#
On
Z
-Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Self-Adjoint Extensions with Symmetries

#### 2.1. G-Invariant Self-Adjoint Extensions

**Definition**

**1.**

**Definition**

**2.**

- (i)
- $\mathbb{1}+U$ is invertible; and
- (ii)
- $-1$ belongs to the spectrum $\sigma \left(U\right)$ of the operator U but it is not an accumulation point of $\sigma \left(U\right)$.

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1**

**.**Let G be a group and $V:G\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\mathcal{U}\left({L}^{2}(\Omega )\right)$ a topological traceable representation of G, with unitary trace $v:G\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\mathcal{U}\left({L}^{2}(\partial \Omega )\right)$ along the boundary $\partial \Omega $. Let $U\in \mathcal{U}\left({L}^{2}(\partial \Omega )\right)$ be an admissible unitary operator with spectral gap at $-1$ defining a self-adjoint extension ${\mathbf{T}}_{b}$ of the Laplace–Beltrami operator. Assume that the self-adjoint extension corresponding to Neumann boundary conditions ($\dot{\psi}=0$) is G-invariant. Then, we have the following cases:

- (i)
- If $\left[v\left(g\right),U\right]=0$ for all $g\in G$, then ${\mathbf{T}}_{b}$ is G-invariant;
- (ii)
- Consider the decomposition of the boundary Hilbert space ${L}^{2}(\partial \Omega )=W\oplus {W}^{\perp}$, where W is the eigenspace relative to the eigenvalue $-1$ of the operator U, and denote by P the orthogonal projection onto W. If ${\mathbf{T}}_{b}$ is G-invariant and $P\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}{H}^{1/2}(\partial \Omega )\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{H}^{1/2}(\partial \Omega )$ is continuous, then $\left[v\left(g\right),U\right]=0$ for all $g\in G$.

#### 2.2. Additional Remarks on Symmetries and Self-Adjoint Extensions

**Theorem**

**2**

**.**Let $\mathbf{T}:\mathcal{D}\left(\mathbf{T}\right)\subset \mathcal{H}\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\mathcal{H}$ be a closed, symmetric and G-invariant operator with equal deficiency indices. Let ${\mathbf{T}}_{K}$ be the self-adjoint extension defined by the unitary operator $K\in \mathcal{U}\left({\mathcal{N}}_{+},{\mathcal{N}}_{-}\right)$. Then, ${\mathbf{T}}_{K}$ is G-invariant iff $V\left(g\right)K\xi =KV\left(g\right)\xi ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \xi \in {\mathcal{N}}_{+},\phantom{\rule{0.222222em}{0ex}}g\in G$.

**Example**

**1.**

**Lemma**

**1.**

## 3. Quantum Circuits: Adjacency and Local Symmetries

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Definition**

**5.**

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**3.**

**Proof.**

## 4. Global Symmetries on the Graph. $\mathbb{Z}$-invariance

**Theorem**

**3.**

- (i)
- The value of the parameter ${\delta}_{i}$ is the same at each vertex, δ.
- (ii)
- The relative phases between the boundary data associated with the loop is the same in all the vertices, i.e., ${\alpha}_{1}^{i}-{\alpha}_{2}^{i}$ does not depend on the vertex i.

**Proof.**

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) The intervals ${I}_{e}$ for an $\Omega $ made out of three intervals; and (

**b**) the associated graph if we connect on one side ${a}_{1}$ with ${a}_{2}$, ${b}_{1}$ and ${b}_{3}$, and in the other side ${b}_{2}$ with ${a}_{3}$. The first of the two connections is represented with the graph vertex labelled by a and the second is represented with the vertex b.

**Figure 4.**Real and imaginary parts of a generalised eigenfunction for ${\alpha}_{j}^{i}=0$ and several values of k. For each of the images, the upper row shows the value on the loops while the lower row shows the value in the chain.

**Figure 5.**Value of a generalised eigenfunction for $k=1/\pi $, ${\alpha}_{3}^{i}={\alpha}_{1}^{i}=0$, and ${\alpha}_{2}^{i}=0.9/\pi $. The upper row shows the value on the loops while the lower row shows the value in the chain.

**Figure 6.**Value of a generalised eigenfunction for $k=1/\pi $, ${\alpha}_{3}^{i}=\pi $, ${\alpha}_{1}^{i}=0$, and ${\alpha}_{2}^{i}=0.9/\pi $. The upper row shows the value on the loops while the lower row shows the value in the chain.

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Balmaseda, A.; Di Cosmo, F.; Pérez-Pardo, J.M.
On *Symmetry* **2019**, *11*, 1047.
https://doi.org/10.3390/sym11081047

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Balmaseda A, Di Cosmo F, Pérez-Pardo JM.
On *Symmetry*. 2019; 11(8):1047.
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**Chicago/Turabian Style**

Balmaseda, Aitor, Fabio Di Cosmo, and Juan Manuel Pérez-Pardo.
2019. "On *Symmetry* 11, no. 8: 1047.
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