bibkey: andradeseverini2018qgraphs
authors: Fabiano M. Andrade; Simone Severini
year: 2018
title: "Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs"
doi: 10.1103/PhysRevA.98.062107
url: https://arxiv.org/abs/1808.01154v2
claim: "For a quantum star graph the Schrödinger approach gives the quantum evolution map U^S = S D and the Green's function approach gives U^G = D S, with S the vertex scattering matrix and D the diagonal edge-propagation matrix; the paper shows they have the same eigenvalues and states, as Claim 1, that they are unitarily similar, checked for n = 2, 3, 4, 5 with Specht's theorem, a proof for general n being not known."
strata_touched:
  - D5/S3/Quantum/Transport/StarGraphEvolutionMapsUnitarilySimilar
license: citation-only
triage: anchor

Andrade and Severini, Unitary equivalence between the Green’s function and Schrödinger approaches for quantum graphs

F. M. Andrade and S. Severini, Phys. Rev. A 98, 062107 (2018); arXiv:1808.01154v2. Quotations are from the v2 TeX source.

An edge of length ${\ell }_{ij}$ contributes $z_{ij}=e^{ik{\ell }_{ij}}$. For the star graph $S_n$ the Schrödinger approach gives the quantum evolution map $U_{S_n}^S(k)=S_{S_n}(k)D_{S_n}(k)$, with $D_{S_n}(k)=\mathrm{diag}(z_{12},z_{12},\dots ,z_{1n},z_{1n})$ and $S_{S_n}(k)$ built from the vertex scattering amplitudes $r_i$, $r_1$, $t_1$; the Green’s function approach gives $U_{S_n}^G(k)=D_{S_n}(k)S_{S_n}(k)$. The paper proves that the two maps have the same eigenvalues with multiplicity, and states:

The quantum evolution maps $U_{S_n}^S(k)$ and $U_{S_n}^G(k)$ are unitarily similar.

Although a proof for a general $n$ is not known, we checked this for $n=2,3,4,5$ by using Specht’s theorem.

Verified locator

Source: https://arxiv.org/abs/1808.01154v2

• DOI: https://doi.org/10.1103/PhysRevA.98.062107 (Physical Review A 98(6), article 062107; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/1808.01154v2. The e-print member qgl.tex has 661 lines and md5 26fb87cb2c04c5a6eb12f5cf0c7c575d. Locations:
  • $z_{ij}$, l. 298; $U_{S_n}^S$, $S_{S_n}$ and $D_{S_n}$, l. 383–405; $U_{S_n}^G$, l. 472–474;
  • the eigenvalue theorem, l. 483–504; Claim 1 and the remark on general $n$, l. 522–530.