﻿ Atkinson类型的具有分布势函数的Sturm-Liouville问题的逆谱问题

# Atkinson类型的具有分布势函数的Sturm-Liouville问题的逆谱问题Inverse Sturm-Liouville Problems with Distribution Potentials of Atkinson Type

Abstract: In this paper, the inverse Sturm-Liouville problems with distribution potentials of Atkinson type are studied. We use the conclusions of the inverse eigenvalue problems of Jacobi matrix and cyclic Jacobi matrix to obtain the corresponding inverse Sturm-Liouville problems with distribution po-tentials which have a finite spectrum.

1. 引言

2015年，闫军在其博士论文中详细介绍了具有分布势函数的S-L问题的有限谱理论 [7] ，给出了具有分布势函数的S-L问题的矩阵表示。2016年，唐松林在其硕士论文中也对具有分布势函数的S-L问题的多种带有转移条件的情况进行研究 [8] 。近年来，具有分布势函数的S-L问题引起了一大批数学工作者的关注与讨论，相关成果可参见 [9] [10] [11] [12] 。

2. 预备知识

$VX=\lambda WX,$ (1)

$-{\left(p\left[{y}^{\prime }+sy\right]\right)}^{\prime }+sp\left[{y}^{\prime }+sy\right]+qy=\lambda wy,\text{\hspace{0.17em}}\text{\hspace{0.17em}}I=\left(a,b\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\infty (2)

$r=\frac{1}{p},q,w,s\in L\left(I,ℝ\right),$ (3)

$-{\left({y}^{\left[1\right]}\right)}^{\prime }+s{y}^{\left[1\right]}+qy=\lambda wy,\text{\hspace{0.17em}}\text{\hspace{0.17em}}I=\left(a,b\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\infty

$AY\left(a\right)+BY\left(b\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}Y={\left[y,{y}^{\left[1\right]}\right]}^{\text{T}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}A,B\in {M}_{2}\left(ℝ\right),$ (4)

$rank\left(A,B\right)=2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}AE{A}^{*}=BE{B}^{*},\text{\hspace{0.17em}}\text{\hspace{0.17em}}E=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],$

$\begin{array}{l}\mathrm{cos}\alpha y\left(a\right)-\mathrm{sin}\alpha {y}^{\left[1\right]}\left(a\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le \alpha <\text{π},\\ \mathrm{cos}\beta y\left(b\right)-\mathrm{sin}\beta {y}^{\left[1\right]}\left(b\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\beta \le \text{π};\end{array}$ (5)

$Y\left(b\right)=kY\left(a\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}K=\left({k}_{ij}\right)\in S{L}_{2}\left(ℝ\right),$ (6)

$a={a}_{0}<{a}_{1}<{a}_{2}<\cdots <{a}_{n}=b,$ (7)

$\begin{array}{l}在\left[{a}_{2i},{a}_{2i+1}\right]上,\text{\hspace{0.17em}}\text{\hspace{0.17em}}r=s=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m,\\ {\int }_{{a}_{2i+1}}^{{a}_{2i+2}}{\text{e}}^{2{\int }_{{a}_{2i+1}}^{t}s\left(u\right)\text{d}u}r\left(t\right)\text{d}t>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{{a}_{2i+1}}^{{a}_{2i+2}}s\left(t\right)\text{d}t\ne 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1;\end{array}$ (8)

$在\left[{a}_{2i+1},{a}_{2i+2}\right]上,\text{\hspace{0.17em}}\text{\hspace{0.17em}}q=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1;$ (9)

$\begin{array}{l}在\left[{a}_{2i+1},{a}_{2i+2}\right]上,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}w=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,m-1,\\ {\int }_{{a}_{2i}}^{{a}_{2i+1}}w\left(t\right)\text{d}t>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m.\end{array}$ (10)

3. 具有分布势函数的Sturm-Liouville问题的矩阵表示

$\begin{array}{l}{r}_{2i+1}={\int }_{{a}_{2i+1}}^{{a}_{2i+2}}{\text{e}}^{2{\int }_{{a}_{2i+1}}^{t}s\left(u\right)\text{d}u}r\left(t\right)\text{d}t>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{2i+1}={\int }_{{a}_{2i+1}}^{{a}_{2i+2}}s\left(t\right)\text{d}t\ne 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1,\\ {q}_{2i}={\int }_{{a}_{2i}}^{{a}_{2i+1}}q\left(t\right)\text{d}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{2i}={\int }_{{a}_{2i}}^{{a}_{2i+1}}w\left(t\right)\text{d}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m.\end{array}$ (11)

$\stackrel{¯}{p}\left(t\right)=\left\{\begin{array}{l}\frac{{a}_{2i+2}-{a}_{2i+1}}{{r}_{2i+1}{s}_{2i+1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i+1},{a}_{2i+2}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1;\\ \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i},{a}_{2i+1}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m,\end{array}\text{ }\left({s}_{2i+1}\ne 0\right),$

$\stackrel{¯}{p}\left(t\right)=\left\{\begin{array}{l}\frac{{a}_{2i+2}-{a}_{2i+1}}{{r}_{2i+1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i+1},{a}_{2i+2}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1;\\ \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i},{a}_{2i+1}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m,\end{array}\text{ }\left({s}_{2i+1}=0\right),$

$\stackrel{¯}{s}\left(t\right)=\left\{\begin{array}{l}\frac{{s}_{2i+1}}{{a}_{2i+2}-{a}_{2i+1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i+1},{a}_{2i+2}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1;\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i},{a}_{2i+1}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m,\end{array}$

$\stackrel{¯}{q}\left(t\right)=\left\{\begin{array}{l}\frac{{q}_{2i}}{{a}_{2i+1}-{a}_{2i}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i},{a}_{2i+1}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m;\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i+1},{a}_{2i+2}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1,\end{array}$

$\stackrel{¯}{w}\left(t\right)=\left\{\begin{array}{l}\frac{{w}_{2i}}{{a}_{2i+1}-{a}_{2i}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i},{a}_{2i+1}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m;\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{a}_{2i+1},{a}_{2i+2}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1.\end{array}$

$-{\left(\stackrel{¯}{p}\left[{y}^{\prime }+\stackrel{¯}{s}y\right]\right)}^{\prime }+\stackrel{¯}{s}\stackrel{¯}{p}\left[{y}^{\prime }+\stackrel{¯}{s}y\right]+\stackrel{¯}{q}y=\lambda \stackrel{¯}{w}y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}I=\left(a,b\right),$ (12)

${P}_{\alpha \beta }=\left[\begin{array}{ccccc}\frac{1}{{r}_{1}}+\mathrm{cot}\alpha & -\frac{{\text{e}}^{{s}_{1}}}{{r}_{1}}& & & \\ -\frac{{\text{e}}^{{s}_{1}}}{{r}_{1}}& \frac{{\text{e}}^{2{s}_{1}}}{{r}_{1}}+\frac{1}{{r}_{3}}& -\frac{{\text{e}}^{{s}_{3}}}{{r}_{3}}& & \\ & \cdots & \cdots & \cdots & \\ & & -\frac{{\text{e}}^{{s}_{2m-3}}}{{r}_{2m-3}}& \frac{{\text{e}}^{2{s}_{2m-3}}}{{r}_{2m-3}}+\frac{1}{{r}_{2m-1}}& -\frac{{\text{e}}^{{s}_{2m-1}}}{{r}_{2m-1}}\\ & & & -\frac{{\text{e}}^{{s}_{2m-1}}}{{r}_{2m-1}}& \frac{{\text{e}}^{2{s}_{2m-1}}}{{r}_{2m-1}}-\mathrm{cot}\beta \end{array}\right],$ (13)

$\begin{array}{l}{Q}_{\alpha \beta }=diag\left({q}_{0},{q}_{2},\cdots ,{q}_{2m-2},{q}_{2m}\right),\\ {W}_{\alpha \beta }=diag\left({w}_{0},{w}_{2},\cdots ,{w}_{2m-2},{w}_{2m}\right).\end{array}$ (14)

$\begin{array}{l}{\left({P}_{\alpha \beta }+{Q}_{\alpha \beta }\right)}_{1}={P}_{0\beta }+{Q}_{0\beta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left({P}_{\alpha \beta }+{Q}_{\alpha \beta }\right)}^{1}={P}_{\alpha \pi }+{Q}_{\alpha \pi },\\ {\left({P}_{\alpha \pi }+{Q}_{\alpha \pi }\right)}_{1}={P}_{0\pi }+{Q}_{0\pi },\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left({P}_{0\beta }+{Q}_{0\beta }\right)}^{1}={P}_{0\pi }+{Q}_{0\pi }.\end{array}$

${P}_{I}=\left[\begin{array}{ccccc}\frac{1}{{r}_{1}}-\frac{{k}_{11}}{{k}_{12}}& -\frac{{\text{e}}^{{s}_{1}}}{{r}_{1}}& & & \frac{1}{{k}_{12}}\\ -\frac{{\text{e}}^{{s}_{1}}}{{r}_{1}}& \frac{{\text{e}}^{2{s}_{1}}}{{r}_{1}}+\frac{1}{{r}_{3}}& -\frac{{\text{e}}^{{s}_{3}}}{{r}_{3}}& & \\ & \cdots & \cdots & \cdots & \\ & & -\frac{{\text{e}}^{{s}_{2m-3}}}{{r}_{2m-3}}& \frac{{\text{e}}^{2{s}_{2m-3}}}{{r}_{2m-3}}+\frac{1}{{r}_{2m-1}}& -\frac{{\text{e}}^{{s}_{2m-1}}}{{r}_{2m-1}}\\ \frac{1}{{k}_{12}}& & & -\frac{{\text{e}}^{{s}_{2m-1}}}{{r}_{2m-1}}& \frac{{\text{e}}^{2{s}_{2m-1}}}{{r}_{2m-1}}-\frac{{k}_{22}}{{k}_{12}}\end{array}\right],$ (15)

$\begin{array}{l}{Q}_{I}=diag\left({q}_{0},{q}_{2},\cdots ,{q}_{2m-2},{q}_{2m}\right),\\ {W}_{I}=diag\left({w}_{0},{w}_{2},\cdots ,{w}_{2m-2},{w}_{2m}\right).\end{array}$ (16)

${P}_{\theta }=\left[\begin{array}{ccccc}-{k}_{11}{k}_{21}+\frac{1}{{r}_{1}}+{k}_{11}^{2}\frac{{\text{e}}^{{s}_{2m-1}}}{{r}_{2m-1}}& -\frac{{\text{e}}^{{s}_{1}}}{{r}_{1}}& & & -{k}_{11}\frac{{\text{e}}^{{s}_{2m-1}}}{{r}_{2m-1}}\\ -\frac{{\text{e}}^{{s}_{1}}}{{r}_{1}}& \frac{{\text{e}}^{2{s}_{1}}}{{r}_{1}}+\frac{1}{{r}_{3}}& -\frac{{\text{e}}^{{s}_{3}}}{{r}_{3}}& & \\ & \cdots & \cdots & \cdots & \\ & & -\frac{{\text{e}}^{{s}_{2m-5}}}{{r}_{2m-5}}& \frac{{\text{e}}^{2{s}_{2m-5}}}{{r}_{2m-5}}+\frac{1}{{r}_{2m-3}}& -\frac{{\text{e}}^{{s}_{2m-3}}}{{r}_{2m-3}}\\ -{k}_{11}\frac{{\text{e}}^{{s}_{2m-1}}}{{r}_{2m-1}}& & & -\frac{{\text{e}}^{{s}_{2m-3}}}{{r}_{2m-3}}& \frac{{\text{e}}^{2{s}_{2m-3}}}{{r}_{2m-3}}+\frac{1}{{r}_{2m-1}}\end{array}\right],$ (17)

$\begin{array}{l}{Q}_{\theta }=diag\left({q}_{0}+{k}_{11}^{2}{q}_{2m},{q}_{2},\cdots ,{q}_{2m-2}\right),\\ {W}_{\theta }=diag\left({w}_{0}+{k}_{11}^{2}{w}_{2m},{w}_{2},\cdots ,{w}_{2m-2}\right).\end{array}$ (18)

4. 矩阵的逆特征值问题

$J=\left[\begin{array}{ccccc}{c}_{1}& {d}_{1}& & & \\ {d}_{1}& {c}_{2}& {d}_{2}& & \\ & \cdots & \cdots & \cdots & \\ & & {d}_{k-2}& {c}_{k-1}& {d}_{k-1}\\ & & & {d}_{k-1}& {c}_{k}\end{array}\right].$ (19)

${\lambda }_{1}<{\mu }_{1}<{\lambda }_{2}<{\mu }_{2}<\cdots <{\lambda }_{k-1}<{\mu }_{k-1}<{\lambda }_{k}.$ (20)

$W=diag\left({w}_{1},\cdots ,{w}_{k}\right)$ 是一个对角矩阵，其中 ${w}_{i}>0,i=1,\cdots ,k$ 。则存在唯一的负Jacobi矩阵 $M\in {\mathbb{M}}_{k}$ 使得： $\sigma \left(M,W\right)=\left\{{\lambda }_{i}:i=1,\cdots ,k\right\}$$\sigma \left({M}_{1},{W}_{1}\right)=\left\{{\mu }_{i}:i=1,\cdots ,k-1\right\}$

${J}_{c}=\left[\begin{array}{ccccc}{c}_{1}& {d}_{1}& & & {d}_{k}\\ {d}_{1}& {c}_{2}& {d}_{2}& & \\ & \cdots & \cdots & \cdots & \\ & & {d}_{k-2}& {c}_{k-1}& {d}_{k-1}\\ {d}_{k}& & & {d}_{k-1}& {c}_{k}\end{array}\right].$ (21)

i) ${\lambda }_{1}\le {\mu }_{1}\le {\lambda }_{2}\le {\mu }_{2}\le \cdots \le {\lambda }_{k-1}\le {\mu }_{k-1}\le {\lambda }_{k}$

ii) ${\mu }_{i}\ne {\mu }_{j}$$i\ne j$

iii) $\exists d>0$ ，对于 $j=1,\cdots ,k-1$

$\underset{i=1}{\overset{k}{\prod }}|{\mu }_{j}-{\lambda }_{i}|\ge 2d\left[1+{\left(-1\right)}^{k+1-j}\right].$ (22)

$W=diag\left({w}_{1},\cdots ,{w}_{k}\right)$ 是一个对角矩阵，其中 ${w}_{i}>0,\text{\hspace{0.17em}}i=1,\cdots ,k$ 。则存在唯一的负循环Jacobi矩阵 $N\in {\mathbb{M}}_{k}$ ，使得 ${\prod }_{i=1}^{k}{d}_{i}=d$ ，并且： $\sigma \left(N,W\right)=\left\{{\lambda }_{i}:i=1,\cdots ,k\right\}$$\sigma \left({N}_{1},{W}_{1}\right)=\left\{{\mu }_{i}:i=1,\cdots ,k-1\right\}$

5. 主要结论及其证明

a) 存在函数 $r,q\in L\left(I,ℝ\right)$ 满足(8)和(9)使得S-L问题(2)，(5)及其等价类的谱为

$\sigma \left(\alpha ,\beta \right)=\left\{{\lambda }_{i}:i=1,\cdots ,k\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left(0,\beta \right)=\left\{{\mu }_{i}:i=1,\cdots ,k-1\right\}.$

b) 存在函数 $r,q\in L\left(I,ℝ\right)$ 满足(8)和(9)使得S-L问题(2)，(5)及其等价类的谱为

$\sigma \left(\alpha ,\beta \right)=\left\{{\lambda }_{i}:i=1,\cdots ,k\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left(\alpha ,\text{π}\right)=\left\{{\mu }_{i}:i=1,\cdots ,k-1\right\}.$

${s}_{2i+1}={\int }_{{a}_{2i+1}}^{{a}_{2i+2}}s\left(t\right)\text{d}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m-1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{2i}={\int }_{{a}_{2i}}^{{a}_{2i+1}}w\left(t\right)\text{d}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,m,$

${W}_{\alpha \beta }=diag\left({w}_{0},{w}_{2},\cdots ,{w}_{2m}\right).$

$\sigma \left(M,W\right)=\left\{{\lambda }_{i}:i=1,\cdots ,m+1\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left({M}_{1},{W}_{1}\right)=\left\{{\mu }_{i}:i=1,\cdots ,m\right\}.$

$\begin{array}{l}{r}_{2i-1}=-\frac{{\text{e}}^{{s}_{2i-1}}}{{d}_{i}},\text{\hspace{0.17em}}i=1,\cdots ,m;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{0}={c}_{1}-\frac{1}{{r}_{1}}-\mathrm{cot}\alpha ,\\ {q}_{2i}={c}_{i+1}-\frac{{\text{e}}^{2{s}_{2i-1}}}{{r}_{2i-1}}-\frac{1}{{r}_{2i-1}},\text{\hspace{0.17em}}i=1,\cdots ,m-1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{2m}={c}_{m+1}-\frac{{\text{e}}^{2{s}_{2m-1}}}{{r}_{2m-1}}+\mathrm{cot}\beta .\end{array}$

$\begin{array}{l}\sigma \left({P}_{\alpha \beta }+{Q}_{\alpha \beta },{W}_{\alpha \beta }\right)=\left\{{\lambda }_{i}:i=1,\cdots ,m+1\right\},\\ \sigma \left({P}_{0\beta }+{Q}_{0\beta },{W}_{0\beta }\right)=\left\{{\mu }_{i}:i=1,\cdots ,m\right\}.\end{array}$

$\sigma \left(\alpha ,\beta \right)=\left\{{\lambda }_{i}:i=1,\cdots ,m+1\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left(0,\beta \right)=\left\{{\mu }_{i}:i=1,\cdots ,m\right\}.$

b) 证明方法与a)相同，只需利用推论4.1，引理4.1，引理3.2和注3.1即可，故省略证明细节。

$m=k-1$ 。则对于区间 $I=\left[a,b\right]$ $\left(-\infty 上的任何划分(7)，任何函数 $s\in L\left(I,ℝ\right)$ 满足(8)以及任何函数 $w\in L\left(I,ℝ\right)$ 满足(10)，我们有以下结论：

a) 对于 $\forall \beta \in \left(0,\text{π}\right)$$\exists K=\left({k}_{ij}\right)\in S{L}_{2}\left(ℝ\right)$ 满足 ${k}_{12}<0$$\mathrm{cot}\beta ={k}_{22}/{k}_{12}$ ，并且存在函数 $r,q\in L\left(I,ℝ\right)$ 满足(8)和(9)，使得S-L问题(2)，(6)及其等价类的谱为

$\sigma \left(K\right)=\left\{{\lambda }_{i}:i=1,\cdots ,k\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left(0,\beta \right)=\left\{{\mu }_{i}:i=1,\cdots ,k-1\right\}.$

b) 对于 $\forall \alpha \in \left(0,\text{π}\right)$$\exists K=\left({k}_{ij}\right)\in S{L}_{2}\left(ℝ\right)$ 满足 ${k}_{12}<0$$\mathrm{cot}\alpha =-{k}_{11}/{k}_{12}$ ，并且存在函数 $r,q\in L\left(I,ℝ\right)$ 满足(8)和(9)，使得S-L问题(2)，(6)及其等价类的谱为

$\sigma \left(K\right)=\left\{{\lambda }_{i}:i=1,\cdots ,k\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left(\alpha ,\pi \right)=\left\{{\mu }_{i}:i=1,\cdots ,k-1\right\}.$

${s}_{2i+1}={\int }_{{a}_{2i+1}}^{{a}_{2i+2}}s\left(t\right)\text{d}t,\text{\hspace{0.17em}}i=0,1,\cdots ,m-1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{2i}={\int }_{{a}_{2i}}^{{a}_{2i+1}}w\left(t\right)\text{d}t,\text{\hspace{0.17em}}i=0,1,\cdots ,m,$

${W}_{I}=diag\left({w}_{0},{w}_{2},\cdots ,{w}_{2m}\right).$

$\sigma \left(N,W\right)=\left\{{\lambda }_{i}:i=1,\cdots ,m+1\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left({N}_{1},{W}_{1}\right)=\left\{{\mu }_{i}:i=1,\cdots ,m\right\}.$

$\begin{array}{l}{q}_{0}={c}_{1}-\frac{1}{{r}_{1}}+\frac{{k}_{11}}{{k}_{12}}，\text{\hspace{0.17em}}\\ {q}_{2i}={c}_{i+1}-\frac{{\text{e}}^{2{s}_{2i-1}}}{{r}_{2i-1}}-\frac{1}{{r}_{2i-1}},\text{\hspace{0.17em}}i=1,\cdots ,m-1,\\ {q}_{2m}={c}_{m+1}-\frac{{\text{e}}^{2{s}_{2m-1}}}{{r}_{2m-1}}+\frac{{k}_{22}}{{k}_{12}}.\end{array}$

$\begin{array}{l}\sigma \left({P}_{I}+{Q}_{I},{W}_{I}\right)=\left\{{\lambda }_{i}:i=1,\cdots ,m+1\right\},\\ \sigma \left({P}_{0\beta }+{Q}_{0\beta },{W}_{0\beta }\right)=\left\{{\mu }_{i}:i=1,\cdots ,m\right\}.\end{array}$

$\sigma \left(K\right)=\left\{{\lambda }_{i}:i=1,\cdots ,m+1\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left(0,\beta \right)=\left\{{\mu }_{i}:i=1,\cdots ,m\right\}.$

b) 证明方法与a)相同，利用推论4.3，引理4.2，3.2，3.3和注3.1即可。证明细节略。

$\sigma \left(K\right)=\left\{{\lambda }_{i}:i=1,\cdots ,k\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma \left(0,\text{π}\right)=\left\{{\mu }_{i}:i=1,\cdots ,k-1\right\}.$

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[11] Savchuk, A.M. and Shkalikov, A.A. (2003) Sturm-Liouville Operators with Distribution Potentials. Transactions of the Moscow Mathematical Society, 64, 143-192.

[12] Savchuk, A.M. and Shkalikov, A.A. (2006) On the Eigenvalues of the Sturm-Liouville Operator with Potentials from Sobolev Spaces. Mathematical Notes, 80, 814-832.
https://doi.org/10.1007/s11006-006-0204-6

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