【作者】 李艳；

【导师】 郭玉劲；

【作者基本信息】 华中师范大学 ， 基础数学， 2025， 博士

【摘要】 由多个相互作用的量子粒子构成的系统称为量子多体系统.根据粒子的统计性质,量子多体系统分为玻色系统和费米系统.近年来,R.L.Frank、M.Lewin、E.Lieb、P.L.Lions等国际著名数学家通过深入研究量子多体系统,取得了一系列原创且具有深远意义的分析成果.受他们研究工作的启发,通过等价地分析两类L2-临界变分问题,本文研究玻色系统和费米系统基态解的存在性、极限行为、涡旋非存在性等性质.具体而言,全文共分为四章:第一章主要概述玻色系统和费米系统及其对应的L2-临界变分问题的相关背景和研究现状,介绍相关的预备知识,并阐述本文的主要结果.第二章主要研究吸引力作用下旋转玻色系统的基态解.通过采用能量估计方法和爆破分析理论,本章证明在旋转非调和位势阱V(x)=ω(|x|2+k|x|4)(ω,k>0)中玻色系统基态解的极限行为,其中V(x)的旋转速度是变化的.该结果解决了国际数学家大会45分钟报告人M.Lewin等人在2018年提出的公开问题,这是本文的创新点之一.通过分析位势能量估计得到精细的基态能量估计是本章的主要研究难点.第三章进一步分析上述吸引力作用下旋转玻色系统基态解的涡旋非存在性,其中位势阱V(x)=ω(|x|2+k|x|4)(ω,k>0),且V(x)的旋转速度是变化的.通过研究旋转玻色系统基态解的精细展式,本章证明基态解在一个充分大区域内的涡旋非存在性,其中区域的半径与旋转速度有关.此外,本章克服了拉格朗日乘子的出现带来的本质性困难.第四章主要研究吸引力作用下费米系统的基态解.本章应用费米系统的变分原理,证明某些条件下基态解的存在性.构造适当的标准正交函数族证明某些条件下基态解的非存在性,是本章的研究难点之一.通过采用多体费米系统的能量估计方法和爆破分析理论,本章也证明了该费米系统基态解的极限行为,且基态解的质量集中在位势阱的最平坦全局极小值点处.更多还原

【Abstract】 A system consisting of multiple interacting quantum particles is called the quantum many-body system.According to the statistical properties of particles,quantum many-body systems can be classified into both Bose systems and Fermi systems.In recent years,mathematicians such as R.L.Frank,M.Lewin,E.Lieb and P.L.Lions made a series of original and profoundly significant contributions in the quantum many-body systems.Inspired by their works,by equivalently studying two different classes of L2-critical variational problems,this thesis mainly investigates the existence,limiting behavior,non-existence of vortices and some other properties of ground states for Bose systems and Fermi systems.Specifically,this thesis consists of the following four chapters.In Chapter 1,we mainly summarize the backgrounds and related research progresses of Bose systems and Fermi systems,and the corresponding L2-critical variational problems.Some related preliminary results and the main results of this thesis are also stated in this chapter.In Chapter 2,we focus on the ground states of attractive rotating Bose systems.Applying the energy method and blow-up analysis,we obtain the limiting behavior of ground states for the Bose system with the anharmonic potential V(x)=ω(|x|2+k|x|4),where ω>0,k>0,and the rotational velocity of V(x)varies.As one of the innovations in this thesis,this solves an open question proposed by the ICM 45 minutes speaker M.Lewin and his collaborators in 2018.The main difficulty of this chapter is to obtain the refined energy estimates by analyzing the potential energy.In Chapter 3,we further investigate the non-existence of vortices for ground states of the above attractive rotating Bose system with the potential V(x)=ω0(|x|2+k|x|4),where ω>0,k>0,and the rotational velocity of V(x)varies.By analyzing the refined expansions of ground states,we prove in this chapter the non-existence of vortices in a large region,where the radius of this region depends on the rotational velocity.Moreover,we overcome the difficulties caused by the appearance of the Lagrange multiplier.In Chapter 4,we mainly study ground states of Fermi systems with attractive interactions.Applying the variational principle of many-body Fermi systems,we prove the existence of ground states for the system under certain conditions.Constructing suitable orthogonal functions to prove the non-existence of ground states under certain conditions is one of the difficulties in this chapter.By the energy method and blow-up analysis of many-body Fermi systems,we also prove the limiting behavior of ground states for the system.Moreover,we show that ground states of the system concentrate at the flattest minimum points of V(x).更多还原