## Local Solvability and Stability of the Inverse Spectral Problems for the Discontinuous Sturm-Liouville Problem with the Mixed Given Data

Guo Yan, Xu Xiaochuan,*

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044; Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science and Technology, Nanjing 210044; Jiangsu International Joint Laboratory on System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing 210044

 基金资助: 国家自然科学基金(11901304)

 Fund supported: NSFC(11901304)

Abstract

This paper studies inverse spectral problems for the Sturm-Liouville operator on $(0,1)$ with the Robin boundary conditions and a discontinuity at $x=d\in(0,\frac{1}{2}]$. Suppose that the known data contains one subspectrum, the potential function on $(d,1)$ as well as partial parameters in the right boundary condition and the discontinuous conditions. The paper proves the local solvability and stability for the inverse problems of recovering the potential function on $(0,d)$ and the parameter in left boundary condition, where the known potential and the parameter in the right boundary condition are allowed to contain errors.

Keywords： Sturm-Liouville operator; Discontinuity condition; Local solvability; Stability

Guo Yan, Xu Xiaochuan. Local Solvability and Stability of the Inverse Spectral Problems for the Discontinuous Sturm-Liouville Problem with the Mixed Given Data[J]. Acta Mathematica Scientia, 2024, 44(4): 859-870

## 1 引言

$-y''(x)+q(x)y(x)=\lambda y(x),\quad 0<x<1,$
$y'(0)-hy(0)=0,\quad y'(1)+Hy(1)=0,$
$y(d+0)=a_1y(d-0),\quad y'(d+0)=a_1^{-1}y'(d-0)+a_2y(d-0),$

$\Delta(\lambda)=a_1\varphi(d_1,\lambda)\psi'(d_2,\lambda)+\frac{1}{a_1}\varphi'(d_1,\lambda)\psi(d_2,\lambda)+a_2\varphi(d_1,\lambda)\psi(d_2,\lambda).$

$\rho_n=\sqrt{\lambda_n}=n \pi+\frac{(-1)^n a+b}{n \pi}+\frac{\beta_n}{n},$

$a=\frac{a_2}{a_1+a_1^{-1}}+\frac{a_1-a_1^{-1}}{a_1+a_1^{-1}}\left(\omega_2-\omega_1\right), \quad b=\frac{a_2}{a_1+a_1^{-1}}+\omega_1+\omega_2,\quad \omega_2=H+\frac{1}{2}\int_{0}^{d_2} q_2(t){\rm d}t.$

$d\ne 1/2$ 时, 特征值的渐近性较为复杂, 其首项为两个正弦函数之差的零点[24]. 不失一般性假设$\lambda_{n}<\lambda_{n+1}$, $\lambda_{n}>0$, $n\geq0$. 当方程 (1.1) 的某特征值$\lambda=-a_0$$(a_0>0) 时, 利用谱的平移, 方程两边同时增加b_0$$b_0>a_0$, 使得$\lambda>0$, 可保证特征值都为正数.

$\left(g,w\right)_{\mathcal{H}}=\int_{0}^{d}(g_1(x)w_1(x)+g_2(x)w_2(x)){\rm d}x,\quad \|w\|_{\mathcal{H}}^{2}=\int_{0}^{d}(w_1^{2}(x)+w_2^{2}(x)){\rm d}x,$

$\Delta(\lambda)=(K(\cdot),\nu(\cdot,\lambda))_{\mathcal{H}}-f(\lambda).$

$K(x):=\left( \begin{matrix} K_{1}(x) \\ K_{2}(x) \\ \end{matrix} \right),\quad \nu(x,\lambda):=\left( \begin{matrix} (a_1\psi'(1-d,\lambda)+a_2\psi(1-d,\lambda))\frac{\sin(\rho x)}{\rho} \\ a_1^{-1}\psi(1-d,\lambda)\cos(\rho x) \\ \end{matrix} \right),$
$\begin{split} f(\lambda)=&-\frac{1}{\rho}(a_1\psi'(1-d,\lambda)+a_2\psi(1-d,\lambda))(\rho\cos(\rho d)+\omega_1\sin(\rho d))\\&+a_1^{-1}\psi(1-d,\lambda)(-\rho\sin(\rho d)+\omega_1\cos(\rho d)).\end{split}$

$\lambda=\lambda_{n}$, 由$\Delta(\lambda_n)=0$

$\left(K,\nu_{n}\right)_{\mathcal{H}}=f_{n},\quad n\geq0,$

$\nu_{n}(x):=\nu(x,\lambda_n),\quad f_n=f(\lambda_n),\quad \quad n\geq0.$

$K(x)=\sum_{n=0}^{\infty}(K,\nu_n)\nu_n^{*}(x)=\sum_{n=0}^{\infty}f_n\nu_n^{*}(x),$

## 3 主要定理及其证明

$\Lambda:=\bigg(\sum_{n\in I}(n+1)^2\left|\rho_n-\tilde{\rho}_n\right|^{2}\bigg)^{\frac{1}{2}},$

$Q:=\left|H-\tilde{H}\right|+\|q_2-\tilde{q}_{2}\|_{L^2(0,d_2)},$

$\Lambda\leq\varepsilon,$
$\tilde{H}+\frac{1}{2}\int_{0}^{\frac{1}{2}}\tilde{q}_{2}(t){\rm d}t=H+\frac{1}{2}\int_{0}^{\frac{1}{2}}q_{2}(t){\rm d}t,\quad Q\leq\varepsilon,$

$\Lambda\leq\varepsilon,$
$\tilde{H}+\frac{1}{2}\int_{0}^{d_2}\tilde{q}_{2}(t){\rm d}t=H+\frac{1}{2}\int_{0}^{d_2}q_{2}(t){\rm d}t,\quad Q\leq\varepsilon,$

$T(x,x)=\tilde{H}-H+\frac{1}{2}\int_{0}^{x}(\tilde{q}_2(t)-q_2(t)){\rm d}t,\quad T_0(x,x)=H+\frac{1}{2}\int_{0}^{x}q_2(t){\rm d}t.$

$\max_{0\leq t\leq x}\left|T(x,t)\right|\leq CQ,\quad \left\|T_x(x,\cdot)\right\|_{L^2(0,x)}\leq CQ,\quad \left\|T_t(x,\cdot)\right\|_{L^2(0,x)}\leq CQ,$

$|\tilde{H}|+\|\tilde{q}_2\|_{L^2(0,d_2)}\leq |H|+\|q_2\|_{L^2(0,d_2)}+\varepsilon.$

$T(d_2,d_2)=0.$

$\tilde{\psi}(d_2,\lambda)-\psi(d_2,\lambda)=\frac{1}{\rho}\int_{0}^{d_2}P(t)\sin\rho t{\rm d}t,$
$\tilde{\psi}'(d_2,\lambda)-\psi'(d_2,\lambda)=\int_{0}^{d_2}P_1(t)\cos\rho t{\rm d}t,$

$P(t)=T(d_2,t)T_0(t,t)-T_t(d_2,t)-\int_{t}^{d_2}T(d_2,s)T_{0t}(s,t){\rm d}s,$
$P_1(t)=T_x(d_2,t)+\int_{t}^{d_2}T_x(d_2,s)T_0(s,t){\rm d}s.$

$\left|{\psi}^{(j)}(x,\lambda)\right|\leq C(|\rho|+1)^j{\rm e}^{|\mathrm{Im}\rho|x},\quad \left|\tilde{\psi}^{(j)}(x,\lambda)\right|\leq C(|\rho|+1)^j{\rm e}^{|\mathrm{Im}\rho|x},\quad j=0,1.$

$\sum_{n\in I}\left|\int_{0}^{d_2}P(t)\sin \rho_n t{\rm d}t\right|^2\leq C\|P\|_{L^2(0,d_2)}^2\leq CQ^2,$
$\sum_{n\in I}\left|\int_{-d_2}^{d_2}P_2(t){\rm e}^{\mathrm{i}\rho_nt}{\rm d}t\right|^2\leq C\|P_2\|_{L^2(-d_2,d_2)}^2\leq CQ^2.$

$\sum_{n\in I}\left\|\nu_{n}-\tilde{\nu}_n\right\|_{\mathcal{H}}^{2}=\sum_{n\in I}\bigg(\int_{0}^{d_2}(p_{n,1}^{2}(x)+p_{n,2}^{2}(x)){\rm d}x\bigg) \leq C(\Lambda^2+Q^2),$

$\tilde{\Delta}(\lambda):=(\tilde K(\cdot), \tilde \nu(\cdot,\lambda))_{\mathcal{H}}-\tilde f(\lambda).$

$\tilde{\Delta}(\tilde{\lambda}_n)=0,\quad n\in I.$

${\tilde{\varphi}}(d_1,\lambda):=\cos\rho d_1+\omega_1 \frac{\sin \rho d_1}{\rho}-\int_0^{d_1} \tilde{K}_1(t)\frac{\sin \rho t}{\rho}{\rm d}t,$
${\tilde{\varphi}}'(d_1,\lambda):=-\rho\sin\rho d_1+\omega_1 \cos \rho d_1+\int_0^{d_1} \tilde{K}_2(t)\cos \rho t{\rm d}t.$

$\tilde{\Delta}(\lambda)=a_1\tilde{\varphi}(d_1,\lambda)\tilde{\psi}'(d_2,\lambda)+\frac{1}{a_1}\tilde{\varphi}'(d_1,\lambda)\tilde{\psi}(d_2,\lambda)+a_2\tilde{\varphi}(d_1,\lambda)\tilde{\psi}(d_2,\lambda).$

$f(\sqrt{\tilde{\mu}_{0,n}})=f(\sqrt{\mu_{0,n}})+\dot{f}(\theta_n)(\sqrt{\tilde{\mu}_{0,n}}-\sqrt{\mu_{0,n}}),$

$f(\sqrt{\tilde{\mu}_{0,n}})-\tilde{f}_0(\sqrt{\tilde{\mu}_{0,n}})=\frac{1}{\sqrt{\tilde{\mu}_{0,n}}}\int_0^{d} (K_1-\tilde{K}_1)(t)\sin (\sqrt{\tilde{\mu}_{0,n}}t){\rm d}t=\dot{f}(\theta_n)(\sqrt{\tilde{\mu}_{0,n}}-\sqrt{\mu_{0,n}}).$

$\sqrt{\tilde{\mu}_{0,n}}-\sqrt{\mu_{0,n}}=\frac{1}{\dot{f}(\theta_n)\sqrt{\tilde{\mu}_{0,n}}}\int_0^{d} (K_1-\tilde{K}_1)(t)\sin (\sqrt{\tilde{\mu}_{0,n}}t){\rm d}t.$

$\left|\sqrt{\tilde{\mu}_{0,n}}-\sqrt{\mu_{0,n}}\right| \leq \frac{C}{n+1}\left|\int_0^{d} (K_1-\tilde{K}_1)(t)\sin ((2n+1)\pi t){\rm d}t\right|+\frac{C\|K_1-\tilde{K}_1\|_{L^2(0,d)}}{(n+1)^2},$
$\bigg(\sum_{n\ge0}(n+1)^2\left|\sqrt{\mu_{0,n}}-\sqrt{\tilde{\mu}_{0,n}}\right|^{2}\bigg)^{\frac{1}{2}}\le C\|K_1-\tilde{K}_1\|_{L^2(0,d)}.$

$\bigg(\sum_{n\ge0}\left|\mu_{0,n}-\tilde{\mu}_{0,n}\right|^{2}\bigg)^{\frac{1}{2}}\le C\|K_1-\tilde{K}_1\|_{L^2(0,d)},$

$\bigg(\sum_{n\ge0}\left|\mu_{1,n}-\tilde{\mu}_{1,n}\right|^{2}\bigg)^{\frac{1}{2}}\le C\|K_2-\tilde{K}_2\|_{L^2(0,d)}.$

\begin{align*}K-\tilde{K}&=\sum_{n\in I} f_n\chi_n-\sum_{n\in I} \tilde{f}_n\tilde{\chi}_n=(A^*)^{-1}\sum_{n\in I}f_ne_n-(\tilde{A}^*)^{-1}\sum_{n\in I}\tilde{f}_ne_n\notag\\& =(A^*)^{-1}\sum_{n\in I}(f_n-\tilde{f}_n)e_n+\left((A^*)^{-1}-(\tilde{A}^*)^{-1}\right)\sum_{n\in I}\tilde{f}_ne_n.\end{align*}

\begin{align*}\left\|K-\tilde{K}\right\|_{\mathcal{H}}&\leq \left\|(A^*)^{-1}\right\|\bigg|\sum_{n\in I}(f_n-\tilde{f}_n)e_n\bigg|+\left\|(A^*)^{-1}-(\tilde{A}^*)^{-1}\right\|\bigg|\sum_{n\in I}\tilde{f}_ne_n\bigg|\notag \leq C (F+\mathcal{V}),\end{align*}

$\nu_{2k}^0(x)=(-1)^k\frac{1}{a_1}\left(\begin{matrix} 0 \\\cos(2k\pi x) \\\end{matrix}\right),\quad\nu_{2k+1}^0(x)=(-1)^{k+1}a_1\left(\begin{matrix} \sin((2k+1)\pi x) \\ 0 \\\end{matrix}\right),\quad k\geq0.$

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