On one-dimensional Bose gases with two- and (critical) attractive three-body interactions
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915095200661504 |
|---|---|
| author | Nguyen, Dinh-Thi Ricaud, Julien |
| author_facet | Nguyen, Dinh-Thi Ricaud, Julien |
| contents | We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $a N^{α-1} U(N^α(x-y))$ and an attractive three-body interaction potential of the form $-b N^{2β-2} W(N^β(x-y,x-z))$, where $a\in\mathbb{R}$, $b,α>0$, $0<β<1$, $U, W \geq 0$, and $\int_{\mathbb{R}}U(x) \mathop{}\!\mathrm{d}x = 1 = \iint_{\mathbb{R}^2} W(x,y) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$. The system is stable either for any $a\in\mathbb{R}$ as long as $b<\mathfrak{b} := 3π^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schrödinger equation) or for $a \geq 0$ when $b=\mathfrak{b}$. In the former case, fixing $b \in (0,\mathfrak{b})$, we prove that in the mean-field limit the many-body system exhibits the Bose$\unicode{x2013}$Einstein condensation on the cubic-quintic NLS ground states. When assuming $b=b_N \nearrow \mathfrak{b}$ and $a=a_N \to 0$ as $N \to\infty$, with the former convergence being slow enough and "not faster" than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case $b=\mathfrak{b}$ fixed, we obtain the convergence of many-body energy for small $β$ when $a > 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04515 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On one-dimensional Bose gases with two- and (critical) attractive three-body interactions Nguyen, Dinh-Thi Ricaud, Julien Mathematical Physics Quantum Gases 81V70, 35J10, 35Q55, 82B10, 82D05 We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $a N^{α-1} U(N^α(x-y))$ and an attractive three-body interaction potential of the form $-b N^{2β-2} W(N^β(x-y,x-z))$, where $a\in\mathbb{R}$, $b,α>0$, $0<β<1$, $U, W \geq 0$, and $\int_{\mathbb{R}}U(x) \mathop{}\!\mathrm{d}x = 1 = \iint_{\mathbb{R}^2} W(x,y) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$. The system is stable either for any $a\in\mathbb{R}$ as long as $b<\mathfrak{b} := 3π^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schrödinger equation) or for $a \geq 0$ when $b=\mathfrak{b}$. In the former case, fixing $b \in (0,\mathfrak{b})$, we prove that in the mean-field limit the many-body system exhibits the Bose$\unicode{x2013}$Einstein condensation on the cubic-quintic NLS ground states. When assuming $b=b_N \nearrow \mathfrak{b}$ and $a=a_N \to 0$ as $N \to\infty$, with the former convergence being slow enough and "not faster" than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case $b=\mathfrak{b}$ fixed, we obtain the convergence of many-body energy for small $β$ when $a > 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$. |
| title | On one-dimensional Bose gases with two- and (critical) attractive three-body interactions |
| topic | Mathematical Physics Quantum Gases 81V70, 35J10, 35Q55, 82B10, 82D05 |
| url | https://arxiv.org/abs/2210.04515 |