# Oscillation Properties of Singular Quantum Trees

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions

#### 2.1. Metric Graphs

#### 2.2. Differential Operators

#### 2.3. Boundary and Interface Conditions

**Definition**

**1.**

**Remark**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. The Prüfer Angle for Quantum Trees

**Example**

**1.**

**Definition**

**4.**

**Remark**

**2.**

## 4. The Sturm Comparison Theorem for Quantum Trees

**Lemma**

**1.**

- (i)
- neither $y(\xb7;{\lambda}_{1})$ nor $y(\xb7;{\lambda}_{2})$ vanishes on $\mathrm{int}\left(\mathrm{\Gamma}\right)$;
- (ii)
- for every non-root boundary vertex $v\in \partial \mathrm{\Gamma}$,$$\underset{x\to v}{lim\; sup}\frac{{y}^{\left[1\right]}(x;{\lambda}_{1})}{y(x;{\lambda}_{1})}\le \underset{x\to v}{lim\; sup}\frac{{y}^{\left[1\right]}(x,{\lambda}_{2})}{y(x;{\lambda}_{2})}.$$

**Remark**

**3.**

**Proof.**

- (i′)
- $\theta (\xb7;{\lambda}_{j})$ do not assume values $\pi k$, $k\in \mathbb{Z}$, on $\mathrm{int}\left(\mathrm{\Gamma}\right)$;
- (ii′)
- $\theta (v;{\lambda}_{1})\ge \theta (v;{\lambda}_{2})$ for every non-root boundary vertex $v\in \partial \left(\mathrm{\Gamma}\right)$.

**Theorem**

**1**

**.**Assume that $y(\xb7;{\lambda}_{1})$ and $y(\xb7;{\lambda}_{2})$ are non-degenerate solutions of the equations $\ell y={\lambda}_{j}y$ with ${\lambda}_{1}<{\lambda}_{2}$ such that

- (a)
- at the root vertex ${v}_{0}$,$$\underset{x\to {v}_{0}}{lim\; sup}\frac{{y}^{\left[1\right]}(x;{\lambda}_{1})}{y(x;{\lambda}_{1})}\ge \underset{x\to {v}_{0}}{lim\; sup}\frac{{y}^{\left[1\right]}(x,{\lambda}_{2})}{y(x;{\lambda}_{2})};$$
- (b)
- for every non-root boundary vertex $v\in \partial \mathrm{\Gamma}$,$$\underset{x\to v}{lim\; sup}\frac{{y}^{\left[1\right]}(x;{\lambda}_{1})}{y(x;{\lambda}_{1})}\le \underset{x\to v}{lim\; sup}\frac{{y}^{\left[1\right]}(x,{\lambda}_{2})}{y(x;{\lambda}_{2})}$$

**Proof.**

## 5. Oscillation Theorems for Quantum Trees

#### 5.1. General Spectral Properties of a Quantum Tree

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

#### 5.2. Sturm Oscillation Theory in the Generic Case

**Definition**

**5.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**2**

**.**Assume the quantum tree $(\mathrm{\Gamma},\mathcal{L})$ is generic and denote by ${y}_{n}$ the (unique up to a multiplicative constant) eigenfunction corresponding to the eigenvalue ${\lambda}_{n}$. Then ${y}_{n}$ has n interior zeros and every nodal domain of ${y}_{n}$ contains exactly one zero of ${y}_{n+1}$.

**Proof.**

## 6. Spectral Properties of Non-Generic Quantum Trees

#### 6.1. The Special Solution and the Special Prüfer Angle

**Lemma**

**6.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**3.**

- (A1)
- for every fixed $x\in \mathrm{int}\left(\mathrm{\Gamma}\right)\cup \left\{{v}_{0}\right\}$, $\varphi (x;\lambda )$ is a continuous strictly decreasing function of $\lambda \in \mathbb{R}$;
- (A2)
- there is $\mu \in \mathbb{R}$ such that $\varphi (x;\lambda )\in (0,\pi )$ for all $x\in \mathrm{int}\left(\mathrm{\Gamma}\right)\cup \left\{{v}_{0}\right\}$ and all $\lambda <\mu $.

- (A3)
- ${lim}_{\lambda \to -\infty}\varphi (x;\lambda )=\pi $ for every fixed $x\in \mathrm{int}\left(\mathrm{\Gamma}\right)\cup \left\{{v}_{0}\right\}$;

- (A4)
- $\varphi (x;{\mu}_{*})>0$ on $\mathrm{int}\left(\mathrm{\Gamma}\right)$ and $\varphi ({v}_{0};{\mu}_{*})=0$.

**Proof.**

**Remark**

**4.**

**Lemma**

**7.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

#### 6.2. Eigenvalue Multiplicities

**Theorem**

**4.**

**Proof.**

- (i)
- for some $j\in \{1,\dots ,m\}$ it holds that ${\varphi}_{{\gamma}_{j}}(b;\lambda )=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}\pi $;
- (ii)
- none of the numbers ${\varphi}_{{\gamma}_{j}}(b;\lambda )$ vanishes modulo $\pi $.

**Corollary**

**7.**

## 7. Conclusions and Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Oscillation and Comparison Theorems for Singular Sturm-Liouville Operators

**Proposition**

**A1.**

**Proposition**

**A2.**

**Lemma**

**A1.**

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Homa, M.; Hryniv, R.
Oscillation Properties of Singular Quantum Trees. *Symmetry* **2020**, *12*, 1266.
https://doi.org/10.3390/sym12081266

**AMA Style**

Homa M, Hryniv R.
Oscillation Properties of Singular Quantum Trees. *Symmetry*. 2020; 12(8):1266.
https://doi.org/10.3390/sym12081266

**Chicago/Turabian Style**

Homa, Monika, and Rostyslav Hryniv.
2020. "Oscillation Properties of Singular Quantum Trees" *Symmetry* 12, no. 8: 1266.
https://doi.org/10.3390/sym12081266