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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 757616, 14 pages
Research Article

The Gross-Pitaevskii Model of Spinor BEC

Department of Mathematics, Sichuan University, Chengdu 610064, China

Received 4 December 2011; Accepted 15 February 2012

Academic Editor: Wan-Tong Li

Copyright © 2012 Dongming Yan and Tian Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Gross-Pitaevskii model of spinor Bose-Einstein condensates is studied. Using the abstract results obtained for infinite-dimensional Hamilton system, we establish the mathematical theory for the model of spinor BEC. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity, are also proved.

1. Introduction

After the first remarkable experiments concerning the observation of Bose-Einstein condensate (BEC) in dilute gases of alkali atoms such as 87Rb [1], 23Na [2], and 7Li [3] the interest in this phenomenon has revived [4, 5]. On the mathematical side, most of the work has concentrated on the Gross-Pitaevskii (GP) model of BEC, which is usually referred to as nonlinear Schödinger equation (NLSE) (cf. [614] and references therein). There are also many pieces of the literature on the spinor BEC ([1519]). In the spinor BEC case, the constituent bosons have internal degrees of freedom, such as spin, the quantum state, and its properties becomes more complex [20]. What has made the alkali spinor BEC particularly interesting is that optical and magnetic fields can be used to probe and manipulate the system.

In [15], Ho shows that in an optical trap the ground states of spin-1 bosons such as 23Na, 39K, and 87Rb can be either ferromagnetic or polar states, depending on the scattering lengths in different angular momentum channels. In [17], Pu et al. discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor BEC for a dilute atomic vapor confined in an optical trap. Their results go beyond the mean field picture and are developed within a fully quantized framework. In [19], Zou and Mathis propose a three-step scheme for generating the maximally entangled atomic Greenberger-Horne-Zeilinger (GHZ) states in a spinor BEC by using strong classical laser fields to shift atom level and drive single-atom Raman transition. Their scheme can be directly used to generate the maximally entangled states between atoms with hyperfine spin 0 and 1.

In this paper, we want to establish the mathematical theory of the GP model of spinor BEC in which the internal degrees of freedom of atoms are also under consideration. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity are also proved.

2. Gross-Pitaevskii Model for Spinor BEC

In this section we derive GP equation for spinor BEC, that is, the following equation:𝑖𝜕𝜓1𝜕𝑡=22𝑚Δ𝜓1+𝑉(𝑥)𝜓1+𝑔𝑛||Ψ||2𝜓1+𝑔𝑠𝜓1𝜓20+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓1,𝑖𝜕𝜓0𝜕𝑡=22𝑚Δ𝜓0+𝑉(𝑥)𝜓0+𝑔𝑛||Ψ||2𝜓0+2𝑔𝑠𝜓1𝜓1𝜓0+𝑔𝑠||𝜓1||2+||𝜓1||2𝜓0,𝑖𝜕𝜓1𝜕𝑡=22𝑚Δ𝜓1+𝑉(𝑥)𝜓1+𝑔𝑛||Ψ||2𝜓1+𝑔𝑠𝜓1𝜓20+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓1.(2.1)

We consider the GP model for 𝐹=1 spinor BEC. Particles of 𝐹=1 have three quantum states: magnetic quantum number 𝑚=1,0,1. The corresponding wave function of these three quantum states are denote by𝜓Ψ=1,𝜓0,𝜓1.(2.2) Here the physical meaning of |𝜓𝑖|2 is the density of 𝑚=𝑖 particles (𝑖=1,0,1). The corresponding Hamilton energy functional of 𝐹=1 spinor BEC is as follows:𝐸Ψ,Ψ=Ω2||||2𝑚Ψ2||Ψ||+𝑉(𝑥)2+12𝑔𝑛||Ψ||4+12𝑔𝑠||Ψ||𝑆Ψ2𝑑𝑥,(2.3) where 𝑚 is the boson mass, is Planck constant, 𝑉() is the external trapping potential, |Ψ|2 is the density of dilute bosonic atoms, 𝑔𝑛 is the interaction constant between atoms, and 𝑔𝑠 is the spin exchange interaction constant𝑔𝑛=4𝜋2𝑚𝑎0+2𝑎23,𝑔𝑠=4𝜋2𝑚𝑎2𝑎03,(2.4)𝑎0 and 𝑎2 are the scattering length, and 𝑆=𝑆𝑥𝑖+𝑆𝑦𝑗+𝑆𝑧𝑘 is the spin operator:𝑆𝑥=12010101010,𝑆𝑦=120𝑖0𝑖0𝑖0𝑖0,𝑆𝑧=.100000001(2.5) We adopt the following notations:Ψ𝜓𝑆Ψ=1,𝜓0,𝜓1𝑆𝑥𝑖+𝑆𝑦𝑗+𝑆𝑧𝑘𝜓1𝜓0𝜓1,||Ψ||𝑆Ψ2=||Ψ𝑆𝑥Ψ||2+||Ψ𝑆𝑦Ψ||2+||Ψ𝑆𝑧Ψ||2=12||𝜓1𝜓0+𝜓0𝜓1+𝜓1+𝜓1𝜓0||2+12||𝜓1𝜓0+𝜓0𝜓1𝜓1+𝜓1𝜓0||2+||𝜓1||2||𝜓1||22.(2.6) By calculation we can get||Ψ||𝑆Ψ2=||𝜓1||4+||𝜓1||4||𝜓21||2||𝜓1||2||𝜓+20||2||𝜓1||2||𝜓+20||2||𝜓1||2+2𝜓20𝜓1𝜓1+2𝜓02𝜓1𝜓1.(2.7)

By using Lagrange multiplier theorem, from the Hamilton energy functional 𝐸 (see (2.3) and the total particle number𝑁=Ω||Ψ||2𝜓𝑑𝑥,Ψ=1,𝜓0,𝜓1(2.8) is conservative, and we can obtain the steady state GP equation of spinor BEC as follows:𝜇𝜓𝑘=𝛿𝛿𝜓𝑘𝐸Ψ,Ψ,𝑘=1,0,1,(2.9) where 𝜇 is the chemical potential. Furthermore, according to general rules of quantum mechanics from steady state GP equation, we can get the dynamical model as follows:𝑖𝜕𝜓𝑘=𝛿𝜕𝑡𝛿𝜓𝑘𝐸Ψ,Ψ,𝑘=1,0,1,(2.10) where 𝑖=1. From ((2.3) and (2.7), we can obtain the concrete expression of (2.10) as (2.1).

In the spinor BEC 𝑔𝑛 and 𝑔𝑠 we have following physical meaning:𝑔𝑛>0,correspondingtotherepulsiveinteractionbetweenatoms,<0,correspondingtotheattractiveinteractionbetweenatoms,𝑔𝑠>0,correspondingtotheantiferromagneticstates,<0,correspondingtotheferromagneticstates.(2.11)

3. Equivalent Form of Spinor BEC

Let 𝜓𝑘=𝜓1𝑘+𝑖𝜓2𝑘. In this section we will show that GP equation (2.1) is equivalent to the following quantum Hamilton systems (see [21]):𝜕𝜓1𝑘=𝜓𝜕𝑡𝛿𝐹1,𝜓2𝛿𝜓2𝑘,𝜕𝜓2𝑘𝜓𝜕𝑡=𝛿𝐹1,𝜓2𝛿𝜓1𝑘,𝑘=1,0,1,(3.1) where 𝜓1=(𝜓11,𝜓10,𝜓11),𝜓2=(𝜓21,𝜓20,𝜓21), and the energy functional defined as𝐹𝜓1,𝜓2=1𝐸2Ψ,Ψ,(𝐸see(2.3)).(3.2)

In fact, on the one hand, splitting real and imaginary parts of (2.1), we can obtain 𝜕𝜓11=𝜕𝑡22𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓21+𝑔𝑠2𝜓11𝜓10𝜓20𝜓21𝜓102+𝜓21𝜓202,𝜕𝜓21=𝜕𝑡22𝑚Δ𝑉(𝑥)𝑔𝑛||Ψ||2𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓11𝑔𝑠2𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓11𝜓202,𝜕𝜓10=𝜕𝑡22𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓1||2𝜓20+2𝑔𝑠𝜓11𝜓21𝜓10+𝜓11𝜓21𝜓10𝜓11𝜓11𝜓20+𝜓21𝜓21𝜓20,𝜕𝜓20=𝜕𝑡22𝑚Δ𝑉(𝑥)𝑔𝑛||Ψ||2𝑔𝑠||𝜓1||2+||𝜓1||2𝜓102𝑔𝑠𝜓11𝜓11𝜓10𝜓21𝜓21𝜓10+𝜓11𝜓21𝜓20+𝜓21𝜓11𝜓20,𝜕𝜓11=𝜕𝑡22𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓21+𝑔𝑠2𝜓11𝜓10𝜓20𝜓21𝜓102+𝜓21𝜓202,𝜕𝜓21=𝜕𝑡22𝑚Δ𝑉(𝑥)𝑔𝑛||Ψ||2𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓11𝑔𝑠2𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓11𝜓202.(3.3) On the other hand, it is easy to check that 𝜓𝛿𝐹1,𝜓2𝛿𝜓21=122𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓21+𝑔𝑠2𝜓11𝜓10𝜓20𝜓21𝜓102+𝜓21𝜓202,𝜓𝛿𝐹1,𝜓2𝛿𝜓11=122𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓11+𝑔𝑠2𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓11𝜓202,𝜓𝛿𝐹1,𝜓2𝛿𝜓20=122𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓1||2𝜓20+2𝑔𝑠𝜓11𝜓21𝜓10+𝜓11𝜓21𝜓10𝜓11𝜓11𝜓20+𝜓21𝜓21𝜓20,𝜓𝛿𝐹1,𝜓2𝛿𝜓10=122𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓1||2𝜓10+2𝑔𝑠𝜓11𝜓11𝜓10𝜓21𝜓21𝜓10+𝜓11𝜓21𝜓20+𝜓21𝜓11𝜓20,𝜓𝛿𝐹1,𝜓2𝛿𝜓21=122𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓21+𝑔𝑠2𝜓11𝜓10𝜓20𝜓21𝜓102+𝜓21𝜓202,𝜓𝛿𝐹1,𝜓2𝛿𝜓11=122𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓11+𝑔𝑠2𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓11𝜓202.(3.4) Consequently, GP equation (2.1) is equivalent to (3.1).

4. Infinite Dimensional Hamilton System

In this section, we consider the following infinite-dimensional Hamilton system𝑑𝑢𝑑𝑡=𝐷𝑣𝐹(𝑢,𝑣),𝑑𝑣𝑑𝑡=𝐷𝑢𝐹(𝑢,𝑣),𝑢(0)=𝜑,𝑣(0)=𝜓,(𝑢,𝑣)𝑋1×𝑋2(4.1) where 𝑋𝑋𝑖𝐻(𝑖=1,2) is dense, 𝑋 is linear space, 𝑋1,𝑋2 is reflexive Banach space, 𝐻 is Hilbert space, 𝐹𝑋1×𝑋2𝑅1  is 𝐶1 functional, and 𝐷𝐹=(𝐷𝑢𝐹,𝐷𝑣𝐹) is derived operator.

Remark 4.1. Infinite-dimensional Hamilton system (4.1) not only has some kind of beauty in its own form, but also many equations can be written as (4.1). For example, Schödinger equation, Weyl equations, and Dirac equations can be written as (4.1). Hence, it is worth to study the infinite-dimensional Hamilton system (4.1), also see [21].

Definition 4.2. One says (𝑢,𝑣)𝑋1×𝑋2 is a weak solution of Hamilton system (4.1), provided𝑢,̃𝑢𝐻̃+𝑣,𝑣𝐻=𝑡0𝐷𝑢̃𝐹(𝑢,𝑣),𝑣𝐷𝑣𝐹(𝑢,𝑣),̃𝑢𝑑𝑡+𝜑,̃𝑢𝐻̃+𝜓,𝑣𝐻,(4.2) for every ̃𝑢𝑋2,̃𝑣𝑋1.
Let 𝐹𝑋1×𝑋2𝑅1 satisfy ||||||||(𝑢,𝑣)𝑋1×𝑋2𝐹(𝑢,𝑣)(or𝐹(𝑢,𝑣)).(4.3) Then we have the following existence theorem.

Theorem 4.3 (see [21]). Assume that 𝐹 satisfies condition (4.3) and 𝐷𝐹𝑋1×𝑋2(𝑋1×𝑋2) is weakly continuous, then for any (𝜑,𝜓)𝑋1×𝑋2, there exists one global weak solution of equation (4.1) (𝑢,𝑣)𝐿(0,),𝑋1×𝑋2.(4.4) Furthermore, 𝐹(𝑢,𝑣) is a conservative quantity for weak solution (𝑢,𝑣), that is, 𝐹(𝑢(𝑡),𝑣(𝑡))=𝐹(𝜑,𝜓),𝑡>0.(4.5)

Proof. We prove the existence of global solution for (4.1) in 𝐿((0,),𝑋1×𝑋2) by standard Galerkin method. Choose 𝑒𝑘𝑘=1,2,𝑋(4.6) as orthonormal basis of space 𝐻. Set 𝑋𝑛,𝑋𝑛 as follows: 𝑋𝑛=𝑛𝑘=1𝛼𝑘𝑒𝑘𝛼𝑘𝑅1,𝑋,1𝑘𝑛𝑛=𝑛𝑘=1𝛽𝑘(𝑡)𝑒𝑘𝛽𝑘()𝐶1[.0,),1𝑘𝑛(4.7)
Consider the ordinary equations as follows: 𝑑𝑥𝑘(𝑡)=𝑑𝑡𝐷𝑣𝐹𝑢𝑛,𝑣𝑛,𝑒𝑘,𝑑𝑦𝑘(𝑡)=𝐷𝑑𝑡𝑢𝐹𝑢𝑛,𝑣𝑛,𝑒𝑘,𝑥𝑘(0)=𝜑,𝑒𝑘𝐻,𝑦𝑘(0)=𝜓,𝑒𝑘𝐻,(𝑘=1,,𝑛),(4.8) where 𝑢𝑛=𝑛𝑘=1𝑥𝑘(𝑡)𝑒𝑘,𝑣𝑛=𝑛𝑘=1𝑦𝑘(𝑡)𝑒𝑘.
By the theory of ordinary equations, there exists only one local solution of (4.8): 𝑥1(𝑡),𝑦1(𝑡),,𝑥𝑛(𝑡),𝑦𝑛(𝑡),0𝑡𝜏.(4.9)
From (4.8) we can obtain the equality 𝑢𝑛,̃𝑢𝑛𝐻+𝑣𝑛,̃𝑣𝑛𝐻=𝑡0𝐷𝑢𝐹𝑢𝑛,𝑣𝑛,̃𝑣𝑛𝐷𝑣𝐹𝑢𝑛,𝑣𝑛,̃𝑢𝑛𝑑𝑡+𝜑,̃𝑢𝑛𝐻̃𝑣+𝜓,𝑛𝐻(4.10) holds true for any ̃𝑢𝑛,̃𝑣𝑛𝑋𝑛. Moreover, equality 𝑡0𝑑𝑢𝑛𝑑𝑡,̃𝑢𝑛𝐻+𝑑𝑣𝑛,̃𝑣𝑑𝑡𝑛𝐻𝑑𝑡=𝑡0𝐷𝑢𝐹𝑢𝑛,𝑣𝑛,̃𝑣𝑛𝐷𝑣𝐹𝑢𝑛,𝑣𝑛,̃𝑢𝑛𝑑𝑡(4.11) holds true for any ̃𝑢𝑛,̃𝑣𝑛𝑋𝑛.
Putting (̃𝑢𝑛,̃𝑣𝑛)=(𝑑𝑣𝑛/𝑑𝑡,𝑑𝑢𝑛/𝑑𝑡) in (4.11), we obtain that 0=𝑡0𝐷𝑢𝐹𝑢𝑛,𝑣𝑛,𝑑𝑢𝑛+𝐷𝑑𝑡𝑣𝐹𝑢𝑛,𝑣𝑛,𝑑𝑣𝑛=𝑑𝑡𝑑𝑡𝑡0𝑑𝐹𝑢𝑑𝑡𝑛,𝑣𝑛𝑑𝑡,(4.12) which implies 𝐹𝑢𝑛,𝑣𝑛𝜑=𝐹𝑛,𝜓𝑛,(4.13) where 𝜑𝑛=𝑛𝑘=1𝜑,𝑒𝑘𝐻𝑒𝑘,𝜓𝑛=𝑛𝑘=1𝜓,𝑒𝑘𝐻𝑒𝑘.(4.14)
From (4.3) and (4.10), we deduce that {(𝑢𝑛,𝑣𝑛)}𝑛=1 is bounded in 𝐿((0,),𝑋1×𝑋2). Therefore there exists a subsequence; we still write it as {(𝑢𝑛,𝑣𝑛)}𝑛=1, such that 𝑢𝑛,𝑣𝑛(𝑢,𝑣)in𝑋1×𝑋2,a.e.𝑡(0,).(4.15)
According to 𝐷𝐹𝑋1×𝑋2(𝑋1×𝑋2) being weakly continuous and (4.10), (4.15), we know the following equality 𝑢,̃𝑢𝐻̃+𝑣,𝑣𝐻=𝑡0𝐷𝑢̃𝐹(𝑢,𝑣),𝑣𝐷𝑣𝐹(𝑢,𝑣),̃𝑢𝑑𝑡+𝜑,̃𝑢𝐻̃+𝜓,𝑣𝐻(4.16) holds true for any ̃̃𝑢,𝑣𝑛=1𝑋𝑛. Since 𝑛=1𝑋𝑛 is dense in 𝑋1 and 𝑋2, equality (4.16) holds true for all ̃(̃𝑢,𝑣)𝑋1×𝑋2, which implies that (𝑢,𝑣)𝐿((0,),𝑋1×𝑋2) is a global weak solution of (4.1).
Next, we prove 𝐹(𝑢,𝑣) is a conservative quantity for weak solution (𝑢,𝑣). From (4.16), for all >0 we have 𝑢(𝑡+)𝑢(𝑡),̃𝑢𝐻̃+𝑣(𝑡+)𝑣(𝑡),𝑣𝐻=𝑡𝑡+𝐷𝑢̃𝐹(𝑢(𝜏),𝑣(𝜏)),𝑣𝐷𝑣𝐹(𝑢(𝜏),𝑣(𝜏)),̃𝑢𝑑𝜏.(4.17) Putting ̃𝑢=Δ̃𝑣=(𝑣(𝑡+)𝑣(𝑡)),𝑣=Δ𝑢=𝑢(𝑡+)𝑢(𝑡)(4.18) in (4.17), we obtain that 10=𝑡𝑡+𝐷𝑢𝐹(𝑢(𝜏),𝑣(𝜏)),Δ𝑢+𝐷𝑣𝐹(𝑢(𝜏),𝑣(𝜏)),Δ𝑣𝑑𝜏=𝐹(𝑢(𝑡+),𝑣(𝑡+))𝐹(𝑢(𝑡),𝑣(𝑡)).(4.19) Therefore, 𝐹(𝑢,𝑣) is a conservative quantity for weak solution (𝑢,𝑣). The proof is completed.

Theorem 4.4 (see [21]). Let 𝑋1,𝑋2 be Hilbert space and 𝐹𝑋1×𝑋2𝑅  be 𝐶1 functional. Then a 𝐶1 functional 𝐺𝑋1×𝑋2𝑅 is a conservative quantity for the infinite-dimensional Hamilton system (4.1) if and only if the following equality 𝛿𝐺(𝑢,𝑣),𝛿𝑢𝛿𝐹(𝑢,𝑣)𝛿𝑣𝑋1×𝑋2=𝛿𝐺(𝑢,𝑣),𝛿𝑣𝛿𝐹(𝑢,𝑣)𝛿𝑢𝑋1×𝑋2(4.20) holds true for any (𝑢,𝑣)𝑋1×𝑋2.

Proof. Let (𝑢,𝑣) be a solution of (4.1). Then we have 𝑑𝑑𝑡𝐺(𝑢,𝑣)=𝛿𝐺(𝑢,𝑣),𝛿𝑢𝑑𝑢𝑑𝑡𝑋1×𝑋2+𝛿𝐺(𝑢,𝑣),𝛿𝑣𝑑𝑣𝑑𝑡𝑋1×𝑋2=𝛿𝐺(𝑢,𝑣),𝛿𝑢𝛿𝐹(𝑢,𝑣)+𝛿𝑣𝛿𝐺(𝑢,𝑣),𝛿𝑣𝛿𝐹(𝑢,𝑣),𝛿𝑢(4.21) which imply that (𝑑/𝑑𝑡)𝐺(𝑢,𝑣)=0 if and only if equality (4.20) holds true. The proof is completed.

5. The Existence of Global Solution of Spinor BEC

In this section we consider the Gross-Pitaevskii equation of spinor BEC (2.10) under the Dirichlet boundary condition, to wit the following initial boundary problem:𝑖𝜕Ψ=𝛿𝜕𝑡𝛿Ψ𝐸Ψ,ΨΨ||,𝑥Ω,𝜕Ω=0,Ψ(𝑥,0)=Ψ0(𝑥),(5.1) where Ω𝑅𝑛(1𝑛3) is a domain. When Ω=𝑅𝑛, then (5.1) become Cauchy problem. By applying Theorem 4.3, we can obtain the following theorem.

Theorem 5.1. Assume that 𝑉𝐿2(Ω) and 𝑔𝑛>max{0,2𝑔𝑠}, then for any Ψ0𝐻1(Ω,𝒞3), there exists one global weak solution of problem (5.1) Ψ𝐶0[0,),𝐿2Ω,𝒞3𝐿(0,),𝐻1Ω,𝒞3.(5.2)

Remark 5.2. If 𝑔𝑠=0, then (5.1) reduce to the GP equation of BEC. Theorem 5.1 is also consistent with the experiments in repulsive case. In the situation of repulsive interaction, solutions to the GP equation of BEC are well defined for all times [12, 13, 20], which corresponds to the emergence of the BEC.

Proof. Let 𝐻=𝐿2(Ω,𝑅6),𝐻1=𝐻1(Ω,𝑅6). Firstly, we need to verify condition (4.3) in Theorem 4.3. From Section 2, we know that 𝐹𝜓1,𝜓2=12Ω2||||2𝑚Ψ2||Ψ||+𝑉(𝑥)2+12𝑔𝑛||Ψ||4+12𝑔𝑠||Ψ||𝑆Ψ2𝑑𝑥,(5.3) where Ψ=(𝜓1,𝜓0,𝜓1),𝜓𝑘=𝜓1𝑘+𝑖𝜓2𝑘(𝑘=1,0,1), and ||Ψ||𝑆Ψ2=||𝜓1||4+||𝜓1||4||𝜓21||2||𝜓1||2||𝜓+20||2||𝜓1||2||𝜓+20||2||𝜓1||2+2𝜓20𝜓1𝜓1+2𝜓02𝜓1𝜓1||𝜓21||4+||𝜓0||4+||𝜓1||4.(5.4) Hence, when 𝑔𝑛>max{0,2𝑔𝑠}, we have Ω𝑔𝑛||Ψ||4+𝑔𝑠||Ψ||𝑆Ψ2𝑑𝑥𝜆Ω||Ψ||4𝑑𝑥,(5.5) where 𝜆=𝑔𝑛max{0,2𝑔𝑠}>0. Therefore, we deduce 𝐹𝜓1,𝜓2𝜓1,𝜓2𝐻1,(5.6) which implies that condition (4.3) holds true.
Next we need to verify the continuous condition in Theorem 4.3. Let operator 𝐷𝐹:𝐻1𝐻1 be defined by Ψ=1𝐷𝐹(Ψ),Ω2Ψ2𝑚ΨΨ+𝑉(𝑥)Ψ+𝑔𝑛||Ψ||2ΨΨ+𝑔𝑠ΨΨ𝑆Ψ𝑆Ψ𝑑𝑥.(5.7) For any Ψ𝐶0(Ω,𝑅6) and Ψ𝑛Φ in 𝐻1, we have lim𝑛Ψ𝐷𝐹𝑛,Ψ=Ψ.𝐷𝐹(Φ),(5.8) Since 𝐶0(Ω,𝑅6) is dense in 𝐻1, equality (5.8) holds true for all Ψ𝐻1, which implies that 𝐷𝐹𝐻1𝐻1 is weakly continuous.
Therefore, according to Theorem 4.3, there exists a global weak solution of (5.1). The proof is completed.

6. The Conservative Quantities of Spinor BEC

In this section we will discuss the conservative quantities of spinor BEC. Let 𝐸 be defined as ((2.3), 𝑁,𝑀 as follows:𝑁=Ω||𝜓1||2+||𝜓0||2+||𝜓1||2𝑑𝑥,𝑀=Ω||𝜓1||2||𝜓1||2𝑑𝑥.(6.1) Then by using the same method as the proof of Theorem 4.4, we will prove the following theorem.

Theorem 6.1. Hamilton energy 𝐸, the total particle number 𝑁, and magnetization intensity 𝑀 are conservative quantities for problem (5.1).

Proof. Firstly, from (3.1) and (3.2) we can get 12𝑑𝐸Ψ,Ψ=𝜓𝑑𝑡𝑑𝐹1,𝜓2=𝜓𝑑𝑡𝛿𝐹1,𝜓2𝛿𝜓1,𝜕𝜓1+𝜓𝜕𝑡𝛿𝐹1,𝜓2𝛿𝜓2,𝜕𝜓2=𝜕𝑡𝑘=1,0,1𝜓𝛿𝐹1,𝜓2𝛿𝜓1𝑘,𝜕𝜓1𝑘+𝜓𝜕𝑡𝛿𝐹1,𝜓2𝛿𝜓2𝑘,𝜕𝜓2𝑘=𝜕𝑡𝑘=1,0,1𝜕𝜓2𝑘,𝜕𝑡𝜕𝜓1𝑘+𝜕𝑡𝜕𝜓1𝑘,𝜕𝑡𝜕𝜓2𝑘𝜕𝑡=0,(6.2) which imply that the energy 𝐸 is a conservative quantity for problem (5.1).
Secondly, by using (3.3) we can get the following equalities: 𝑑𝑁=𝑑𝑑𝑡𝑑𝑡Ω𝑘=1,0,1||𝜓1𝑘||2+||𝜓2𝑘||2𝑑𝑥=2Ω𝑘=1,0,1𝜓1𝑘𝜕𝜓1𝑘𝜕𝑡+𝜓2𝑘𝜕𝜓2𝑘𝜕𝑡𝑑𝑥=0,(6.3) which imply that the total particle number 𝑁 is a conservative quantity for problem (5.1).
At last, we show 𝑀 is a conservative quantity for problem (5.1). Let 𝑋1=𝑋2=𝐻1(Ω,𝑅3), then 𝐸𝑋1×𝑋2𝑅1denedby(2.3),𝑀𝑋1×𝑋2𝑅1denedby(6.1)(6.4) are both functional. Let 𝜓1=(𝜓11,𝜓10,𝜓11),𝜓2=(𝜓21,𝜓20,𝜓21). It is easy to check that 𝛿𝑀𝛿𝜓1=2𝜓11,0,2𝜓11,𝛿𝑀𝛿𝜓2=2𝜓21,0,2𝜓21,𝛿𝐸𝛿𝜓1=𝛿𝐸𝛿𝜓11,𝛿𝐸𝛿𝜓10,𝛿𝐸𝛿𝜓11,𝛿𝐸𝛿𝜓2=𝛿𝐸𝛿𝜓21,𝛿𝐸𝛿𝜓20,𝛿𝐸𝛿𝜓21.(6.5) From ((2.3) and (2.7), we have 𝛿𝐸𝛿𝜓21=22𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓21+𝑔𝑠2𝜓11𝜓10𝜓20𝜓21𝜓102+𝜓21𝜓202,𝛿𝐸𝛿𝜓11=22𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓11+𝑔𝑠2𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓11𝜓202,𝛿𝐸𝛿𝜓21=22𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓21+𝑔𝑠2𝜓11𝜓10𝜓20𝜓21𝜓102+𝜓21𝜓202,𝛿𝐸𝛿𝜓11=22𝑚Δ+𝑉(𝑥)+𝑔𝑛||Ψ||2+𝑔𝑠||𝜓1||2+||𝜓0||2||𝜓1||2𝜓11+𝑔𝑠2𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓11𝜓202.(6.6) Combining (6.5), (6.6) with (3.1), (3.2), we can get following equalities: 𝑑𝑀=𝜓𝑑𝑡𝛿𝑀1,𝜓2𝛿𝜓1,𝜕𝜓1𝜕𝑡𝑋1×𝑋2+𝜓𝛿𝑀1,𝜓2𝛿𝜓2,𝜕𝜓2𝜕𝑡𝑋1×𝑋2=𝜓𝛿𝑀1,𝜓2𝛿𝜓1,𝛿𝐹𝛿𝜓2𝑋1×𝑋2𝜓𝛿𝑀1,𝜓2𝛿𝜓2,𝛿𝐹𝛿𝜓1𝑋1×𝑋2=12Ω2𝜓11𝜕𝐸𝜕𝜓21𝜓11𝜕𝐸𝜕𝜓21𝜓21𝜕𝐸𝜕𝜓11+𝜓21𝜕𝐸𝜕𝜓11=𝑔𝑑𝑥𝑠Ω𝜓112𝜓11𝜓10𝜓20𝜓21𝜓102𝜓202𝜓112𝜓11𝜓10𝜓20𝜓21𝜓102𝜓202𝜓212𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓202+𝜓212𝜓21𝜓10𝜓20+𝜓11𝜓102𝜓202𝑑𝑥=0,(6.7) which imply that the magnetization intensity 𝑀 is a conservative quantity for problem (5.1). The proof is completed.


The authors would like to thank anonymous reviewers for their careful reading and many valuable comments that greatly improved the presentation of this paper. This work is supported by NSFC 11171236.


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