# Partial Inverse Sturm-Liouville Problems

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Hochstadt–Lieberman Problem and Generalizations

#### 2.1. Uniqueness Theorems

**Problem 1**

**Theorem 1**

**Theorem 2**

**Theorem 3**

**Theorem 4**

**Problem 2**

**Theorem 5**

**Problem 3**

**Theorem 6**

**Theorem 7**

#### 2.2. Solvability Conditions and Constructive Solution

**Theorem 8**

- 1.
- The function $h\left(t\right)$ has a bounded derivative, and $h\left(0\right)=0$.
- 2.
- The function $p\left(x\right)$ is bounded on the segment $[0,{\textstyle \frac{1}{2}}]$.
- 3.
- The following inequality holds:$$\underset{0\le t\le 1}{sup}}|{h}^{\prime}\left(t\right)|+\frac{1}{4}{\displaystyle \underset{0\le x\le \frac{1}{2}}{sup}}|p\left(x\right)|<\frac{1}{2}.$$

**Problem 4**

**Theorem 9**

- 1.
- Problem 4 is solvable for the mixed spectral data $({\sigma}_{0},\Lambda )$ if and only if ${\varphi}_{0}\in {\Pi}_{\Lambda}$.
- 2.
- If ${\varphi}_{0}\in {\Pi}_{\Lambda}$, then the solution of Problem 4 is unique, that is, there exists a unique $\sigma \in Re\phantom{\rule{0.166667em}{0ex}}{L}_{2}(0,1)$ and a unique $h\in \mathbb{R}$ such that σ is an extension of ${\sigma}_{0}$ and the spectrum of ${T}_{\sigma ,h}$ coincides with ${\Lambda}^{2}={\left\{{\lambda}_{n}^{2}\right\}}_{n\ge 0}$.

**Theorem 10**

**Method 1**

- 1.
- Find $\Delta \left(\lambda \right)$ by the formula$$\Delta \left(\lambda \right)=\pi ({\lambda}_{0}-\lambda ){\displaystyle \prod _{n=1}^{\infty}}\frac{{\lambda}_{n}-\lambda}{{n}^{2}}.$$
- 2.
- Construct the functions $\Theta \left(\lambda \right)$ and $\Xi \left(\lambda \right)$ using (15) and find their zeros ${\theta}_{n}$, ${\xi}_{n}$, $n\ge 0$.
- 3.
- Calculate the numbers$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& d\left({\xi}_{n}\right)=\Delta \left({\xi}_{n}\right){S}^{\prime}(\pi /2,{\xi}_{n})+\sqrt{{\xi}_{n}}sin({\sqrt{\xi}}_{n}\pi /2),\hfill \\ & {d}_{0}\left({\theta}_{n}\right)=-\Delta \left({\theta}_{n}\right)S(\pi /2,{\theta}_{n})-cos({\sqrt{\theta}}_{n}\pi /2).\hfill \end{array}$$
- 4.
- By interpolation, find the functions$$d\left(\lambda \right)={\displaystyle \sum _{n=0}^{\infty}}d\left({\xi}_{n}\right)\frac{\Xi \left(\lambda \right)}{(\lambda -{\xi}_{n}){\Xi}^{\prime}\left({\xi}_{n}\right)},\phantom{\rule{1.em}{0ex}}{d}_{0}\left(\lambda \right)={\displaystyle \sum _{n=0}^{\infty}}{d}_{0}\left({\theta}_{n}\right)\frac{\Theta \left(\lambda \right)}{(\lambda -{\theta}_{n}){\Theta}^{\prime}\left({\theta}_{n}\right)}.$$
- 5.
- Let ${\Delta}_{1}\left(\lambda \right)=-\sqrt{\lambda}sin(\sqrt{\lambda}\pi /2)+d\left(\lambda \right)$, ${\Delta}_{1}^{0}\left(\lambda \right)=cos(\sqrt{\lambda}\pi /2)+{d}_{1}\left(\lambda \right)$.
- 6.
- Recover $q\left(x\right)$ on $(\pi /2,\pi )$ and H from the Weyl function $M\left(\lambda \right)=-{\displaystyle \frac{{\Delta}_{1}^{0}\left(\lambda \right)}{{\Delta}_{1}\left(\lambda \right)}}$.

**Theorem 11**

- 1.
- ${\lambda}_{k}=-{\lambda}_{k}$,
- 2.
- $-\infty <{\lambda}_{1}^{2}<{\lambda}_{2}^{2}<\dots <{\lambda}_{k}^{2}<\dots $,
- 3.
- ${\lambda}_{k}={\displaystyle \frac{\pi k}{a}}+{\displaystyle \frac{K}{\pi k}}+{\displaystyle \frac{{\beta}_{k}}{k}}$.Here, $K\in \mathbb{R}$, and ${\left\{{\beta}_{k}\right\}}_{k\in {\mathbb{Z}}_{0}}\in {l}_{2}$.If the function $\frac{{s}_{2}(\sqrt{\lambda},a/2)}{{s}_{2}^{\prime}(\sqrt{\lambda},a/2)}$ belongs to the Nevanlinna class, then there exists a real-valued function ${q}_{2}\left(x\right)\in {L}_{2}[0,a/2]$ such that the spectrum of the problems (16) and (17) generated by ${q}_{1}$ and ${q}_{2}$ coincides with ${\left\{{\lambda}_{k}\right\}}_{k\in {\mathbb{Z}}_{0}}$.

#### 2.3. McLaughlin–Polyakov Problem

**Problem 5**

**Theorem 12**

**Theorem 13**

## 3. Partial Inverse Problems on Graphs

#### 3.1. Star-Shaped Graphs

**Problem 6**

**Problem 7**

**Problem 8**

**Problem 9**

**Theorem 14**

**Condition 1.**

**Problem 10**

**Theorem 15**

**Theorem 16**

- 1.
- All the eigenvalues ${\left\{{\lambda}_{nk}\right\}}_{n\in \mathbb{N},\phantom{\rule{0.166667em}{0ex}}k=1,2}$ are distinct;
- 2.
- ${\lambda}_{nk}>0$, $n\in \mathbb{N}$, $k=1,2$;
- 3.
- ${S}_{j}(\pi ,{\lambda}_{nk})\ne 0$, $j=\overline{1,m}$, $n\in \mathbb{N}$, $k=1,2$;
- 4.
- ${z}_{1}\ne {\omega}_{j}$, $j=\overline{1,m}$;
- 5.
- ${S}_{1}(\pi ,0)\ne 0$, ${S}_{1}^{\prime}(\pi ,0)\ne 0$.

**Problem 11**

- Eigenvalue asymptotics;
- Uniqueness;
- A constructive solution.

**Problem 12**

**Theorem 17**

**Theorem 18**

#### 3.2. Simple Graphs with Loops

**Problem 13**

**Problem 14**

**Problem 15**

- Eigenvalue asymptotics;
- Uniqueness;
- Algorithm;
- The solution of the inverse periodic problem with a singular potential.

**Problem 16**

**Problem 17**

- Eigenvalue asymptotics;
- Uniqueness;
- A constructive solution.

#### 3.3. Graphs of a General Structure

- The reconstruction of the potentials on an arbitrary tree graph (graph without cycles) by several spectra, while the potential on one edge is known a priori (see [78]).
- For a tree graph, the reconstruction of the potentials on a connected subtree from parts of several spectra, while the potentials on the remaining edges are known a priori (see [39]).

**Theorem 19**

**Theorem 20**

- 1.
- If ${e}_{f}$ is a boundary edge, the spectra ${\Lambda}_{0}$ and ${\Lambda}_{k}$, ${v}_{k}\in \partial \mathcal{G}\setminus \{{v}_{f},{v}_{r}\}$, uniquely determine the potential q on the whole graph $\mathcal{G}$.
- 2.
- If ${e}_{f}$ is an internal edge, the spectra ${\Lambda}_{0}$ and ${\Lambda}_{k}$ ${v}_{k}\in \partial \mathcal{G}\setminus \{{v}_{r1},{v}_{r2}\}$, where ${v}_{r1}\in \partial {P}_{1}$ and ${v}_{r2}\in \partial {P}_{2}$, uniquely determine the potential q on the whole graph $\mathcal{G}$.

**Problem 18**

**Problem 19**

- Uniqueness in the general case;
- A constructive solution for rationally dependent edge lengths.

**Problem 20**

- A uniqueness theorem;
- A constructive solution;
- Sufficient conditions for global solvability;
- Local solvability and stability.

**Condition 2.**

**Condition 3.**

- 1.
- All the values ${\left\{{r}_{k}\right\}}_{k\in \mathcal{K}}$ from (44) are distinct.
- 2.
- ${r}_{k}\notin \{0,{\textstyle \frac{1}{2}}\}$, $k\in \mathcal{K}$.
- 3.
- For each $k\in \mathcal{K}$, there exists $s\in \mathcal{K}$ such that ${r}_{k}+{r}_{s}=1$.
- 4.
- $|\mathcal{K}|=4{n}_{1}$.

**Theorem 21**

## 4. Unified Approach

#### 4.1. Sturm-Liouville Problem with Entire Functions in a Boundary Condition

**Problem 21**

**Problem 22.**

**Theorem 22**

**Theorem 23**

**Method 2**

- 1.
- 2.
- For the basis ${\left\{{v}_{n}\right\}}_{n\ge 0}$, find the biorthonormal basis ${\left\{{v}_{n}^{*}\right\}}_{n\ge 0}$, that is, ${({v}_{n},{v}_{k}^{*})}_{\mathcal{H}}={\delta}_{nk}$, $n,k\ge 0$.
- 3.
- Construct the element $u\in \mathcal{H}$ satisfying (52) using the formula$$u={\displaystyle \sum _{n=0}^{\infty}}\overline{{w}_{n}}{v}_{n}^{*}.$$
- 4.
- Using the elements of $u\left(t\right)=[\overline{N\left(t\right)},\overline{K\left(t\right)}]$, solve Problem 22 with the Cauchy data and find q.

**Theorem 24**

- 1.
- (Separation) and (Complete C) together imply (Complete).
- 2.
- (Separation), (Asymptotics), and (Basis C) together imply (Basis).

**Theorem 25**

**Theorem 26**

**Theorem 27**

- The necessary and sufficient conditions of uniqueness;
- A constructive solution;
- Simple sufficient conditions for uniqueness and the algorithm;
- Sufficient conditions for the global solvability;
- Local solvability and stability.

- Uniqueness;
- A constructive solution;
- Simple sufficient conditions for uniqueness and the algorithm;
- Application to Hochstadt–Lieberman-type problems.

#### 4.2. Applications to Partial Inverse Problems

- The Hochstadt–Lieberman problem (Problem 1);
- The McLaughlin–Polyakov problem (Problem 5);
- A partial inverse problem on a star-shaped graph (Problem 9);
- A partial inverse problem on a graph of an arbitrary structure (Problem 20).

**Problem 23**

**Corollary 1**

**Corollary 2.**

**Corollary 3.**

- 1.
- ${\left\{{\lambda}_{n}\right\}}_{n\ge 1}$ satisfies the asymptotics (21).
- 2.
- The zeros ${\left\{{\theta}_{nj}\right\}}_{n\ge 1,\phantom{\rule{0.166667em}{0ex}}j=1,2}$ of the functions ${\eta}_{j}\left(\lambda \right)$, $j=1,2$, defined by (49) and (50) using the functions $K,N\in {L}_{2}(0,\alpha )$, which are constructed by Method 2, are real and interlace in the sense of (57).

**Theorem 28**

## 5. Other Types of Operators

- Integro-differential operators;
- Functional differential operators with a constant delay;
- Higher-order differential operators;
- Matrix Sturm-Liouville operators.

**Theorem 29**

**Problem 24**

**Theorem 30**

**Problem 25**

**Theorem 31**

**Theorem 32**

**Theorem 33**

**Theorem 34**

**Theorem 35**

## 6. Conclusions

**Problem 26.**

**Problem 27.**

**Problem 28.**

**Problem 29.**

**Problem 30.**

**Problem 31.**

**Problem 32.**

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Bondarenko, N.P.
Partial Inverse Sturm-Liouville Problems. *Mathematics* **2023**, *11*, 2408.
https://doi.org/10.3390/math11102408

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Bondarenko NP.
Partial Inverse Sturm-Liouville Problems. *Mathematics*. 2023; 11(10):2408.
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Bondarenko, Natalia P.
2023. "Partial Inverse Sturm-Liouville Problems" *Mathematics* 11, no. 10: 2408.
https://doi.org/10.3390/math11102408