The Gross-Pitaevskii Model of Spinor BEC

Abstract

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. [6–14] and references therein). There are also many pieces of the literature on the spinor BEC ([15–19]). 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:𝑖ℏ𝜕𝜓¹𝜕𝑡=−2ℏ𝑚²Δ𝜓¹+𝑉(𝑥)𝜓₁+𝑔_𝑛||Ψ||²𝜓₁+𝑔_𝑠𝜓^∗−1𝜓²⁰𝑖ℏ𝜕𝜓₀+𝑔_𝑠||𝜓¹||²+||𝜓₀||²−||𝜓₋₁||²𝜓¹,𝜕𝑡=−2ℏ𝑚²Δ𝜓⁰+𝑉(𝑥)𝜓₀+𝑔_𝑛||Ψ||²𝜓₀+2𝑔_𝑠𝜓₁𝜓₋₁𝜓^∗0𝑖ℏ𝜕𝜓₋₁+𝑔_𝑠||𝜓¹||²+||𝜓₋₁||²𝜓⁰,𝜕𝑡=−2ℏ𝑚²Δ𝜓⁻¹+𝑉(𝑥)𝜓₋₁+𝑔_𝑛||Ψ||²𝜓₋₁+𝑔_𝑠𝜓^∗1𝜓²⁰+𝑔_𝑠||𝜓⁻¹||²+||𝜓₀||²−||𝜓₁||²𝜓⁻¹.(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 byΨ=𝜓₁,𝜓₀,𝜓₋₁.(2.2) Here the physical meaning of |𝜓_𝑖|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:𝐸Ψ,Ψ^∗=_Ω2ℏ𝑚²||∇Ψ||²+𝑉(𝑥)||Ψ||²+12𝑔^𝑛||Ψ||⁴+12𝑔^𝑠||Ψ^∗𝑆Ψ||²𝑑𝑥,(2.3) where 𝑚 is the boson mass, ℏ is Planck constant, 𝑉(⋅) is the external trapping potential, |Ψ|² is the density of dilute bosonic atoms, 𝑔_𝑛 is the interaction constant between atoms, and 𝑔_𝑠 is the spin exchange interaction constant𝑔_𝑛=4𝜋ℏ²𝑚𝑎⁰+2𝑎₂3,𝑔^𝑠=4𝜋ℏ²𝑚𝑎²−𝑎₀3,(2.4)𝑎₀ and 𝑎₂ are the scattering length, and 𝑆=𝑆_𝑥𝑖⃗+𝑆_𝑦𝑗⃗+𝑆_𝑧𝑘⃗ is the spin operator:𝑆_𝑥=1√2010101010,𝑆^𝑦=1√200𝑖−0𝑖𝑖−00𝑖,𝑆^𝑧=100000−001.(2.5) We adopt the following notations:Ψ^∗𝑆Ψ=𝜓^∗1,𝜓^∗0,𝜓^∗−1𝑆_𝑥𝑖⃗+𝑆_𝑦𝑗⃗+𝑆_𝑧𝑘⃗𝜓𝜓¹₀||Ψ_∗𝜓₋₁,𝑆Ψ||²=||Ψ^∗𝑆_𝑥Ψ||²+||Ψ^∗𝑆_𝑦Ψ||²+||Ψ^∗𝑆_𝑧Ψ||²=12||𝜓^∗1𝜓₀+𝜓^∗0𝜓₁+𝜓₋₁+𝜓^∗−1𝜓₀||²+12||−𝜓^∗1𝜓₀+𝜓^∗0𝜓₁−𝜓₋₁+𝜓^∗−1𝜓₀||²+||𝜓₁||²−||𝜓₋₁||²².(2.6) By calculation we can get||Ψ^∗𝑆Ψ||²=||𝜓₁||⁴+||𝜓₋₁||⁴−2||𝜓₁||²||𝜓₋₁||²+2||𝜓₀||²||𝜓₁||²+2||𝜓₀||²||𝜓₋₁||²+2𝜓²⁰𝜓^∗1𝜓^∗−1+2𝜓^∗02𝜓₁𝜓₋₁.(2.7)

By using Lagrange multiplier theorem, from the Hamilton energy functional 𝐸 (see (2.3) and the total particle number𝑁=_Ω||Ψ||²𝑑𝑥,Ψ=𝜓₁,𝜓₀,𝜓₋₁(2.8) is conservative, and we can obtain the steady state GP equation of spinor BEC as follows:𝜇𝜓_𝑘=𝛿𝛿𝜓^∗𝑘𝐸Ψ,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:𝑖ℏ𝜕𝜓^𝑘𝜕𝑡=𝛿𝜓𝛿^∗𝑘𝐸Ψ,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 𝑔_𝑛 and 𝑔_𝑠 we have following physical meaning:𝑔_𝑛>0,<0,correspondingtotherepulsiveinteractionbetweenatoms,𝑔_𝑠correspondingtotheattractiveinteractionbetweenatoms,>0,<0,correspondingtotheantiferromagneticstates,correspondingtotheferromagneticstates.(2.11)

3. Equivalent Form of Spinor BEC

Let 𝜓_𝑘=𝜓^1𝑘+𝑖𝜓^2𝑘. In this section we will show that GP equation (2.1) is equivalent to the following quantum Hamilton systems (see [21]):𝜕𝜓^1𝑘𝜕𝑡=𝛿𝐹𝜓¹,𝜓²𝛿𝜓^2𝑘𝜕𝜓_2𝑘,𝜕𝑡=−𝛿𝐹𝜓¹,𝜓²𝛿𝜓^1𝑘,𝑘=1,0,−1,(3.1) where 𝜓¹=(𝜓¹¹,𝜓¹⁰,𝜓¹⁻¹),𝜓²=(𝜓²¹,𝜓²⁰,𝜓²⁻¹), and the energy functional defined as𝐹𝜓¹,𝜓²=12ℏ𝐸Ψ,Ψ^∗,(𝐸see(2.3)).(3.2)

In fact, on the one hand, splitting real and imaginary parts of (2.1), we can obtain ℏ𝜕𝜓¹¹𝜕𝑡=−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₀||²−||𝜓₋₁||²𝜓²¹ℏ𝜕𝜓₂₁+𝑔_𝑠2𝜓¹⁻¹𝜓¹⁰𝜓²⁰−𝜓²⁻¹𝜓¹⁰²+𝜓²⁻¹𝜓²⁰²,𝜕𝑡=2ℏ𝑚²Δ−𝑉(𝑥)−𝑔^𝑛||Ψ||²−𝑔_𝑠||𝜓¹||²+||𝜓₀||²−||𝜓₋₁||²𝜓¹¹ℏ𝜕𝜓₁₀−𝑔_𝑠2𝜓²⁻¹𝜓¹⁰𝜓²⁰+𝜓¹⁻¹𝜓¹⁰²−𝜓¹⁻¹𝜓²⁰²,𝜕𝑡=−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₋₁||²𝜓²⁰ℏ𝜕𝜓₂₀+2𝑔_𝑠𝜓¹¹𝜓²⁻¹𝜓¹⁰+𝜓¹⁻¹𝜓²¹𝜓¹⁰−𝜓¹¹𝜓¹⁻¹𝜓²⁰+𝜓²¹𝜓²⁻¹𝜓²⁰,𝜕𝑡=2ℏ𝑚²Δ−𝑉(𝑥)−𝑔^𝑛||Ψ||²−𝑔_𝑠||𝜓¹||²+||𝜓₋₁||²𝜓¹⁰ℏ𝜕𝜓₁₋₁−2𝑔_𝑠𝜓¹¹𝜓¹⁻¹𝜓¹⁰−𝜓²⁻¹𝜓²¹𝜓¹⁰+𝜓¹¹𝜓²⁻¹𝜓²⁰+𝜓²¹𝜓¹⁻¹𝜓²⁰,𝜕𝑡=−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓⁻¹||²+||𝜓₀||²−||𝜓₁||²𝜓²⁻¹ℏ𝜕𝜓₂₋₁+𝑔_𝑠2𝜓¹¹𝜓¹⁰𝜓²⁰−𝜓²¹𝜓¹⁰²+𝜓²¹𝜓²⁰²,𝜕𝑡=2ℏ𝑚²Δ−𝑉(𝑥)−𝑔^𝑛||Ψ||²−𝑔_𝑠||𝜓⁻¹||²+||𝜓₀||²−||𝜓₁||²𝜓¹⁻¹−𝑔_𝑠2𝜓²¹𝜓¹⁰𝜓²⁰+𝜓¹¹𝜓¹⁰²−𝜓¹¹𝜓²⁰².(3.3) On the other hand, it is easy to check that 𝛿𝐹𝜓¹,𝜓²𝛿𝜓²¹=1ℏ−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₀||²−||𝜓₋₁||²𝜓²¹𝛿𝐹𝜓₁+𝑔_𝑠2𝜓¹⁻¹𝜓¹⁰𝜓²⁰−𝜓²⁻¹𝜓¹⁰²+𝜓²⁻¹𝜓²⁰²,,𝜓²𝛿𝜓¹¹=1ℏ−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₀||²−||𝜓₋₁||²𝜓¹¹𝛿𝐹𝜓₁+𝑔_𝑠2𝜓²⁻¹𝜓¹⁰𝜓²⁰+𝜓¹⁻¹𝜓¹⁰²−𝜓¹⁻¹𝜓²⁰²,,𝜓²𝛿𝜓²⁰=1ℏ−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₋₁||²𝜓²⁰𝛿𝐹𝜓₁+2𝑔_𝑠𝜓¹¹𝜓²⁻¹𝜓¹⁰+𝜓¹⁻¹𝜓²¹𝜓¹⁰−𝜓¹¹𝜓¹⁻¹𝜓²⁰+𝜓²¹𝜓²⁻¹𝜓²⁰,,𝜓²𝛿𝜓¹⁰=1ℏ−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₋₁||²𝜓¹⁰𝛿𝐹𝜓₁+2𝑔_𝑠𝜓¹¹𝜓¹⁻¹𝜓¹⁰−𝜓²⁻¹𝜓²¹𝜓¹⁰+𝜓¹¹𝜓²⁻¹𝜓²⁰+𝜓²¹𝜓¹⁻¹𝜓²⁰,,𝜓²𝛿𝜓²⁻¹=1ℏ−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓⁻¹||²+||𝜓₀||²−||𝜓₁||²𝜓²⁻¹𝛿𝐹𝜓₁+𝑔_𝑠2𝜓¹¹𝜓¹⁰𝜓²⁰−𝜓²¹𝜓¹⁰²+𝜓²¹𝜓²⁰²,,𝜓²𝛿𝜓¹⁻¹=1ℏ−2ℏ𝑚²Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓⁻¹||²+||𝜓₀||²−||𝜓₁||²𝜓¹⁻¹+𝑔_𝑠2𝜓²¹𝜓¹⁰𝜓²⁰+𝜓¹¹𝜓¹⁰²−𝜓¹¹𝜓²⁰².(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=−𝐷^𝑣𝑑𝑣𝐹(𝑢,𝑣),𝑑𝑡=𝐷^𝑢𝑢(0)=𝜑,𝐹𝑣((𝑢0,)𝑣=),𝜓,(𝑢,𝑣)∈𝑋¹×𝑋₂(4.1) where 𝑋⊂𝑋_𝑖⊂𝐻(𝑖=1,2) is dense, 𝑋 is linear space, 𝑋₁,𝑋₂ is reflexive Banach space, 𝐻 is Hilbert space, 𝐹∶𝑋₁×𝑋₂→𝑅¹  is 𝐶¹ functional, and 𝐷𝐹=(𝐷_𝑢𝐹,𝐷_𝑣𝐹) 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), provided⟨𝑢,𝑢⟩̃_𝐻+⟨𝑣,𝑣⟩̃_𝐻=^𝑡0⟨𝐷_𝑢𝐹(𝑢,𝑣),𝑣⟩̃−⟨𝐷_𝑣𝐹(𝑢,𝑣),𝑢⟩̃𝑑𝑡+⟨𝜑,𝑢⟩̃_𝐻+⟨𝜓,𝑣⟩̃_𝐻,(4.2) for every 𝑢̃∈𝑋₂,𝑣̃∈𝑋₁.
Let 𝐹∶𝑋₁×𝑋₂→𝑅¹ satisfy ||||(𝑢,𝑣)||||_𝑋1×𝑋2⟶∞⟺𝐹(𝑢,𝑣)⟶∞(or−𝐹(𝑢,𝑣)⟶∞).(4.3) Then we have the following existence theorem.

Theorem 4.3 (see [21]). Assume that 𝐹 satisfies condition (4.3) and 𝐷𝐹∶𝑋₁×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 𝑒_𝑘∣𝑘=1,2,…⊂𝑋(4.6) as orthonormal basis of space 𝐻. Set 𝑋_𝑛,𝑋_𝑛 as follows: 𝑋_𝑛=^𝑛_𝑘=1𝑋𝛼_𝑘𝑒_𝑘∣𝛼_𝑘∈𝑅¹,1≤𝑘≤𝑛_𝑛,=^𝑛_𝑘=1𝛽_𝑘(𝑡)𝑒_𝑘∣𝛽_𝑘(⋅)∈𝐶¹[0,∞),1≤𝑘≤𝑛.(4.7)
Consider the ordinary equations as follows: 𝑑𝑥_𝑘(𝑡)𝑑𝑡=−𝐷^𝑣𝑑𝑦_𝑘𝐹𝑢_𝑛,𝑣_𝑛,𝑒_𝑘,(𝑡)𝑑𝑡=𝐷^𝑢𝑥_𝑘𝐹𝑢_𝑛,𝑣_𝑛,𝑒_𝑘,𝑦_𝑘(0)=⟨𝜑,𝑒_𝑘⟩_𝐻,(0)=⟨𝜓,𝑒_𝑘⟩_𝐻,(𝑘=1,…,𝑛),(4.8) where 𝑢_𝑛=∑^𝑛𝑘=1𝑥_𝑘(𝑡)𝑒_𝑘,𝑣_𝑛=∑^𝑛𝑘=1𝑦_𝑘(𝑡)𝑒_𝑘.
By the theory of ordinary equations, there exists only one local solution of (4.8): 𝑥₁(𝑡),𝑦₁(𝑡),…,𝑥_𝑛(𝑡),𝑦_𝑛(𝑡),0≤𝑡≤𝜏.(4.9)
From (4.8) we can obtain the equality ⟨𝑢_𝑛,𝑢_𝑛̃⟩_𝐻+⟨𝑣_𝑛,𝑣_𝑛̃⟩_𝐻=+^𝑡0⟨𝜑,𝐷𝑢_𝑛̃_𝑢𝐹𝑢_𝑛,𝑣_𝑛,𝑣_𝑛̃−𝐷_𝑣𝐹𝑢_𝑛,𝑣_𝑛,𝑢_𝑛̃𝑑𝑡⟩_𝐻+⟨𝜓,𝑣_𝑛̃⟩_𝐻(4.10) holds true for any 𝑢_𝑛̃,𝑣_𝑛̃∈𝑋_𝑛. Moreover, equality ^𝑡0𝑑𝑑𝑢𝑡^𝑛,𝑢^𝑛̃_𝐻+𝑑𝑣^𝑛𝑑𝑡,𝑣^𝑛̃_𝐻𝑑𝑡=^𝑡0𝐷_𝑢𝐹𝑢_𝑛,𝑣_𝑛,𝑣_𝑛̃−𝐷_𝑣𝐹𝑢_𝑛,𝑣_𝑛,𝑢_𝑛̃𝑑𝑡(4.11) holds true for any 𝑢_𝑛̃,𝑣_𝑛̃∈𝑋_𝑛.
Putting (𝑢_𝑛̃,𝑣_𝑛̃)=(−𝑑𝑣_𝑛/𝑑𝑡,𝑑𝑢_𝑛/𝑑𝑡) in (4.11), we obtain that 0=^𝑡0𝐷^𝑢𝐹𝑢_𝑛,𝑣_𝑛,𝑑𝑢^𝑛𝑑𝑡+𝐷^𝑣𝐹𝑢_𝑛,𝑣_𝑛,𝑑𝑣^𝑛=𝑑𝑡𝑑𝑡_𝑡0𝑑𝑑𝑡𝐹𝑢^𝑛,𝑣_𝑛𝑑𝑡,(4.12) which implies 𝐹𝑢_𝑛,𝑣_𝑛=𝐹𝜑_𝑛,𝜓_𝑛,(4.13) where 𝜑_𝑛=^𝑛_𝑘=1⟨𝜑,𝑒_𝑘⟩_𝐻𝑒_𝑘,𝜓_𝑛=^𝑛_𝑘=1⟨𝜓,𝑒_𝑘⟩_𝐻𝑒_𝑘.(4.14)
From (4.3) and (4.10), we deduce that {(𝑢_𝑛,𝑣_𝑛)}^∞𝑛=1 is bounded in 𝐿^∞((0,∞),𝑋₁×𝑋₂). Therefore there exists a subsequence; we still write it as {(𝑢_𝑛,𝑣_𝑛)}^∞𝑛=1, such that 𝑢_𝑛,𝑣_𝑛⇀(𝑢,𝑣)in𝑋₁×𝑋₂,a.e.𝑡∈(0,∞).(4.15)
According to 𝐷𝐹∶𝑋₁×𝑋₂→(𝑋₁×𝑋₂)^∗ being weakly continuous and (4.10), (4.15), we know the following equality ⟨𝑢,𝑢⟩̃_𝐻+⟨𝑣,𝑣⟩̃_𝐻=^𝑡0⟨𝐷_𝑢𝐹(𝑢,𝑣),𝑣⟩̃−⟨𝐷_𝑣𝐹(𝑢,𝑣),𝑢⟩̃𝑑𝑡+⟨𝜑,𝑢⟩̃_𝐻+⟨𝜓,𝑣⟩̃_𝐻(4.16) holds true for any 𝑢,̃𝑣̃∈⋃^∞𝑛=1𝑋_𝑛. Since ⋃^∞𝑛=1𝑋_𝑛 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 ℎ>0 we have ⟨𝑢(𝑡+ℎ)−𝑢(𝑡),𝑢⟩̃_𝐻+⟨𝑣(𝑡+ℎ)−𝑣(𝑡),𝑣⟩̃_𝐻=^𝑡𝑡+ℎ⟨𝐷_𝑢𝐹(𝑢(𝜏),𝑣(𝜏)),𝑣⟩̃−⟨𝐷_𝑣𝐹(𝑢(𝜏),𝑣(𝜏)),𝑢⟩̃𝑑𝜏.(4.17) Putting 𝑢̃=−Δ_ℎ𝑣=−(𝑣(𝑡+ℎ)−𝑣(𝑡)),𝑣̃=Δ_ℎ𝑢=𝑢(𝑡+ℎ)−𝑢(𝑡)(4.18) in (4.17), we obtain that 0=1ℎ^𝑡𝑡+ℎ⟨𝐷_𝑢=𝐹(𝑢(𝑡+ℎ)𝐹,𝑣((𝑢𝑡(+𝜏)ℎ,)𝑣)−(𝜏)𝐹),(Δ𝑢(_ℎ𝑡𝑢)⟩,𝑣+(𝑡⟨)𝐷)._𝑣𝐹(𝑢(𝜏),𝑣(𝜏)),Δ_ℎ𝑣⟩𝑑𝜏(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 𝐹∶𝑋₁×𝑋₂→𝑅  be 𝐶¹ functional. Then a 𝐶¹ functional 𝐺∶𝑋₁×𝑋₂→𝑅 is a conservative quantity for the infinite-dimensional Hamilton system (4.1) if and only if the following equality 𝛿𝐺𝛿(𝑢𝑢,𝑣),𝛿𝐹𝛿(𝑣𝑢,𝑣)_𝑋1×𝑋2=𝛿𝐺(𝑢,𝑣)𝛿𝑣,𝛿𝐹𝛿(𝑢𝑢,𝑣)_𝑋1×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)=𝛿𝐺𝛿(𝑢𝑢,𝑣),𝑑𝑑𝑢𝑡_𝑋1×𝑋2+𝛿𝐺(𝑢,𝑣)𝛿𝑣,𝑑𝑑𝑣𝑡_𝑋1×𝑋2=−𝛿𝐺(𝑢,𝑣)𝛿𝑢,𝛿𝐹𝛿(𝑣𝑢,𝑣)+𝛿𝐺𝛿(𝑣𝑢,𝑣),𝛿𝐹𝛿(𝑢𝑢,𝑣),(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:𝑖ℏ𝜕Ψ𝜕𝑡=𝛿Ψ𝛿^∗Ψ𝐸||Ψ,Ψ^∗,𝑥∈Ω,_𝜕ΩΨ(𝑥,0)==Ψ0₀,(𝑥),(5.1) where Ω⊂𝑅^𝑛(1≤𝑛≤3) is a domain. When Ω=𝑅^𝑛, then (5.1) become Cauchy problem. By applying Theorem 4.3, we can obtain the following theorem.

Theorem 5.1. Assume that 𝑉∈𝐿²(Ω) and 𝑔_𝑛>max{0,−2𝑔_𝑠}, then for any Ψ₀∈𝐻¹(Ω,𝒞³), there exists one global weak solution of problem (5.1) Ψ∈𝐶⁰[0,∞),𝐿²Ω,𝒞³∩𝐿^∞(0,∞),𝐻¹Ω,𝒞³.(5.2)

Remark 5.2. If 𝑔_𝑠=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 𝐹𝜓¹,𝜓²=12ℏ_Ω2ℏ𝑚²||∇Ψ||²+𝑉(𝑥)||Ψ||²+12𝑔^𝑛||Ψ||⁴+12𝑔^𝑠||Ψ^∗𝑆Ψ||²𝑑𝑥,(5.3) where Ψ=(𝜓₁,𝜓₀,𝜓₋₁),𝜓_𝑘=𝜓^1𝑘+𝑖𝜓^2𝑘(𝑘=1,0,−1), and ||Ψ^∗𝑆Ψ||²=||𝜓₁||⁴+||𝜓₋₁||⁴−2||𝜓₁||²||𝜓₋₁||²+2||𝜓₀||²||𝜓₁||²+2||𝜓₀||²||𝜓₋₁||²+2𝜓²⁰𝜓^∗1𝜓^∗−1+2𝜓^∗02𝜓₁𝜓₋₁≤2||𝜓₁||⁴+||𝜓₀||⁴+||𝜓₋₁||⁴.(5.4) Hence, when 𝑔_𝑛>max{0,−2𝑔_𝑠}, we have _Ω𝑔_𝑛||Ψ||⁴+𝑔_𝑠||Ψ^∗𝑆Ψ||²𝑑𝑥≥𝜆_Ω||Ψ||⁴𝑑𝑥,(5.5) where 𝜆=𝑔_𝑛−max{0,−2𝑔_𝑠}>0. Therefore, we deduce 𝐹𝜓¹,𝜓²⟶∞⟺‖‖𝜓¹,𝜓²‖‖_𝐻1⟶∞,(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:𝐻₁→𝐻₁ be defined by 𝐷𝐹(Ψ),Ψ=ℏ1_Ω2ℏ+𝑚²𝑔∇_𝑛Ψ⋅∇Ψ+𝑉(𝑥)Ψ⋅Ψ||Ψ||²Ψ⋅Ψ+𝑔_𝑠Ψ^∗𝑆ΨΨ^∗𝑆Ψ𝑑𝑥.(5.7) For any Ψ∈𝐶^∞0(Ω,𝑅⁶) and Ψ_𝑛⇀Φ in 𝐻₁, we have lim_𝑛→∞𝐷𝐹Ψ^𝑛,Ψ=𝐷𝐹(Φ),Ψ.(5.8) Since 𝐶^∞0(Ω,𝑅⁶) is dense in 𝐻₁, equality (5.8) holds true for all Ψ∈𝐻₁, which implies that 𝐷𝐹∶𝐻₁→𝐻₁ 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+||𝜓₋₁||²𝑑𝑥,𝑀=_Ω||𝜓¹||²−||𝜓₋₁||²𝑑𝑥.(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 12ℏ𝑑𝐸Ψ,Ψ^∗𝑑𝑡=𝑑𝐹𝜓¹,𝜓²𝑑𝑡=𝛿𝐹𝜓¹,𝜓²𝛿𝜓¹,𝜕𝜓¹𝜕𝑡+𝛿𝐹𝜓¹,𝜓²𝛿𝜓²,𝜕𝜓²=𝜕𝑡_{𝑘=1,0,−1}𝛿𝐹𝜓¹,𝜓²𝛿𝜓^1𝑘,𝜕𝜓^1𝑘𝜕𝑡+𝛿𝐹𝜓¹,𝜓²𝛿𝜓^2𝑘,𝜕𝜓^2𝑘=𝜕𝑡_{𝑘=1,0,−1}=0,−𝜕𝜕𝜓𝑡^2𝑘,𝜕𝜕𝜓𝑡^1𝑘+𝜕𝜕𝜓𝑡^1𝑘,𝜕𝜕𝜓𝑡^2𝑘(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: 𝑑𝑁𝑑𝑡=𝑑𝑑𝑡_Ω_{𝑘=1,0,−1}||𝜓^1𝑘||²+||𝜓^2𝑘||²=2_Ω𝑑𝑥_𝑘=_1,0,−1𝜓^1𝑘𝜕𝜓^1𝑘𝜕𝑡+𝜓^2𝑘𝜕𝜓^2𝑘=0,𝜕𝑡𝑑𝑥(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 𝐸𝑀∶∶𝑋𝑋₁₁×𝑋₂⟶𝑅¹definedby(2.3),×𝑋₂⟶𝑅¹definedby(6.1)(6.4) are both functional. Let 𝜓¹=(𝜓¹¹,𝜓¹⁰,𝜓¹⁻¹),𝜓²=(𝜓²¹,𝜓²⁰,𝜓²⁻¹). It is easy to check that 𝛿𝑀𝛿𝜓¹=2𝜓¹¹,0,−2𝜓¹⁻¹,𝛿𝑀𝛿𝜓²𝛿𝐸=2𝜓²¹,0,−2𝜓²⁻¹,𝛿𝜓¹=𝛿𝐸𝛿𝜓¹¹,𝛿𝐸𝛿𝜓¹⁰,𝛿𝐸𝛿𝜓¹⁻¹,𝛿𝛿𝜓𝐸²=𝛿𝐸𝛿𝜓²¹,𝛿𝐸𝛿𝜓²⁰,𝛿𝐸𝛿𝜓²⁻¹.(6.5) From ((2.3) and (2.7), we have 𝛿𝐸𝛿𝜓²¹=−ℏ²2𝑚Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₀||²−||𝜓₋₁||²𝜓²¹𝛿𝐸+𝑔_𝑠2𝜓¹⁻¹𝜓¹⁰𝜓²⁰−𝜓²⁻¹𝜓¹⁰²+𝜓²⁻¹𝜓²⁰²,𝛿𝜓¹¹=−ℏ²2𝑚Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓¹||²+||𝜓₀||²−||𝜓₋₁||²𝜓¹¹𝛿𝐸+𝑔_𝑠2𝜓²⁻¹𝜓¹⁰𝜓²⁰+𝜓¹⁻¹𝜓¹⁰²−𝜓¹⁻¹𝜓²⁰²,𝛿𝜓²⁻¹=−ℏ²2𝑚Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓⁻¹||²+||𝜓₀||²−||𝜓₁||²𝜓²⁻¹𝛿𝐸+𝑔_𝑠2𝜓¹¹𝜓¹⁰𝜓²⁰−𝜓²¹𝜓¹⁰²+𝜓²¹𝜓²⁰²,𝛿𝜓¹⁻¹=−ℏ²2𝑚Δ+𝑉(𝑥)+𝑔^𝑛||Ψ||²+𝑔_𝑠||𝜓⁻¹||²+||𝜓₀||²−||𝜓₁||²𝜓¹⁻¹+𝑔_𝑠2𝜓²¹𝜓¹⁰𝜓²⁰+𝜓¹¹𝜓¹⁰²−𝜓¹¹𝜓²⁰².(6.6) Combining (6.5), (6.6) with (3.1), (3.2), we can get following equalities: 𝑑𝑀𝑑𝑡=𝛿𝑀𝜓¹,𝜓²𝛿𝜓¹,𝜕𝜓¹𝜕𝑡_𝑋1×𝑋2+𝛿𝑀𝜓¹,𝜓²𝛿𝜓²,𝜕𝜓²𝜕𝑡_𝑋1×𝑋2=𝛿𝑀𝜓¹,𝜓²𝛿𝜓¹,𝛿𝐹𝛿𝜓²_𝑋1×𝑋2−𝛿𝑀𝜓¹,𝜓²𝛿𝜓²,𝛿𝐹𝛿𝜓¹_𝑋1×𝑋2=12ℏ_Ω2𝜓¹¹𝜕𝐸𝜕𝜓²¹−𝜓¹⁻¹𝜕𝐸𝜕𝜓²⁻¹−𝜓²¹𝜕𝐸𝜕𝜓¹¹+𝜓²⁻¹𝜕𝐸=𝑔𝜕𝜓¹⁻¹𝑑𝑥_𝑠ℏ_Ω𝜓¹¹2𝜓¹⁻¹−𝜓₁₋₁𝜓¹⁰𝜓²⁰−𝜓²⁻¹𝜓¹⁰²−𝜓²⁰²−𝜓₂₁2𝜓¹¹𝜓¹⁰𝜓²⁰−𝜓²¹𝜓¹⁰²−𝜓²⁰²+𝜓₂₋₁2𝜓²⁻¹𝜓¹⁰𝜓²⁰+𝜓¹⁻¹𝜓¹⁰²−𝜓²⁰²=0,2𝜓²¹𝜓¹⁰𝜓²⁰+𝜓¹¹𝜓¹⁰²−𝜓²⁰²𝑑𝑥(6.7) which imply that the magnetization intensity 𝑀 is a conservative quantity for problem (5.1). The proof is completed.

Acknowledgments

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.