# Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations

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## Abstract

**:**

## 1. Introduction

#### 1.1. Inverse Spectral Problems—Inverse Scattering Problems and Method

**Theorem 1.**

#### 1.2. Inverse Problems for Hyperbolic Equations

## 2. One-Dimensional Problems

#### 2.1. I.M. Gelfand–B.M. Levitan Equation

#### 2.2. V.A. Marchenko Equation—The Inverse Scattering Method

#### 2.3. Krein Equation

#### 2.4. Boundary-Control Method in One-Dimensional Case

**Lemma 1.**

**Lemma 2.**

**Lemma 3.**

#### 2.5. One-Dimensional Inverse Seismic Problem

## 3. Two-Dimensional Analogs of the Approach

#### 3.1. A Two-Dimensional Analog of Gelfand–Levitan Equation

#### 3.2. A Two-Dimensional Analog of Krein Equation

## 4. Numerical Methods for Solving Gelfand–Levitan and Krein Equations

## 5. Numerical Calculations

## 6. Reconstruction of the Velocity $\mathbf{c}(\mathbf{x},\mathbf{y})$ and the Density $\mathbf{\rho}(\mathbf{x},\mathbf{y})$

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The connection between the inverse spectral problem, the inverse scattering problem and the inverse problem in time domain.

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**MDPI and ACS Style**

Kabanikhin, S.; Shishlenin, M.; Novikov, N.; Prokhoshin, N.
Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations. *Mathematics* **2023**, *11*, 4458.
https://doi.org/10.3390/math11214458

**AMA Style**

Kabanikhin S, Shishlenin M, Novikov N, Prokhoshin N.
Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations. *Mathematics*. 2023; 11(21):4458.
https://doi.org/10.3390/math11214458

**Chicago/Turabian Style**

Kabanikhin, Sergey, Maxim Shishlenin, Nikita Novikov, and Nikita Prokhoshin.
2023. "Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations" *Mathematics* 11, no. 21: 4458.
https://doi.org/10.3390/math11214458