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On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits

A. Balmaseda, F. Di Cosmo, J. M. Pérez-Pardo

TL;DR

The paper investigates how symmetry groups constrain self-adjoint extensions of the Laplace–Beltrami operator on quantum circuits, focusing on $ obreak ext{$Z$}$-invariant constructions built from infinite chains of unit cells. By representing extensions via boundary unitaries $U$ with a spectral gap at $-1$ and employing traceable group representations $V$ with boundary trace $v$, it derives criteria for $G$-invariance, notably the commutativity condition $[v(g),U]=0$, and analyzes local versus global symmetries through a block-structured, quasi-$ ext{delta}$ boundary condition framework. The work provides explicit conditions under which $ obreak ext{$Z$}$-invariance holds (constant $oldsymbol{ extdelta}$ and uniform relative phases) and demonstrates how to compute spectra and generalized eigenfunctions from unit-cell boundary equations. The findings offer a symmetry-guided blueprint for designing quantum circuits with preserved topology and tailored spectral properties, with extensions to more complex unit cells and closed periodic architectures.

Abstract

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group $G$, criteria for the existence of $G$-invariant self-adjoint extensions of the Laplace-Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the Laplace-Beltrami operator on an infinite set of intervals, $Ω$, constituting a quantum circuit, which are invariant under a given action of the group $\mathbb{Z}$. A study of the different unitary representations of the group $\mathbb{Z}$ on the space of square integrable functions on $Ω$ is performed and the corresponding $\mathbb{Z}$-invariant self-adjoint extensions of the Laplace-Beltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples.

On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits

TL;DR

The paper investigates how symmetry groups constrain self-adjoint extensions of the Laplace–Beltrami operator on quantum circuits, focusing on Z-invariant constructions built from infinite chains of unit cells. By representing extensions via boundary unitaries with a spectral gap at and employing traceable group representations with boundary trace , it derives criteria for -invariance, notably the commutativity condition , and analyzes local versus global symmetries through a block-structured, quasi- boundary condition framework. The work provides explicit conditions under which Z-invariance holds (constant and uniform relative phases) and demonstrates how to compute spectra and generalized eigenfunctions from unit-cell boundary equations. The findings offer a symmetry-guided blueprint for designing quantum circuits with preserved topology and tailored spectral properties, with extensions to more complex unit cells and closed periodic architectures.

Abstract

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group , criteria for the existence of -invariant self-adjoint extensions of the Laplace-Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the Laplace-Beltrami operator on an infinite set of intervals, , constituting a quantum circuit, which are invariant under a given action of the group . A study of the different unitary representations of the group on the space of square integrable functions on is performed and the corresponding -invariant self-adjoint extensions of the Laplace-Beltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples.

Paper Structure

This paper contains 7 sections, 8 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Self-adjoint extensions with symmetries
  3. $G$-invariant self-adjoint extensions
  4. Additional remarks on symmetries and self-adjoint extensions
  5. Quantum Circuits: adjacency and local symmetries
  6. Global symmetries on the Graph. $\mathbb{Z}$-invariance
  7. Conclusions and discussion

Key Result

Theorem 2.5

Let $G$ be a group and $V: G \, \rightarrow \, \mathcal{U}(L^2(\Omega))$ a topological traceable representation of $G$, with unitary trace $v: G \, \rightarrow \, \mathcal{U}(L^2(\partial \Omega))$ along the boundary $\partial \Omega$. Let $U \in \mathcal{U}(L^2(\partial \Omega))$ be an admissible u

Figures (6)

  • Figure 1: Subfigure (A) shows the intervals $I_e$ for an $\Omega$ made out of three intervals. On Subfigure (B) we can see 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 2: Representation of the infinite graph associated with $\Omega$.
  • Figure 3: Elementary cell of the graph $\Omega$.
  • 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 $k = 1/\pi$, $\alpha_3^i = \alpha_1^i = 0$, $\alpha_2^i = 0.9 / \pi$. The upper row shows the value on the loops while the lower row shows the value in the chain.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: IbortLledoPerezPardo2015a
  • Theorem 2.6: IbortLledoPerezPardo2015a
  • Example
  • Lemma 2.7
  • Proposition 3.1
  • proof
  • ...and 10 more