# Reconstruction of Differential Operators with Frozen Argument

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (A1)
- the operator A is self-adjoint and has simple discrete spectrum.

- (A2)
- the eigenvalues of A are d-separated, i.e.,$$d=\underset{n\in I}{inf}({\lambda}_{n+1}-{\lambda}_{n})>0.$$

- (a)
- the algebraic multiplicity m of an eigenvalue $\mu \in \sigma \left(B\right)\backslash {\sigma}_{0}\left(B\right)$ coincides with the multiplicity l of $z=\mu $ as a zero of F;
- (b)
- if $\mu \in {\sigma}_{0}\left(B\right)$, then the above multiplicities m and l satisfy the relation $m=l+1$;
- (c)
- for any n-tuple ${z}_{1},{z}_{2},\dots ,{z}_{n}$ of pairwise distinct complex numbers and any n-tuple ${m}_{1},{m}_{2},\dots ,{m}_{n}$ of natural numbers, there exists a rank-one perturbation B of A such that every ${z}_{j}$ is an eigenvalue of B of algebraic multiplicity ${m}_{j}$;
- (d)
- the eigenvalues of B can be enumerated as ${\mu}_{n}$, $n\in I$, in such a way that ${\mu}_{n}-{\lambda}_{n}\to 0$ as $\left|n\right|\to \infty $; in particular, B has at most finitely many non-simple eigenvalues.

## 3. Eigenvalue Distribution of the Operator $\mathit{B}$

- (A3)
- the Fourier coefficients ${a}_{n}$ of $\phi $ are uniformly bounded.

**Definition**

**1.**

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Lemma**

**2.**

- (a)
- for every k with $\left|k\right|>{K}_{\epsilon}$ and every $z\in \overline{{R}_{k}}=\partial {R}_{k}\cup {R}_{k}$$$\underset{\begin{array}{c}\left|n\right|>{K}_{\epsilon}\\ n\ne k\end{array}}{\sum ^{\left(1\right)}}\left|\frac{{c}_{n}}{{\lambda}_{n}-z}\right|<\epsilon ;$$
- (b)
- for every $z\in \mathbb{C}\backslash {Q}_{{K}_{\epsilon}^{\prime}}$$$\underset{\left|n\right|\le {K}_{\epsilon}}{\sum ^{\left(1\right)}}\left|\frac{{c}_{n}}{{\lambda}_{n}-z}\right|<\epsilon .$$

**Proof.**

**Corollary**

**1.**

**Lemma**

**3.**

- (a)
- the function F has exactly one zero in ${R}_{k}$;
- (b)
- the functions ${H}_{k}$ and F have the same number of zeros in ${Q}_{k}$.

**Proof.**

**Remark**

**2.**

**Corollary**

**2.**

**Corollary**

**3.**

**Proof of Theorem 1**

## 4. Inverse Spectral Problem

**Theorem**

**2.**

**Lemma**

**4.**

**Proof.**

**Corollary**

**4.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Proof of Theorem 2**

- 1.
- construct the product $\tilde{F}$ of (17);
- 2.
- then calculate the residua $-{c}_{n}$ of $\tilde{F}$ at the points ${\lambda}_{n}$;
- 3.
- 4.

**Corollary**

**5.**

## 5. Examples and Discussions

**Example**

**1.**

**Example**

**2.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Dobosevych, O.; Hryniv, R.
Reconstruction of Differential Operators with Frozen Argument. *Axioms* **2022**, *11*, 24.
https://doi.org/10.3390/axioms11010024

**AMA Style**

Dobosevych O, Hryniv R.
Reconstruction of Differential Operators with Frozen Argument. *Axioms*. 2022; 11(1):24.
https://doi.org/10.3390/axioms11010024

**Chicago/Turabian Style**

Dobosevych, Oles, and Rostyslav Hryniv.
2022. "Reconstruction of Differential Operators with Frozen Argument" *Axioms* 11, no. 1: 24.
https://doi.org/10.3390/axioms11010024