bibkey: klausen2025spectragenerators
authors: Frederik Ravn Klausen
year: 2025
title: "Spectra of generators of Markovian evolution in the thermodynamic limit: From non-Hermitian to full evolution via tridiagonal Laurent matrices"
doi: 10.48550/arXiv.2206.09879
url: https://arxiv.org/abs/2206.09879v3
claim: "A single-particle Lindbladian on the Hilbert-Schmidt operators over ℓ²(ℤ), with finite-range translation-invariant Hamiltonian and Lindblad operators, is a bounded operator and decomposes as a direct integral of finite-range Laurent operators with finite-range perturbations. Under the additional rank-one hypothesis the paper proves that if 0 lies in the spectrum then the Lindbladian is gapless or has an infinite-dimensional kernel, and lists as a further question whether 0 always lies in the spectrum."
strata_touched:
  - D5/S3/QuantumChannels/LatticeLindbladianZeroSpectrum
license: citation-only
triage: anchor

Klausen, Spectra of generators of Markovian evolution in the thermodynamic limit

F. R. Klausen, arXiv:2206.09879v3 [math-ph] (version 3, 29 September 2025; version 1, June 2022, with A. H. Werner, under the title “Spectra of Lindbladians on the infinite line”). Quotations are from the v3 TeX source, with the source’s macros \Li (a calligraphic L) and \HS (HS) expanded.

The Lindbladian (section02.tex): $\mathcal{H}={\ell }^2(\mathbb{Z})$ with position basis $\{|{\delta }_k⟩\}$ and

$$\mathcal{L}(\rho )=-i[H,\rho ]+G\sum _k{L}_k\rho L_k^{\ast }-\frac{1}{2}(L_k^{\ast }{L}_k\rho +\rho L_k^{\ast }{L}_k),$$

acting on $\mathrm{HS}(\mathcal{H})$, with $H$ bounded self-adjoint, $G>0$ and $L_k\in \mathcal{B}(\mathcal{H})$; the effective Hamiltonian $H_{\mathrm{eff}}=H-\frac{iG}{2}{\sum }_k{L}_k^{\ast }{L}_k$ shows that $G$ multiplies the whole dissipator. The assumptions are (Finite Range) $⟨{\delta }_y,L_k{\delta }_x⟩=0=⟨{\delta }_y,H{\delta }_x⟩$ whenever $\max \{|x-k|,|y-k|\}>r$, (TransInv) $S_1{L}_k{S}_{-1}=L_{k+1}$ and $H=S_{1}H{S}_1^{\ast }$, and (Rank 1) $\mathrm{Rank}(L_k)=1$; under the first two, $\mathcal{L}\in \mathcal{B}(\mathrm{HS}(\mathcal{H}))$, and the spectrum is taken there.

Theorem thm:no_gap (section03.tex): if $\mathcal{L}$ satisfies (Finite Range), (TransInv), (Rank 1) and $0\in \sigma (\mathcal{L})$, then $\mathcal{L}$ is gapless or $\dim \ker \mathcal{L}=\infty$. The list of further questions (section04.tex) contains:

Prove that if $\mathcal{L}$ is (Finite Range), (TransInv), and (Rank 1) then $0\in \sigma (\mathcal{L})$, thereby strengthening Theorem (no gap).

Verified locator

Source: https://arxiv.org/abs/2206.09879v3

• DOI: https://doi.org/10.48550/arXiv.2206.09879 (arXiv-issued DataCite DOI; the arXiv record has no journal reference).
• URL: https://arxiv.org/abs/2206.09879v3. The e-print members section02.tex (258 lines, md5 b5b553f1c8203d16c63a5b9330242407), section03.tex (757 lines, md5 a16eacc51b1bb4e77ceafb39b96fbb18) and section04.tex (597 lines, md5 227274cfa5a97ad104ecfdde8c5d4562). Locations:
  • the Lindblad form, section02.tex l. 6–13; the effective Hamiltonian, l. 22;
  • the assumptions, l. 39–50; boundedness on the Hilbert-Schmidt operators, l. 54–57;
  • the vectorization, section03.tex l. 162–182; Theorem thm:no_gap, l. 506–509;
  • the question, section04.tex l. 574.