# An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Results

**Theorem**

**1.**

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Preliminaries

#### 4.2. Representation of Solution on an Interval and Reduced Response Operator

**Theorem**

**2.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Corollary**

**1.**

#### 4.3. Solution of Inverse Problem

**Lemma**

**3.**

**Proof.**

**Step 1**

**Step 2**

**Lemma**

**4.**

**Proof.**

**Case 1:**${\gamma}_{2}\ne {\zeta}_{2}+{\xi}_{1}$.

**Case 2a:**${\gamma}_{2}={\zeta}_{2}+{\xi}_{1}$ and ${\varphi}_{2}\ne {\psi}_{2}{\alpha}_{1}.$ Note that the last inequality can be verified by an observer at this stage. Then ${\gamma}_{2}={\zeta}_{1}+{\xi}_{2}$ and ${\varphi}_{2}={\psi}_{1}{\alpha}_{2}+{\psi}_{2}{\alpha}_{1}.$ and hence

**Case 2b:**${\gamma}_{2}={\zeta}_{2}+{\xi}_{1}$ and ${\varphi}_{2}={\psi}_{2}{\alpha}_{1}.$ Then ${\gamma}_{2}\ne {\zeta}_{1}+{\xi}_{2}$. Note we have not yet solved for $\{{\xi}_{2},{\alpha}_{2}\}$. In this case, we now repeat the matching coefficient argument just used with $\delta (t-{\gamma}_{3})$.

**Case 1:**${\gamma}_{({N}_{p}+1)}\ne {\zeta}_{k}+{\xi}_{j},\phantom{\rule{4pt}{0ex}}\forall j\le p,\phantom{\rule{4pt}{0ex}}\forall k.$ Note that we know ${\left\{{\xi}_{j}\right\}}_{1}^{p}$ and $\left\{{\zeta}_{k}\right\}$, so these inequalities are verifiable. In this case, we must have ${\gamma}_{({N}_{p}+1)}={\zeta}_{1}+{\xi}_{p+1}$ and ${\psi}_{1}{\alpha}_{p+1}={\varphi}_{({N}_{p}+1)}$, so we have determined ${\alpha}_{p+1},{\xi}_{p+1}$ in this case.

**Case 2:**There exists an integer Q and pairs ${\{{\zeta}_{{i}_{n}},{\xi}_{{j}_{n}}\}}_{n=1}^{Q}$, with ${j}_{n}\le p$, such that

**Step 3**Because ${R}_{{k}_{0},j}$ are determined by assumption for $j=2,\dots ,{\mathsf{{\rm Y}}}_{{k}_{0}}-1$, the functions ${u}_{j}^{f}({v}_{{k}_{0}},t)$ are determined. In Step 2, we showed ${u}_{1}^{f}({v}_{{k}_{0}},t)$ is also determined. Hence by (4), ${u}_{{\mathsf{{\rm Y}}}_{{k}_{0}}}^{f}({v}_{{k}_{0}},t)$ is also determined. We can now carry out the argument in Step 2 on the remaining edges ${e}_{{k}_{0},3},\dots ,{e}_{{k}_{0},{\mathsf{{\rm Y}}}_{{k}_{0}}}$ incident on ${v}_{{k}_{0}}$ to determine ${\tilde{R}}_{{k}_{0},j}$ for all j.

**Step 4**For each $j=2,\dots ,{\mathsf{{\rm Y}}}_{{k}_{0}}$, we use Proposition 1, to find the associated ${\ell}_{j},{q}_{j}$ together with the valence of the vertex adjacent to ${v}_{{k}_{0}}$. Careful reading of Steps 2 and 3 shows that we can use ${R}_{0,1}^{T}$ and ${R}_{{k}_{0},j}^{T}$ for any $T>2({\ell}_{1}+{\ell}_{j})$.

**Step 5**Let ${v}_{{k}_{1}},\dots $ be the vertices adjacent to ${v}_{{k}_{0}}$, other than ${\gamma}_{0}$. We now iterate Steps 2–4 for the each of these vertices. Choose for instance ${v}_{{k}_{1}}$. If it were a boundary vertex, this fact would be determined in Step 4, and then this algorithm goes to the next vertex, which we, for convenience, still label ${v}_{{k}_{1}}$. We can thus assume ${v}_{{k}_{1}}$ is an interior vertex. Let us label an incident edge (other than ${e}_{2}:={e}_{{k}_{0},2}$) as ${e}_{3}:={e}_{{k}_{1},3}$, see Figure 6.

**Step 6**Arguing as in Step 5, we determine ${\tilde{R}}_{k,j}$ for all other vertices adjacent to ${v}_{{k}_{0}}$ and their associated edges. The details are left to the reader.

**Steps above 6**Clearly this procedure can be iterated until all edges of our finite graph have been covered.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

VESK | Volterra equation of the second kind |

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**Figure 5.**(

**a**) $\mathsf{\Omega}$. (

**b**) Subtree ${\mathsf{\Omega}}_{{k}_{0}}^{2}$, with ${e}_{1}={e}_{{k}_{0},1}$.

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**MDPI and ACS Style**

Avdonin, S.; Edward, J.
An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions. *Vibration* **2020**, *3*, 448-463.
https://doi.org/10.3390/vibration3040028

**AMA Style**

Avdonin S, Edward J.
An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions. *Vibration*. 2020; 3(4):448-463.
https://doi.org/10.3390/vibration3040028

**Chicago/Turabian Style**

Avdonin, Sergei, and Julian Edward.
2020. "An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions" *Vibration* 3, no. 4: 448-463.
https://doi.org/10.3390/vibration3040028