Scholarlink

<p>In this paper, we investigate the Gibbs measures associated with the focusing nonlinear Schrödinger equation with an anharmonic potential. We establish a dichotomy for normalizability and non-normalizability of the Gibbs measures in one dimension and higher dimensions with radial data. This extends a recent result of the third and fourth authors with Robert and Seong (Focusing Gibbs measures with harmonic potential. Ann Inst Henri Poincaré Probab Stat. <b>61</b>(1), 571–598, 2025), where the focusing Gibbs measures with a harmonic potential were addressed. Notably, in the case of a subharmonic potential, we identify a novel critical nonlinearity (below the usual mass-critical exponent) for which the Gibbs measures exhibit a phase transition. The primary challenge emerges from the limited understanding of eigenvalues and eigenfunctions of the Schrödinger operator with an anharmonic potential. We overcome the difficulty by employing techniques related to a recent work of the first two authors (Invariant Gibbs measures for 1D NLS in a trap. arXiv preprint <a href="http://arxiv.org/abs/2301.02544">arXiv:2301.02544</a>. 2023).</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Statistical mechanics of the radial focusing nonlinear Schrödinger equation in general traps

  • 期刊论文

  • 发表日期：

    2026年5月20日

查看全文
  • Van Duong Dinh,
  • Nicolas Rougerie,
  • Leonardo Tolomeo,
  • Yuzhao Wang

摘要

In this paper, we investigate the Gibbs measures associated with the focusing nonlinear Schrödinger equation with an anharmonic potential. We establish a dichotomy for normalizability and non-normalizability of the Gibbs measures in one dimension and higher dimensions with radial data. This extends a recent result of the third and fourth authors with Robert and Seong (Focusing Gibbs measures with harmonic potential. Ann Inst Henri Poincaré Probab Stat. 61(1), 571–598, 2025), where the focusing Gibbs measures with a harmonic potential were addressed. Notably, in the case of a subharmonic potential, we identify a novel critical nonlinearity (below the usual mass-critical exponent) for which the Gibbs measures exhibit a phase transition. The primary challenge emerges from the limited understanding of eigenvalues and eigenfunctions of the Schrödinger operator with an anharmonic potential. We overcome the difficulty by employing techniques related to a recent work of the first two authors (Invariant Gibbs measures for 1D NLS in a trap. arXiv preprint arXiv:2301.02544. 2023).