\leq\|\chi\|^{t}_{t_{i_0}}, \quad \tau \leq t, $

可得

$ \begin{array}{l} \|\chi\|^{t}_{t_{i_0}}\leq |c_{i_01}|\|\varphi_0\|^{t}_{t_{i_0}}+|c_{i_02}|\|\psi_0\|^{t}_{t_{i_0}}+|\lambda-\lambda_0|r_{i_0}(\|\varphi_0\|^{t}_{t_{i_0}}+|\|\psi_0\|^{t}_{t_{i_0}})\|\chi\|^{t}_{t_{i_0}}\\ %&\leq |c_{1}|\|\varphi_0\|^{t}_{t_i}+|c_{2}|\|\psi_0\|^{t}_{t_i}+|\lambda-\lambda_0|\{\int^{t}_{t_i}2r^2_{i}(|\varphi_0(\tau)^2|+|\psi_0(\tau)|^2)(\|\chi\|^t_{t_i})^2d\tau \}^\frac{1}{2}\\ \leq |c_{i_01}|\|\varphi_0\|^{t}_{t_{i_0}}+|c_{i_02}|\|\psi_0\|^{t}_{t_{i_0}}+2|\lambda-\lambda_0|r^2_i\|\chi\|^t_{t_{i_0}}\\ \leq (|c_{i_01}|+|c_{i_02}|)r_{i_0}+\frac{1}{2}\|\chi\|^t_{t_{i_0}}. \end{array} $

因此

$ \|\chi\|^t_{t_{i_0}}\leq 2(|c_{i_01}|+|c_{i_02}|)r_{i_0}. $

不难看出不等式右端与$t$无关, 令$t\rightarrow\infty$, 即得

$ \chi(t, \lambda)\in H. $

证毕.

推论3.1  若存在$\lambda_0\in {\mathbb{C}}, $使带有不连续条件(1.2)的方程(3.2)有一个不属于$H$的非平凡解, 则对于一切$\lambda_0\in {\mathbb{C}} (\Im\lambda\neq0), $带有不连续条件(1.2)的方程(3.3)仅有一个属于$H$的线性无关解.

引理3.2  算子$C_{\max}$和$C_{\min}$的定义如(2.2)和(2.3)式.则算子$C_{\min}$在点$\infty$处极限点型当且仅当对任意的$f, g\in D(C_{\max})$有

$ [f, g](\infty)=\lim\limits_{i\to \infty}[f_i, g_i]_i(x_i)=\lim\limits_{i\to \infty}h_i\left(f_i(x_i)\overline{g^{'}_i(x_i)}-f^{'}_i(x_i)\overline{g_{i}(x_i)}\right)=0, x_i\in (t_i, t_{i+1}). $

证  设算子$C_{\min}$在$\infty$处是极限点型的.设$\bigtriangleup=[0, t_1-\varepsilon_0], $则$\bigtriangleup$是包含在区间$[0, t_1)$内的有限区间, 这时表达式$Mf$在区间$\bigtriangleup$上是正则的.在$D_{\bigtriangleup}(C_{\max})$内选择这样的函数$z_1, z_2$, 使得

$ z_v^{(k-1)}(0)=\left\{ \begin{array}{ll} 1, &\mbox{在$ v=k$时, }\\ 0, &\mbox{在$ v\neq k$时, } \end{array} \right. $

$ z_v^{(k-1)}(t_1-\varepsilon_0)=0, \ v=1, 2, \ k=1, 2. $

由经典正则的Sturm-Liouville理论知, 这样的函数$z_v$是存在的.我们将它们延拖到区间$I$上, 并且当$t>t_1-\varepsilon_0$时, 它们等于零.那么$z_1, z_2 \in D(C_{\max})$.按照条件, 算子$C_{\min}$的亏指数为$(1, 1)$, 因此$C_{\max}$按模$C_{\min}$的维数等于2.另一方面, 函数$z_1, z_2$按模$C_{\min}$是线性无关的, 故$C_{\max}$中的任何函数可以表示为

$ f=f_0+\alpha_1 z_1+\alpha_2z_2, $

其中$f_0\in D(C_{\min})$, 而$\alpha_1, \ \alpha_2$为常数.因为对任意的$g\in D(C_{\max})$有$[f_0, g](\infty)=0.$此外, 由$z_1, z_2$的构造可知$[z_v, g](\infty)=0, v=1, 2.$因此有

$ [f, g](\infty)=0, \ f, g\in D(C_{\max}). $

下面设对于任意的$f, g\in D(C_{\max})$有$[f, g](\infty)=0.$若算子$C_{\min}$在$\infty$处是极限圆型的, 则算子$C_{\min}$的亏指数为$(2, 2)$.则由定理3.1知带有不连续条件(1.2)的方程$Mf=0$的两个线性无关解$\varphi_1, \varphi_2$都属于$D(C_{\max})$.故我们得到

$ [\varphi_1, \varphi_2](\infty)=\langle M\varphi_1, \varphi_2\rangle - \langle M\varphi_1, \varphi_2\rangle +[\varphi_1, \varphi_2](0)=[\varphi_1, \varphi_2](0)\neq0. $

这与条件矛盾.则算子$C_{\min}$在$\infty$处是极限点型的.

引理3.3  如果存在序列$\{x_i|x_i\in (t_i, t_{i+1}), i\in {\Bbb N})\}, \{y_i|y_i\in (t_i, t_{i+1}), i\in {\Bbb N})\}$满足如下性质:对任意的$f\in D(C_{\max})$存在$\widetilde{f}\in D(C_{\max})$使得

$ f(x_i)=\widetilde{f}(x_i), f'(x_i)=\widetilde{f}'(x_i), \widetilde{f}(y_i)=\widetilde{f}'(y_i)=0, \quad i\in {\Bbb N}, $

则算子$C_{\min}$在$\infty$点处极限点型的.

证  由引理3.2直接可得

$ [f, g](\infty)=\lim\limits_{i\to \infty}[f_i, g_i]_i(x_i)=\lim\limits_{i\to \infty}[\widetilde{f}_i, \widetilde{g}_i]_i(y_i)=0, \quad f, g \in D(C_{\max}). $

证毕.

定理3.2  算子$C_{\max}$和$C_{\min}$的定义如(2.2)和(2.3)式.如果$q\geq0, $则$C_{\min}$在$\infty$点处是极限点型的, 即$C_{\min}$的亏指数为$(1, 1).$

证  令$h\in C_0^{\infty}(0, 1)$且满足$h(\frac{1}{2})=1, \ h'(\frac{1}{2})=0.$定义函数$\phi(t)$如下

$ \begin{array}{l} \phi(t)=\left\{ \begin{array}{ll} h\Big(\frac{t-c_i+\varepsilon_0}{2\varepsilon_0}\Big), &t\in I_i=(c_i-\varepsilon_0, c_i+\varepsilon_0), \quad i\in {\Bbb N}\setminus \{0\}, \\[3mm] 0, & t\in [0, \infty)\setminus \bigcup\limits_{i=1}^{\infty}(c_i-\varepsilon_0, c_i+\varepsilon_0), \end{array} \right. \end{array} $

其中$c_i=\frac{t_i+t_{i+1}}{2}.$易知, $\phi \in C^{\infty}[0, \infty)$且其任意阶导函数是有界的.令$\psi_i=\chi_{I_i}\phi.$对任意的实值函数$f\in D(C_{\max}), $有

$ \begin{array}{l} \|\psi'_if'_i\|_i^2=h_i\int_{I_i}[\psi'^2_i(t)f_i'(t)]f_i'(t){\rm d}t=-h_i\int_{I_i}f_i(t)[\psi_i'^2(t)f_i'(t)]'{\rm d}t\\ =-2h_i\int_{I_i}f_i(t)\psi_i'(t)\psi_i''(t)f_i'(t){\rm d}t+h_i\int_{I_i}\psi_i'^2(t)f_i(t)(M_if_i)(t)\\ -h_i\int_{I_i}q_i(t)\psi_i'^2(t)f_i^2(t){\rm d}t\\ =h_i\int_{I_i}(\psi_i'(t)\psi_i''(t))'f_i^2(t){\rm d}t+h_i\int_{I_i}\psi_i'^2(t)f_i(t)(M_if_i)(t){\rm d}t\\ -h_i\int_{I_i}q_i(t)\psi_i'^2(t)f_i^2(t){\rm d}t\\ \leq \frac{\pi^4}{8\varepsilon_0^4}\|f_i\|_i^2+\frac{\pi^2}{4\varepsilon_0^2}(\|f_i\|_i^2+\|M_if_i\|_i^2)\\ =\Big(\frac{\pi^4}{8\varepsilon_0^4}+\frac{\pi^2}{4\varepsilon_0^2}\Big)\|f_i\|_i^2+\frac{\pi^2}{4\varepsilon_0^2}\|M_if_i\|_i^2. \end{array} $

因此可得

$ \|\phi'f'\|_H^2\leq\Big(\frac{\pi^4}{8\varepsilon_0^4}+\frac{\pi^2}{4 \varepsilon_0^2}\Big)\|f\|_H^2+\frac{\pi^2}{4\varepsilon_0^2}\|Mf\|_H^2, $

即, $\phi'f'\in H.$又因为$\phi f$在点$\{t_i|i\in {\Bbb N}\setminus \{0\}\}$附近取值为零且

$ -(\phi f)''+q\phi f=-\phi'' f-2\phi'f'-\phi Mf \in H, $

所以$\phi f\in D(C_{\max})$.令$\widetilde{f}=\phi f, x_i=c_i$且$y_i=c_i-\varepsilon_0$, 则由引理3.3可推得定理的结论.

进而可得到如下的推论:

推论3.2  如果$q$下有界, 则算子$C_{\min}$在$\infty$处是极限点型的, 即算子$C_{\min}$的亏指数为$(1, 1)$.

下面的定理考虑$q$没有下界的情形.为了使算子$C_{\min}$是极限点型的, 必须对$q$趋于$-\infty$的速度加以某些限制, 并对不连续条件系数和区间的分割加以某些限制.

定理3.3  如果

$ q(t)\geq-C_1|t|^2, \ \beta_{1i}=0, \ \frac{\alpha_{2i}}{\beta_{2i}}\geq -C_2|t_i|, \quad i\geq1, $

$ \begin{equation}\label{tj3.1} \inf\limits_{n\geq1} \frac{1}{\prod\limits_{i=1}^{n}\beta^2 _{2i}}>0, \end{equation} $

(3.5)

其中$C_1, C_2, \alpha$为正实数.则算子$C_{\min}$在$\infty$处是极限点型的, 即算子$C_{\min}$的亏指数为$(1, 1).$

证   (1)设$\varphi$是带有不连续条件的方程$My=0$的属于$H$的解.不妨设$\varphi$是实函数.对任意的$a_i>0, $有

$ \begin{array}{l} \int^{t_i}_{t_{i-1}}\left(\varphi^2(t_i-0)-\varphi^2(t)\right){\rm d}t=\int^{t_i}_{t_{i-1}}\int^{t_i}_{t}2\varphi(t)\varphi'(t){\rm d}t\\ =2(t_i-t_{i-1})\int^{t_i}_{t_{i-1}}|\varphi(t)||\varphi'(t)|{\rm d}t\\ \leq (t_i-t_{i-1})\left(a_i\int^{t_i}_{t_{i-1}}|\varphi(t)|^2{\rm d}t+\frac{1}{a_i}\int^{t_i}_{t_{i-1}}|\varphi'(t)|^2{\rm d}t\right ). \end{array} $

故有

$ \varphi^2(t_i-0)\leq\left(\frac{1}{t_i-t_{i-1}}+a_i\right)\int^{t_i}_{t_{i-1}}|\varphi(t)|^2{\rm d}t+\frac{1}{a_i}\int^{t_i}_{t_{i-1}}|\varphi'(t)|^2{\rm d}t. $

令$V_{i}=-\min\left\{0, \frac{\alpha_{2i}}{\beta_{2i}}\right\}$, 则有

$ \begin{array}{l} \sum^n_{i=1}\frac{V_{i}}{t^2_{i}}h_{i-1}\varphi^2(t_i-0) \leq \sum^n_{i=1}\frac{V_{i}}{t^2_{i}}\left(\frac{1}{t_i-t_{i-1}}+a_i\right)h_{i-1}\int^{t_i}_{t_{i-1}}|\varphi(t)|^2{\rm d}t \\ +\sum^n_{i=1}\frac{V_{i}}{t^2_{i}a_i}h_{i-1}\int^{t_i}_{t_{i-1}}|\varphi'(t)|^2{\rm d}t. \end{array} $

再令$a_i=2C_2t_i, $则我们得到

$ \lim\sup\frac{V_{i}}{a_i}=\lim\sup\frac{-\min\left\{0, \frac{\alpha_{2i}}{\beta_{2i}}\right\} }{2C_2t_i} <1, $

且存在$\alpha\in(0, 1), \ C>0, $使得

$ \begin{array}{l} \sum^n_{i=1}\frac{V_{i}}{t^2_{i}a_i}h_{i-1}\int^{t_i}_{t_{i-1}}| \varphi'(t)|^2{\rm d}t&\leq &\sum^n_{i=1}\frac{V_{i}}{a_i}h_{i-1}\int^{t_i}_{t_{i-1}}\frac{|\varphi'(t)|^2}{t^2}{\rm d}t\\ &\leq &C+(1-\alpha)\sum^{n}_{i=1}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{|\varphi'(t)|^2}{t^2}{\rm d}t. \end{array} $

再由假设$\inf\limits_{i\geq 1}(t_i-t_{i-1}) \geq 4\varepsilon_0$可得

$ \begin{array}{l} \lim\sup\frac{V_{i}}{t^2_i}\left(\frac{1}{t_i-t_{i-1}}+a_i\right)&= &\lim\sup\frac{-\min\left\{0, \frac{\alpha_{2i}}{\beta_{2i}}\right\}}{t^2_i}\left(\frac{1}{t_i-t_{i-1}}+2C_2t_i\right)\\ \leq \frac{C_1}{t_i}\left( \frac{1}{4\varepsilon_0}+2C_2t_i\right) <\infty. \end{array} $

由$\varphi\in H$可知

$ \sum^n_{i=1}\frac{V_{i}}{t^2_{i}}\left(\frac{1}{t_i-t_{i-1}}+a_i\right)h_{i-1}\int^{t_i}_{t_{i-1}}|\varphi(t)|^2{\rm d}t <\infty, $

则

$ \begin{array}{l}\label{bds1} \sum^n_{i=1}\frac{V_{i}}{t^2_{i}}h_{i-1}\varphi^2(t_i-0)\leq \widetilde{C}+(1-\alpha)\sum^n_{i=1}h_{i-1}\int^{t_i}_{t_{i-1}}\frac{|\varphi'(t)|^2}{t^2}{\rm d}t, \end{array} $

(3.6)

其中$\widetilde{C}$为正常数.

(2) 利用$q(t)\geq-C_1|t|^2, \ M\varphi=0$, 得到

$ \begin{array}{l} h_n\int^t_{t_n}\frac{\varphi''(s)\varphi(s)}{s^2}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi''(s)\varphi(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi''(s)\varphi(s)}{s^2}{\rm d}s\\ =h_n\int^t_{t_n}\frac{q(s)\varphi^2(s)}{s^2}{\rm d}s+\sum^{n}_{i=2}h_i\int^{t_{i}}_{t_{i-1}}\frac{q(s)\varphi^2(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{q(s)\varphi^2(s)}{s^2}{\rm d}s\\ \geq-C_1\|\varphi\|^2. \end{array} $

另一方面, 对于上述不等式左边利用分部积分和不连续条件可以得到

$ \begin{array}{l} h_n\int^t_{t_n}\frac{\varphi''(s)\varphi(s)}{s^2}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi''(s)\varphi(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi''(s)\varphi(s)}{s^2}{\rm d}s\\ %=h_n\frac{\varphi'(t)\varphi(t)}{t^2}-\frac{\varphi'(c)\varphi(c)}{c^2}+\sum^n_{i=1}\left\{h_{i-1}\frac{\varphi'(t_{i}-0)\varphi(t_{i}-0)}{t^2_i}-h_{i}\frac{\varphi'(t_{i}+0)\varphi(t_{i}+0)}{t^2_i}\right\}\\ =h_n\frac{\varphi'(t)\varphi(t)}{t^2}-\frac{\varphi'(c)\varphi(c)}{c^2}-\sum^n_{i=1}h_{i-1}\frac{\alpha_{2i}}{\beta_{2i}t^2_i}\varphi^2(t_{i}-0)\\ -\left\{h_n\int^t_{t_n}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi'^2(s)}{s^2}{\rm d}s\right\} \\ +2\left\{h_n\int^t_{t_n}\frac{\varphi'(s)\varphi(s)}{s^3}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi'(s)\varphi(s)}{s^3}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi'(s)\varphi(s)}{s^3}{\rm d}s\right\}. \end{array} $

又因为

$ \begin{array}{l} 2\left|h_n\int^t_{t_n}\frac{\varphi'(s)\varphi(s)}{s^3}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi'(s)\varphi(s)}{s^3}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi'(s)\varphi(s)}{s^3}{\rm d}s\right|\\ \leq \frac{2\|\varphi\|}{c^2}\sqrt{\left|h_n\int^t_{t_n}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi'^2(s)}{s^2}{\rm d}s\right|}, \end{array} $

因此结合(3.6)式可得

$ \begin{array}{l} -h_n\frac{\varphi'(t)\varphi(t)}{t^2}+\left\{h_n\int^t_{t_n}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi'^2(s)}{s^2}{\rm d}s\right\} \\ \leq \widetilde{C}+(1-\alpha)\left\{h_n\int^t_{t_n}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\sum^n_{i=1}h_{i-1}\int^{t_i}_{t_{i-1}}\frac{|\varphi'(t)|^2}{t^2}{\rm d}t+\int^{t_{1}}_{c}\frac{\varphi'^2(s)}{s^2}{\rm d}s\right\}+C_1\|\varphi\|^2\\ -\frac{\varphi'(c)\varphi(c)}{c^2}+\frac{2\|\varphi\|}{c^2}\sqrt{\left|h_n\int^t_{t_n}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\sum^{n}_{i=2}h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi'^2(s)}{s^2}{\rm d}s\right|}. \end{array} $

令

$ F(t)=h_n\int^t_{t_n}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\mathop \sum \limits_{i = 2}^n h_{i-1}\int^{t_{i}}_{t_{i-1}}\frac{\varphi'^2(s)}{s^2}{\rm d}s+\int^{t_{1}}_{c}\frac{\varphi'^2(s)}{s^2}{\rm d}s, $

$ M=\widetilde{C}+C_1\|\varphi\|^2-\frac{\varphi'(c)\varphi(c)}{c^2}, $

则

$ \begin{array}{l}\label{bds2} h_n\frac{\varphi'(t)\varphi(t)}{t^2}\geq\alpha F(t)-\frac{2\|\varphi\|}{c^2}\sqrt{F(t)}-M. \end{array} $

(3.7)

若假设$F(t)\rightarrow \infty, \ t\rightarrow \infty, $则存在$n_0, $当$n\geq n_0$时, 有

$ h_n\frac{\varphi'(t)\varphi(t)}{t^2}>0, \ t\in (t_n, t_{n+1}). $

这说明了当$n\geq n_0$时$\varphi^2$在每个区间$(t_n, t_{n+1})$内是单增的.再由条件(3.5)知

$ h_n\varphi^2(t_i-0)=\frac{1}{\prod\limits_{i=1}^{n}\beta^2 _{2i}}\varphi^2(t_1-0)\geq \inf\limits_{n\geq1}\frac{1}{\prod\limits_{i=1}^{n}\beta^2 _{2i}}\varphi^2(t_1-0)>0, \ t\in (t_n, t_{n+1}). $

故有

$ \begin{array}{l} \sum\limits_{n=n_0}^{\infty}h_n\int_{t_n}^{t_{n+1}}\varphi^2(t)&\geq&\sum\limits_{n=n_0}^{\infty}h_n\varphi^2(t_n+0)(t_{n+1}-t_n)\\ &\geq&\sum\limits_{n=n_0}^{\infty}\frac{1}{4\varepsilon_0}\inf\limits_{n\geq1}\frac{1}{\prod\limits_{i=1}^{n}\beta^2 _{2i}}\varphi^2(t_1-0). \end{array} $

这与$\varphi\in H$矛盾.故

$ \int^{t_1}_{c}\frac{|\varphi|^2}{t^2}+\sum^{\infty}_{i=1}h_i\int^{t_{i+1}}_{t_i}\frac{|\varphi|^2}{t^2} <\infty, \ c\in (0, t_1). $

(3) 设$\varphi, \ \phi$是带有不连续条件(1.2)的方程$My=0$满足

$ \varphi'(0)\phi(0)-\varphi(0)\phi'(0)=1 $

的两个线性无关的解.故有

$ \begin{equation}\label{bds3} h_i(\varphi'(t)\phi(t)-\varphi(t)\phi'(t))=1, \ t\in(t_{i}, t_{i+1}). \end{equation} $

(3.8)

构造函数$\widetilde{\varphi}(t), \ \widetilde{\phi}(t)$如下

$ \widetilde{\varphi}(t)=\left\{ \begin{array}{ll} \varphi(t), &t\in [c, t_{1}), \hbox{} \\ \sqrt{h_i}\varphi(t_i+0), &t=t_i+0, \hbox{} \\ \sqrt{h_i}\varphi(t), &t\in (t_i, t_{i+1}), \hbox{} \\ \sqrt{h_i}\varphi(t_{i+1}-0), &t=t_{i+1}-0, \hbox{} \end{array} \right. i\geq1, $

$ \widetilde{\phi}(t)=\left\{ \begin{array}{ll} \phi(t), &t\in [c, t_{1}), \hbox{} \\ \sqrt{h_i}\phi(t_i+0), &t=t_i+0, \hbox{} \\ \sqrt{h_i}\phi(t), &t\in (t_i, t_{i+1}), \hbox{} \\ \sqrt{h_i}\phi(t_{i+1}-0), &t=t_{i+1}-0, \hbox{} \end{array} \right. i\geq1. $

若假定$\varphi, \ \phi\in H, $则$\widetilde{\varphi}(t), \ \frac{\widetilde{\varphi}(t)}{t}, \ \widetilde{\phi}(t), \ \frac{\widetilde{\phi}(t)}{t}\in L^2[c, \infty).$从而结合(3.8)式, 我们得到

$ \begin{array}{l} \int_c^{\infty}\frac{1}{t}{\rm d}t=\int_c^{\infty}\left\{\frac{\widetilde{\varphi}'(t)}{t}\widetilde{\phi}(t)-\widetilde{\varphi}(t)\frac{\widetilde{\phi}'(t)}{t}\right\}{\rm d}t\\ \leq \sqrt{\int_c^{\infty}\frac{\widetilde{\varphi}'^2(t)}{t^2}{\rm d}t}\sqrt{\int_c^{\infty}\widetilde{\phi}^2(t){\rm d}t}+\sqrt{\int_c^{\infty}\widetilde{\varphi}^2(t){\rm d}t}\sqrt{\int_c^{\infty}\frac{\widetilde{\phi}'^2(t)}{t^2}{\rm d}t}\\ &<&\infty. \end{array} $

这与$\frac{1}{t}\notin L^2[c, \infty)$矛盾.故$\varphi, \ \phi$中至少一个不属于$H$.因此算子$C_{\min}$在$\infty$处是极限点型的, 即算子$C_{\min}$的亏指数为$(1, 1)$.