# Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

@article{Oh2021OptimalIT, title={Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus}, author={Tadahiro Oh and Philippe Sosoe and Leonardo Tolomeo}, journal={Inventiones mathematicae}, year={2021} }

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus. In an influential paper, Lebowitz et al. (J Stat Phys 50(3–4):657–687, 1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit…

## 8 Citations

### Gibbs measure for the focusing fractional NLS on the torus

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We study the construction of the Gibbs measures for the focusing mass-critical fractional nonlinear Schrödinger equation on the multi-dimensional torus. We identify the sharp mass threshold for…

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. We consider the Cauchy problem for the fractional nonlinear Schr¨odinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter α > 1, subject to a…

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We study the Gibbs dynamics for the Zakharov-Yukawa system on the twodimensional torus T, namely a Schrödinger-wave system with a Zakharov-type coupling (−∆) . We first construct the Gibbs measure in…

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Using the variational formulation, non- normalizability of the Gibbs measure is proved using an alternative proof of the non-normalizability result for the focusing $\Phi^4_2$-measure by Brydges and Slade (1996).

### Focusing $\Phi^4_3$-model with a Hartree-type nonlinearity

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(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.)
We study a focusing $\Phi^4_3$-model with a…

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Using the variational formulation, non- normalizability of the Gibbs measure is proved using an alternative proof of the non-normalizability result for the focusing $\Phi^4_2$-measure by Brydges and Slade (1996).