本文選題：Bose-Einstein凝聚 + 吸引相互作用��； 參考：《蘭州大學(xué)》2017年博士論文

【摘要】：本文主要研究兩個(gè)具有吸引相互作用的量子系統(tǒng):Bose-Einstein凝聚和玻色星體系統(tǒng).前者為非相對(duì)論量子系統(tǒng),在數(shù)學(xué)上可以用Gross-Pitaevskii能量泛函來描述;而后者為近似相對(duì)論量子系統(tǒng),在數(shù)學(xué)上可以用近似相對(duì)論Hartree能量泛函來描述.我們分別考慮了 M2中Gross-Pitaevskii能量的極小化問題和R3中近似相對(duì)論Hartree方程能量的極小化問題.首先,我們將Guo和Seiringer在[28]中關(guān)于IR2上的吸引型Bose-Einstein凝聚在勢(shì)阱V(x)滿足條件lim|x|→∞V(x)= +∞時(shí)得到的Gross-Pitaevskii能量泛函的極小元的存在性和質(zhì)量集中性結(jié)論推廣到了兩類很重要的外勢(shì):周期外勢(shì)和庫侖型外勢(shì)的情形.當(dāng)外勢(shì)為周期函數(shù)時(shí),我們應(yīng)用集中緊原理證明了當(dāng)相互作用參數(shù)a滿足a* a a* = ‖Q‖22時(shí),Gross-Pitaevskii能量的極小元存在,其中a*≥ 0,Q是非線性方程-△u + u - u3 = 0的唯一徑向?qū)ΨQ正解.進(jìn)一步,再次應(yīng)用集中緊原理我們得到了一個(gè)最佳的Gross-Pitaevskii能量上下界估計(jì),在此能量估計(jì)下,研究了當(dāng)a趨向a*時(shí)極小元的質(zhì)量集中現(xiàn)象,證明了質(zhì)量集中在周期函數(shù)的一個(gè)周期阱的最小值點(diǎn).當(dāng)外勢(shì)為形如-1/|x|1-β (0 ≤β 1)的庫侖型勢(shì)阱時(shí),我們得到了 Gross-Pitaevskii能量的一個(gè)較好的估計(jì),此估計(jì)依賴參數(shù)β.然后我們給出了 Gross-Pitaevskii能量的極小元的存在性定理.特別,當(dāng)0 β 1時(shí),我們發(fā)現(xiàn)Gross-Pitaevskii能量e(a)在臨界值點(diǎn)a*的取值不再是一個(gè)跳躍點(diǎn),表現(xiàn)為:e(a)關(guān)于a是連續(xù)的,遞減的,且對(duì)所有a ≥ a*都有e(a)=-∞.當(dāng)0 β 1時(shí),我們還分析了當(dāng)a趨向于臨界值a*時(shí)Gross-Pitaevskii能量的極小元的漸近行為,證明了極小元函數(shù)在此極限下質(zhì)量會(huì)集中在庫侖勢(shì)阱的奇異點(diǎn).其次,我們研究了與玻色星體的近似相對(duì)論Hartree方程相關(guān)的一個(gè)極小化問題.我們對(duì)Frohlich, Jonsson和Lenzmann [66]關(guān)于IR3上的近似相對(duì)論Hartree方程對(duì)應(yīng)的一個(gè)極小化問題的結(jié)果給出了補(bǔ)充.[66]中證明了當(dāng)粒子數(shù)N小于臨界值Nc(v)時(shí)準(zhǔn)基態(tài)的存在性和一些性質(zhì),我們給出了粒子數(shù)N趨向于臨界值Nc(v)時(shí),準(zhǔn)基態(tài)能量的一個(gè)最佳估計(jì),并分析了準(zhǔn)基態(tài)的漸近行為.
[Abstract]:In this paper, we study two quantum systems with attractive interaction: Bose-Einstein condensate and Bose-Einstein condensate and bosonic system. The former is a non-relativistic quantum system, which can be described mathematically by Gross-Pitaevskii energy functional, while the latter is an approximate relativistic quantum system, which can be described mathematically by an approximate relativistic Hartree energy functional. We consider the minimization of the Gross-Pitaevskii energy in M2 and the energy minimization of the approximate relativistic Hartree equation in R3, respectively. First, In [28], we generalize the existence of minimal elements of the energy functional of Gross-Pitaevskii obtained by Guo and Seiringer in [28] when the IR2 Bose-Einstein condensates in the potential well VX) and the existence of the minimal element of the energy functional of Gross-Pitaevskii obtained by the condition lim x 鈭，

本文編號(hào)：1847419