# Trace Formulae for Second-Order Differential Pencils with a Frozen Argument

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Main Results

**Theorem 1**

**Theorem 2**

**Lemma 1**

**Theorem 3.**

**Proof.**

## 3. Conclusions

- 1.
- Operator ${L}_{\alpha ,\beta}$ is non-selfadjoint which may have complex eigenvalues with multiplicity; however, the method we use allows us dealing with the regularized sum of eigenvalues in the whole meaning.
- 2.
- The regularized trace of ${L}_{\alpha ,\beta}$ depends only on the value of ${q}_{1}\left(x\right)$ at the frozen point a, regardless of the boundary conditions and the potential ${q}_{0}\left(x\right)$.
- 3.
- In the study of inverse spectral problem of ${L}_{\alpha ,\beta}$, the rationality of frozen argument a is important. Whether a is rational leads to different approachs of inverse spectral problem. However, we do not need this distinction while calculating the trace formulae.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Hu, Y.-T.; Şat, M.
Trace Formulae for Second-Order Differential Pencils with a Frozen Argument. *Mathematics* **2023**, *11*, 3996.
https://doi.org/10.3390/math11183996

**AMA Style**

Hu Y-T, Şat M.
Trace Formulae for Second-Order Differential Pencils with a Frozen Argument. *Mathematics*. 2023; 11(18):3996.
https://doi.org/10.3390/math11183996

**Chicago/Turabian Style**

Hu, Yi-Teng, and Murat Şat.
2023. "Trace Formulae for Second-Order Differential Pencils with a Frozen Argument" *Mathematics* 11, no. 18: 3996.
https://doi.org/10.3390/math11183996