---
OA_place: repository
OA_type: green
_id: '22080'
abstract:
- lang: eng
  text: We consider discrete analogues of two well-known open problems regarding invariant
    measures for dispersive PDE, namely, the invariance of the Gibbs measure for the
    continuum (classical) Heisenberg model and the invariance of white noise under
    focusing cubic nonlinear Schrödinger equation. These continuum models are completely
    integrable and connected by the Hasimoto transform; correspondingly, we focus
    our attention on discretizations that are also completely integrable and also
    connected by a discrete Hasimoto transform. We consider these models on the infinite
    lattice ℤ. Concretely, for a completely integrable variant of the classical Heisenberg
    spin chain model (introduced independently by Haldane, Ishimori, and Sklyanin)
    we prove the existence and uniqueness of solutions for initial data following
    a Gibbs law (which we show is unique) and show that the Gibbs measure is preserved
    under these dynamics. In the setting of the focusing Ablowitz--Ladik system, we
    prove invariance of a measure that we will show is the appropriate discrete analogue
    of white noise. We also include a thorough discussion of the Poisson geometry
    associated to the discrete Hasimoto transform introduced by Ishimori that connects
    the two models studied in this article.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yannis
  full_name: Angelopoulos, Yannis
  last_name: Angelopoulos
- first_name: Rowan
  full_name: Killip, Rowan
  last_name: Killip
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
citation:
  ama: Angelopoulos Y, Killip R, Vişan M. Invariant measures for integrable spin chains
    and an integrable discrete nonlinear Schrödinger equation. <i>SIAM Journal on
    Mathematical Analysis</i>. 2020;52(1):135-163. doi:<a href="https://doi.org/10.1137/19m1265314">10.1137/19m1265314</a>
  apa: Angelopoulos, Y., Killip, R., &#38; Vişan, M. (2020). Invariant measures for
    integrable spin chains and an integrable discrete nonlinear Schrödinger equation.
    <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial &#38; Applied
    Mathematics. <a href="https://doi.org/10.1137/19m1265314">https://doi.org/10.1137/19m1265314</a>
  chicago: Angelopoulos, Yannis, Rowan Killip, and Monica Vişan. “Invariant Measures
    for Integrable Spin Chains and an Integrable Discrete Nonlinear Schrödinger Equation.”
    <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial &#38; Applied
    Mathematics, 2020. <a href="https://doi.org/10.1137/19m1265314">https://doi.org/10.1137/19m1265314</a>.
  ieee: Y. Angelopoulos, R. Killip, and M. Vişan, “Invariant measures for integrable
    spin chains and an integrable discrete nonlinear Schrödinger equation,” <i>SIAM
    Journal on Mathematical Analysis</i>, vol. 52, no. 1. Society for Industrial &#38;
    Applied Mathematics, pp. 135–163, 2020.
  ista: Angelopoulos Y, Killip R, Vişan M. 2020. Invariant measures for integrable
    spin chains and an integrable discrete nonlinear Schrödinger equation. SIAM Journal
    on Mathematical Analysis. 52(1), 135–163.
  mla: Angelopoulos, Yannis, et al. “Invariant Measures for Integrable Spin Chains
    and an Integrable Discrete Nonlinear Schrödinger Equation.” <i>SIAM Journal on
    Mathematical Analysis</i>, vol. 52, no. 1, Society for Industrial &#38; Applied
    Mathematics, 2020, pp. 135–63, doi:<a href="https://doi.org/10.1137/19m1265314">10.1137/19m1265314</a>.
  short: Y. Angelopoulos, R. Killip, M. Vişan, SIAM Journal on Mathematical Analysis
    52 (2020) 135–163.
das_tickbox: '1'
date_created: 2026-06-19T08:26:32Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2026-06-30T12:21:20Z
day: '01'
doi: 10.1137/19m1265314
extern: '1'
external_id:
  arxiv:
  - '1807.08801'
intvolume: '        52'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1807.08801
mathsc:
- 35Q55
- 35Q51
- 35Q82
month: '01'
oa: 1
oa_version: Preprint
page: 135-163
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  eissn:
  - 1095-7154
  issn:
  - 0036-1410
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Invariant measures for integrable spin chains and an integrable discrete nonlinear
  Schrödinger equation
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 52
year: '2020'
...