質量重分配及其在自旋-1玻色-愛因斯坦凝聚基態上之應用

國立台灣大學理學院數學所 博士論文

Department of Mathematics College of Science

National Taiwan University Doctoral Dissertation

質量重分配及其在自旋-1 玻色-愛因斯坦凝聚基態上之應用

Mass Redistribution and Its Applications to the Ground States of Spin-1 Bose-Einstein Condensates

林立人 Liren Lin

指導教授：陳宜良 博士 Advisor: I-Liang Chern Ph.D.

中華民國 102 年 7 月

July, 2013

誌 誌 誌 謝 謝 謝

感謝我的指導教授陳宜良老師。這篇論文的產生就從他將自旋玻色-愛因斯坦凝聚

介紹給我開始。 先是一個小問題(雖然一開始也不覺得小)，然後發現越來越多有趣的 事情。 回想起來，我能夠知道這個問題真是非常幸運。 從大三、大四、碩士一直到

博士班的頭幾年，不停地換領域，找不到一個可以下定決心的問題， 我可以說是沒有

一樣東西是專精的。代數會一些(差不多都忘了)，幾何會一些。 一直到為了考資格考 才終於認真學了實分析。 本來想做應用數學也好，到頭來卻又覺得自己還是擁有一個

純數學家的靈魂。 混到了博三，偏微分方程的近代理論只知道一些最皮毛的事。 索

伯列夫空間，弱解等概念還說得出來，其他的基本上是零。 就在這情形下，老師告訴 我自旋玻色-愛因斯坦凝聚的某個計算結果。 我於是沒有任何武器，直接面對這個有 三個分量，看起來相當可怕的系統。 也因為這個系統看起來太複雜了，什麼都不會的 我，只能從一些最天真的想法去做。 而就因為這樣，才讓我發現到問題背後原來有一 個非常簡單的道理。 在這之中，還是由於老師堅信某些“幸運”的背後必定有些重要的 事情在其中， 讓我發現質量重分配這個方法。否則，我的工作可能就只是一些幸運的 猜測， 然後在一堆莫名的計算中驗證了所要證明的結果。

除此之外，在博士期間，常常聽老師分享他對各式各樣數學的看法，包含跟我的

研究有關與無關的， 拓展了我的視野。這些想法雖然很多離我很遠，有些確實給了一

些可以實行的想法。 很多想法在短時間之內也還沒辦法去弄清楚，似乎讓未來有做不 完的事。

還有非常重要的，由於老師交遊廣闊，也讓我認識了許多學者。 包括新加坡國立

大學的包維柱教授，他與他的合作者們的數值模擬工作在我的研究上給了我許多想 法。 還有新竹教育大學的陳人豪教授，因為老師的邀請特別來到台大一起討論。 像 我這麼不擅與不認識的人往來的人，這些機會對我來說都是非常難得的。

摘 摘 摘 要 要 要

自 旋-1玻 色-愛 因 斯 坦 凝 聚 是 一 類 特 殊 的 ， 含 有 三 個 分 量 函 數 的 系 統 。 通 常 以 Ψ = (ψ₁, ψ₀, ψ−1) 表示。 它的行為由一個能量泛函 E[Ψ] 及兩個限制條件所描述。

這兩個限制分別是原子數守恆與磁化量守恆，也就是說R |Ψ|² 及R (|ψ1|²− |ψ−1|²) 是兩個固定的數。 而所謂的基態即指在這兩個條件之下， 使能量 E 達到最小的狀態

Ψ。 要解釋這篇論文所討論的問題，我們還得指出，根據能量 E 的表達式裡的某個

參數的正負號， 自旋-1玻色愛因斯坦凝聚被分成順磁性與反磁性兩類。 這兩類系統

表現出來的行為有本質上的不同。 這篇論文裡的工作，其動機來自於兩個現象，恰好 一個屬於順磁性，另一個反磁性。

1. 任何順磁性系統中的基態，必定滿足下列形式

Ψ = (γ1ψ, γ0ψ, γ−1ψ),

其中 γj 皆為常數，而 ψ 為函數。 這個形式稱作單模近似。由其名稱即可知，這原本

只被視為一種簡化的假設。 然而在後來的研究中卻發現，對順磁性系統的基態來說，

此形式是完全正確，而非只是近似。

2. 考慮外加一個均勻磁場的情形。若將磁場的強度由零慢慢增加，當強度超過某個特

定的數值時， 反磁性系統的基態會經歷一個從 ψ0 ≡ 0到 ψ0 6= 0 的分歧。

雖然這兩個現象很早就已經在數值模擬中被發現，但在我們的研究之前，還沒有 一個真正嚴格的數學證明。 這篇論文包含我們在 [16, 17] 這兩篇論文裡的工作， 它 們分別給出了上面兩個現象的嚴格證明。 比起兩篇原本的論文，在本文中我們盡可 能把所有的細節都交待清楚。 我們的證明方法主要是使用了下面這個原理：質量密 度(也就是 |ψ1|², |ψ0|²及|ψ−1|²)的重分配將必定導致動能的下降。 這個原理可視為某 個廣為人知的梯度的凸性不等式的簡單推廣。 我們將會說明這個原理如何給出解決上 面問題的一個統一的想法。

關關關鍵鍵鍵詞詞詞： 自旋，旋量，玻色-愛因斯坦凝聚系統，薛丁格系統，單模近似， 質量重分

配，分歧

Abstract

Spin-1 Bose-Einstein condensate (BEC) is a special three-component system, writ- ten as Ψ = (ψ1, ψ0, ψ−1). Its behavior is described by an energy functional E[Ψ] with two constraints: the conservation of the number of atoms and the conservation of total magnetization. That isR |Ψ|²andR (|ψ₁|²− |ψ−1|²) are fixed numbers. And a ground state is a minimizer of E under the constraints. To explain the problems considered in this thesis, we remark that according to the sign of a specific parameter in the energy E, spin-1 BECs are classified into two groups: ferromagnetic ones and antiferromagnetic ones. They exhibit rather different behaviors. The works in this thesis are motivated by the following two phenomena.

1. Any ground state of a ferromagnetic system is of the form Ψ = (γ₁ψ, γ₀ψ, γ₋₁ψ),

where γj are constants and ψ a function. This is called single-mode approximation.

According to the name, this form was originally only used as a simplified assumption, while from later studies it is found to be exactly the case for ferromagnetic ground states.

2. When an external magnetic field is applied, the ground state of an antiferromagnetic system undergoes a bifurcation from ψ0 ≡ 0 to ψ₀ 6= 0 as the strength of the magnetic field surpasses a critical value.

Although these phenomena have been well-known from numerical simulations for quite a long time, there were no rigorous mathematical justifications before our inves- tigations. In this thesis, our works [16, 17] on their proofs are given, with more details.

The proofs rely on a principle which says that a redistribution of the mass densities (i.e.

|ψ₁|², |ψ₀|²and |ψ−1|²) will decrease the kinetic energy. This principle can be regarded as a simple generalization of a well-known convexity inequality for gradients. We will show how this principle can give a rather unified approach toward our problems.

Keywords: spin-1, spinor, BEC system, Schr¨odinger system, single-mode approxima- tion, mass redistribution, bifurcation

Contents

口口口試試試委委委員員員審審審定定定書書書 i

誌誌誌謝謝謝 iii

摘摘摘要要要 v

Abstract vii

Contents viii

1 Introduction 1

1.1 Mathematical model of spin-1 BEC . . . 1 1.2 The motivations . . . 4

2 Preliminaries 7

2.1 A reduction of the model . . . 7 2.2 Fundamental properties . . . 10 2.3 Mass redistribution . . . 13

3 The Simplified Characterizations 16

3.1 The single-mode approximation . . . 16 3.2 The vanishing of u0 . . . 19 3.3 Some degenerate situations . . . 20

4 Some Further Properties 23

4.1 Continuity and monotonicity of E_g(M, q) . . . 23

4.1.1 E_g as a function of M . . . 23

4.1.2 E_g as a function of q . . . 27

4.2 u−1is no larger than u₁ . . . 28

4.3 Exponential decay of ground states . . . 32

5 The Bifurcation Phenomenon 35 5.1 Proof of Claim 3 . . . 37

5.2 What remains . . . 41

5.2.1 Estimates of qcfrom the proof . . . 41

5.2.2 The boundedness of qcwith respect to M . . . 42

6 Redistributional Perturbation in a Fixed Admissible Class 44 6.1 Inequality from redistributional perturbation . . . 44

6.1.1 Comparison with previous results . . . 49

6.2 From the viewpoint of the GP system . . . 50

6.2.1 Validity of the equality of (6.6) . . . 51

6.2.2 Discussions on (6.4) . . . 53

7 Discussions of some Open Problems 56 7.1 Uniqueness . . . 56

7.2 Uniform convergence at boundary regimes . . . 58

7.3 Comparison of the decaying rates . . . 59

8 Appendices 61 8.1 Convexity inequality for gradients . . . 61

8.2 Equivalence of the u-model and the Ψ-model . . . 62

8.3 Complements to Section 2.2 . . . 66

8.3.1 Positivity of nonvanishing components . . . 66

8.3.2 Proof of Proposition 2.2 . . . 68

Bibliography 72

Chapter 1 Introduction

When a Bose-Einstein condensate (BEC) of dilute atomic gas is confined by an optical trap, all its hyperfine spin states can be active simultaneously. In this situation, a spin-f BEC is described by a (2f + 1)-component order parameter

Ψ = (ψ_f, ψ_{f −1}, ..., ψ−f),

where the components ψ_jare complex-valued functions in the mean-field theory. Since the first realization of such spinor BECs in 1997 [19] (spin-1 BEC of²³Na), their rich structures have drawn great interest and a lot of researches.

This thesis focuses on some facts exhibited by the ground states of spin-1 BEC which have been well-known from numerical simulations for a long time. The aim is to provide rigorous mathematical justifications of them, based on a principle which says that a redistribution of the mass densities between different components will decrease the kinetic energy. Before further discussion, we shall first introduce the mathematical model.

1.1 Mathematical model of spin-1 BEC

A spin-1 BEC, as mentioned above, is described by a three-component vector function Ψ = (ψ₁, ψ₀, ψ−1), where each ψ_j is a complex-valued function on R³. We leave the specification of the suitable function space for Ψ to §2.1, although it should be very clear from the energy functional given below. Also note that we consider Ψ as

being independent of time since we will only be interested in ground states. For the dynamical law, see e.g. [15].

The energy of the system is

E[Ψ] = E_kin[Ψ] + E_pot[Ψ] + E₀[Ψ] + E₁[Ψ] + E_Zee[Ψ],

where¹

Ekin[Ψ] =

Z X

j

|∇ψj|²

Epot[Ψ] = Z

V (x)|Ψ|² E₀[Ψ] =

Z

β₀|Ψ|⁴ E1[Ψ] =

Z

β1|Ψ^∗F Ψ|² E_Zee[Ψ] =

Z

p(|ψ₁|²− |ψ−1|²) + q(|ψ₁|²+ |ψ−1|²).

V (x) is a real-valued function, and β₀, β₁, p, q are real constants. In the definition of E₁[Ψ], Ψ is regarded as a column vector and Ψ^∗ is its conjugate transpose. F stands for the triple (Fx, Fy, Fz) of 3 × 3 matrices given by

F_x = 1

√2







0 1 0 1 0 1 0 1 0





, F_y = i

√2







0 −1 0

1 0 −1

0 1 0





, F_z =







1 0 0

0 0 0

0 0 −1





. Thus

Ψ^∗F Ψ = (Ψ^∗F_xΨ, Ψ^∗F_yΨ, Ψ^∗F_zΨ).

The notation |Ψ| denotes the Euclidean length (P

j|ψj|²)^1/2, and similarly for |∇ψj| and |Ψ^∗F Ψ|.

Physically, V represents a state-independent trap potential, the terms with coeffi- cients β0 and β1 describe the interactions between the atoms, p and q are the linear

1

Remark on notation. When the domain of an integration is not specified, it’s understood to

be R

. Also, the dummy variable x as well as the differential dx in integrals are almost never

written explicitly. Nevertheless, we shall sometimes retain the variable x for the trap potential

V . This convention seems better in some places.

and quadratic Zeeman effects induced by an external uniform magnetic field, and the components of F are called the spin-1 Pauli matrices.

Besides the energy, the system is described to have the following two conserved quantities:

Number of atoms N [Ψ] = Z

|Ψ|², Total magnetization M[Ψ] =

Z

|ψ1|²− |ψ−1|²
.

And a ground state is a minimizer of E under fixed N and M. By normalization, we can assume N [Ψ] = 1. And M[Ψ] = M for some constant M . Note that |M[Ψ]| ≤ N [Ψ] for every state Ψ, so we must have |M | ≤ 1. Due to the symmetry of the roles of ψ1 and ψ−1, we will only consider 0 ≤ M ≤ 1. The general assumptions on the parameters of E are the following:

(A1) V ∈ L^∞_loc(R³), and V (x) tends to infinity as |x| tends to infinity. Precisely²

R→∞lim ( inf

|x|≥RV (x)) = ∞.

Note that in particular V is bounded from below.

(A2) β0 > |β₁| > 0.

(A3) q ≥ 0.

(A4) V ≥ 0 and p = 0.

We give some remarks for these assumptions.

1. (A2) indicates a repulsive nature of the system. (A1) will then guarantee that V (x) traps the system essentially in a localized region, which will be crucial in some places, including the proof of the existence result.

2. I’m not sure whether (A2) must be true in principle, but it holds for real systems as far as I know. (For example spin-1 BEC of²³Na and⁸⁷Rb.) Mathematically,

2

We’ll write inf (and the like) also for ess inf.

the fact |β1| < β₀ is also helpful in the proof of existence when β1 < 0. For β₁ > 0, the assumption β₁ < β₀ is in fact not used in this thesis. By the way, the case β₀ = β₁ = 0 or only β₁ = 0 can also be studied mathematically. We’ll however not consider them since they exhibit no further difficulty but only result in some degenerate situations not of much interest.

3. According to the sign of β₁, spin-1 BECs are classified into two groups: ferro- magneticones for β1 < 0, and antiferromagnetic ones for β1 > 0. The typical examples are respectively ²³Na and⁸⁷Rb. They show very different behaviors from each other.

4. Physically, the values of p and q can be tuned by modifying the applied magnetic field. It’s also possible to make q negative, but we do not consider this case in this thesis.

5. Due to the conservations of N and M, ground states are not changed by shifting the values of V and p by any constants. Hence (A4) causes no loss of generality for our purposes.

Note. The model of spin-1 BEC appeared very soon after the first realization. The ear- liest papers being [9], [18] and [12]. As to the understanding of the model, I however most benefited from [22] and various papers by Dr. Weizhu Bao and his collaborators, for example [3], [2] and [15]. Besides them, the review article [11] is also a reference I consulted from time to time. Our expression of the energy functional is mostly similar to that given in [22], [15] and [11].

1.2 The motivations

The whole study is motivated by two phenomena, pertaining to ferromagnetic and antiferromagnetic systems respectively.

1. For a ferromagnetic system, when there is no external magnetic field (i.e. q = 0), its ground state Ψ obeys the single-mode approximation (SMA)

Ψ = (γ₁ψ, γ₀ψ, γ−1ψ),

where each γ_j is a constant, and ψ is a function independent of j.

2. For an antiferromagnetic system, as q increases from zero, its ground state Ψ undergoes a bifurcation from ψ₀ ≡ 0 to ψ₀ 6= 0 at a critical q_c> 0.

The SMA, as the name indicates, was originally only regarded as an approximation, which was used to simplify the study of spin-1 BEC. As later investigations showed, it turns out to be exactly the case but not only an approximation for ferromagnetic systems (and is in general not suited for antiferromagnetic systems).

These phenomena have been known from numerical simulations for a long time.

For clear declarations and discussions of them, see respectively [21] and [22, 15]. The bifurcation phenomenon was recently also observed in experiments [10]. Nevertheless, there seems to be no sound mathematical reasonings for the validity of these facts be- fore. In theoretical discussions on the bifurcation, like in [22], the researchers usually assume Ψ is a constant vector, which is of course not a satisfactory demonstration.

We will later first consider q = 0. Due to the SMA, a ferromagnetic ground state can be characterized as a one-component system. On the other hand, the antiferromag- netic ground state has only two components since ψ₀ ≡ 0. They will be referred to as simplified characterizations in this thesis. Their proofs first appeared in our paper [16].

It’s interesting that, by using the mentioned redistribution method, they can be proved in almost the same way.

On the other hand, the bifurcation phenomenon can also be deduced by using mass redistribution, while a lot more technical details are involved. The most difficult part is to prove that we do have ψ0 ≡ 0 for some q strictly larger than zero. The proof first appeared in [17], where there are also many relevant discussions on the redistribution method. I am recently also preparing a simplified version, which go straight to the verification of the bifurcation phenomenon.

Note. The outline of the thesis is very clear from the contents. Moreover, I’ll use a few words in the beginning of every chapter or section to indicate what we are going to do.

Chapter 2

Preliminaries

In this chapter we give some preliminaries which are essential for the discussions in the rest of this thesis. In Section 2.1, we introduce a reduction which says that we can simply consider (|ψ₁|, |ψ₀|, |ψ−1|) for our purposes. Many notations are also given in the same section. In Section 2.2, the fundamental facts such as the existence of ground state, the Euler-Lagrange system and its direct corollaries are given. In Section 2.3, the idea of mass redistribution is introduced.

2.1 A reduction of the model

We shall write H¹ for H¹(R³, C), and similarly for other function spaces. Let

B =(ψ₁, ψ₀, ψ−1)

ψ_j ∈ H¹∩ L²_V ∩ L⁴ for each j ,

where L²_V is the V -weighted L²space. That is, a measurable function f belongs to L²_V if kf k²_L2

V :=R V (x)|f |²is finite. Note that L²_V is nothing but L² space with respect to the (σ-finite) measure V (x)dx. By endowing B with the norm

kΨk =X

j

kψ_jk_H¹ + kψ_jk_L²

V + kψ_jk_L⁴

, (2.1)

B is a Banach space. Obviously, B is the appropriate space for our variational model.

Precisely, ground states are minimizers of the following problem:

min E over {Ψ ∈ B | N [Ψ] = 1, M[Ψ] = M } .

For our purposes, we can reduce this model on B to a model on B+, where

B+ = {(u₁, u₀, u−1) ∈ B | uj ≥ 0 for each j } . We give the reduction in the following.

Given Ψ = (ψ₁, ψ₀, ψ−1) ∈ B, we have

E_kin[Ψ] =X

j

|∇ψ_j|² ≥X

j

|∇|ψ_j||²

by the convexity inequality for gradients (Section 8.1). Moreover, let

ψ_j = |ψ_j|e^iθj,

then it’s easy to check that

E1[Ψ] = Z

β1



2|ψ0|²h

|ψ1|²+ |ψ−1|²+ 2|ψ1||ψ−1| cos(θ1− 2θ0 + θ−1) i

+ (|ψ1|²− |ψ−1|²)²

 .

Hence

E₁[Ψ] ≥ Z

β₁



2|ψ₀|²(|ψ₁| − sgn(β₁)|ψ−1|)²+ (|ψ₁|²− |ψ−1|²)²

 ,

where

sgn(β₁) =

( 1 if β₁ > 0

−1 if β₁ < 0.

And the equality holds if

cos (θ₁− 2θ₀+ θ₋₁) ≡ −sgn(β₁). (2.2)

For other parts of the energy, we obviously have

E_pot[Ψ] = E_pot[(|ψ₁|, |ψ₀|, |ψ−1|)], E₀[Ψ] = E₀[(|ψ₁|, |ψ₀|, |ψ₋₁|)], E_Zee[Ψ] = E_Zee[(|ψ₁|, |ψ₀|, |ψ−1|)].

We thus obtain

E[Ψ] ≥ E [(|ψ₁|, |ψ₀|, |ψ−1|)],

where (remember that we have assumed p = 0) E[u] :=

Z  X

j

|∇u_j|²+ V (x)|u|² + β₀|u|⁴

+ β₁h

2u²₀(u₁− sgn(β₁)u−1)²+ (u²₁− u²₋₁)²i

+ q(u²₁+ u²₋₁)



for u = (u₁, u₀, u−1) ∈ B+. Also, note that the conservations of N and M are actually constraints on |ψj| and have nothing to do with the phases. These observations lead us to replace the original variational problem by the following one:

min

u∈AE[u], (2.3)

where the admissible class

A = {u ∈ B+ | N [u] = 1, M[u] = M } .

The validity of using this reduced model is provided by Corollary 8.5, Corollary 8.6 and Corollary 8.8, of which we can say the last one is the only not totally trivial asser- tion. We give careful examinations of them for the sake of being completely rigorous.

For later discussions, one can indeed just forget the original model and focus on (2.3).

We introduce some more notations to conclude this section. Define

E_g = min

u∈AE[u], and

G = {u ∈ A | E [u] = E^g} .

Thus Eg is the ground-state energy, and G is the set of minimizers of (2.3), which are exactly the objects to study. Since many assertions and discussions in this thesis involve different values of M and q, in later parts of this thesis we will write AM (the

admissible class has nothing to do with q), GM,q and Eg(M, q) to specify their values explicitly.

Similar to E, we will use E_kin, E_pot, E₀, E₁, and E_Zee to denote the five parts of E . Moreover, we will use H(u) to denote the integrand of E [u], i.e.

E[u] = Z

H(u).

H_kin, Hpot, etc. are similarly defined for the corresponding parts.

2.2 Fundamental properties

In some aspects our three-component system can be regarded as a generalization of the one-component system studied in [14]. The fundamental properties about the one- component model hold and can be proved similarly for our model. (The uniqueness is however a remarkable exception. See Remark 2.2 below. Detailed discussions are given in §7.1.) We summarize them in the following.

Theorem 2.1. G 6= ∅. u ∈ G is at least of class C¹, and satisfies the Euler-Lagrange system















(µ + λ)u₁ = Lu₁+2β₁u²₀(u₁− sgn(β₁)u−1)+u₁(u²₁− u²₋₁) +qu₁ µu₀ = Lu₀+ 2β₁u₀(u₁− s(β₁)u−1)²

(µ − λ)u−1 = Lu−1+2β1u²₀(u−1− sgn(β1)u1)+u−1(u²₋₁− u²₁) +qu−1

(2.4a) (2.4b) (2.4c) in the sense of distribution, where L = −∆ + V + 2β0|u|², andλ and µ are the La- grange multipliers induced by the constraintsN [u] = 1 and M[u] = M respectively.

Moreover, for eachu_j, eitheru_j ≡ 0 or u_j > 0 on all of R³.

The existence result can be proved by the standard direct method in the calculus of variations, in which one tries to show that a minimizing sequence in A has a subse- quence which weakly converges to an element in G. The only difference from a typical situation is that here the system is on the whole space but not a bounded domain. As a result, we do not have compact embedding H¹ ,→ L² to guarantee that the weak

limit is still in A. Instead, we should use the assumption (A1), which implies that, in some sense, most part of u is really contained in bounded domains, on which compact embedding applies. A precise argument can be given almost the same as in Lemma A.2 of [14]. (See also [1, 6].) Nevertheless, besides the conclusion of existence, some observations from its proof will also be needed later. We give them in Proposition 2.2 below. For convenience we give the proof in Section 8.3. The most important point is that we actually have strong convergence but not only weak convergence for the ex- tracted subsequence of the minimizing sequence. This holds for our model since the norm of B is bounded by a constant multiple of the energy functional.

Proposition 2.2. Let {uⁿ} be a sequence in B+. Suppose N [uⁿ] → 1, M[uⁿ] → M,

and E[uⁿ] is uniformly bounded in n, then {uⁿ} has a subsequence {u^n(k)}^∞_k=1 con- verging weakly to someu^∞ ∈ A, which satisfies

E[u^∞] ≤ lim inf

k→∞ E[u^n(k)].

If we assume further thatE[uⁿ] → E_g, thenu^∞ ∈ G, and u^n(k) → u^∞in the norm of B.

The Euler-Lagrange system (2.4) is called a time-independent Gross-Pitaevskii sys- tem (GP system). We remark that (2.4) is indeed valid not only in the sense of distri- bution, but also when tested by elements in B. In fact, E, N and M are continuously (Fr´echet) differentiable as functionals from B into R, and (2.4), after multiplied by 2, is exactly

µN⁰[u] + λM⁰[u] = E⁰[u].

We omit the verification of this fact. Once (2.4) is obtained, that u ∈ G is continu- ously differentiable follows standard regularity theorem (see e.g. [13], 10.2). And the strict positivity of a nonvanishing component can be obtained by the strong maximum principle. We give the proof of this last assertion also in Section 8.3. We shall usually use this fact tacitly to avoid repeatedly referring to Theorem 2.1.

Corollary 2.3. Let u ∈ G. If 0 < M < 1, then u^j 6= 0 for j = 1, −1.

Proof. SinceR (u²₁ − u²₋₁) = M > 0, u₁ 6= 0, and hence u₁ > 0. To prove u−1 6= 0, assume otherwise, then (2.2c) gives u²₀u₁ = 0, and so u₀ = 0. Thus among the three components only u₁ 6= 0, which implies M = 1 from the constraint N [u] = 1, contradicting to our assumption.

The two-component ground state

Since we will investigate whether u₀ ≡ 0 for u ∈ G, it will be convenient to introduce the two-component admissible class

A^two = {u ∈ A | u⁰ ≡ 0} . Note that for u ∈ A^twothe constraints are equivalent to

Z

u²₁ = 1 + M

2 and

Z

u²₋₁ = 1 − M 2 .

Due to the following uniqueness result, there is no need to introduce the corresponding class of minimizers G^two.

Theorem 2.4. There exists exactly one element z = (z₁, 0, z−1) ∈ A^two which mini- mizes the energyE over A^two. Moreover,z is independent of the value of q ∈ [0, ∞).

Proof. The existence of z can be proved as for the general three-component case. On the other hand, the fact that z is independent of q follows the simple observation that E_Zeeequals the constant q over A^two, and hence plays no role in the minimization. We prove the uniqueness of z in the following.

Given u, v ∈ A^two. Let w ∈ B+be defined by w²_j = (u²_j + v_j²)/2 for j = 1, 0, −1, then w is also in A^two. Let D = (E [u] + E [v])/2 − E [w], then D = D_kin+ D_n+ D_s, where

D_kin = E_kin[u] + E_kin[v]

2 − E_kin[w] =

Z X

j=1,−1

 |∇u_j|²+ |∇v_j|²

2 − |∇w_j|²

 ,

which is nonnegative by the convexity inequality for gradients. Also,

D_n= E_n[u] + E_n[v]

2 − E_n[w] = β₀ 4

Z

|u|² − |v|²
2

≥ 0,

and

D_s= E_s[u] + E_s[v]

2 − E_s[w] = β₁ 4

Z

u²₁− u²₋₁− v₁²+ v₋₁²
2

≥ 0,

as are easily checked. Now assume, moreover, u and v both minimize E over A^two, then we must have Dkin = Dn = Ds = 0. Otherwise we get the contradiction E[w] < (E[u] + E[v])/2. From D_n = 0 and D_s = 0 we then conclude that u = v.

This proves the uniqueness of z.

Remark 2.1. Let u ∈ G. The assertion u0 ≡ 0 is obviously equivalent to u = z.

We will show in Section 3.2 that z ∈ G when q = 0. As a corollary, the assertions in Theorem 2.1 for elements in G also apply to z.

Remark 2.2. The convexity argument used to prove the uniqueness of z is standard.

The idea however fails for general G, due to the term H1(u). Although the unique- ness will not be needed essentially, the lack of it still causes troubles in some of our presentations. See Section 7.1 for more discussions on the uniqueness problem.

2.3 Mass redistribution

Let f = (f₁, f₂, . . . , f_n) be an n-tuple of real-valued function in H¹(R^d), and let g =

|f |. The convexity inequality for gradients (Section 8.1) says

|∇g|² ≤X

k

|∇f_k|².

This fact has a simple while interesting generalization, when f₁², . . . , f_n² do not sum to a single g², but are distributed into multiple parts. To be precise, we give the following definition.

Definition 2.1. Let f be as above, and let g = (g1, g₂, . . . , g_m) be an m-tuple of non- negative functions. We say g is a mass redistribution of f if

g²₁ = a₁₁f₁²+ a₁₂f₂²+ · · · + a_1nf_n² g²₂ = a₂₁f₁²+ a₂₂f₂²+ · · · + a_2nf_n²

...

g_m² = a_m1f₁²+ a_m2f₂²+ · · · + a_mnf_n²,

where a<sub>`k</sub>(` = 1, . . . , m; k = 1, . . . , n) are nonnegative constants satisfying

m

X

`=1

a_`k = 1 for each k = 1, . . . , n.

That is, the coefficients of every column sum to 1.

Note that g = |f | is the only mass redistribution of f for m = 1. For general m, we have the following result.

Proposition 2.5. Let g be a mass redistribution of f as in Definition 2.1, then we have (1) |g| = |f |, and

(2) Pm

`=1|∇g<sub>`</sub>|² ≤Pn

k=1|∇f_k|².

Proof. The first assertion follows directly from the definition of mass redistribution.

For the second assertion, apply the convexity inequality for gradients to g_` =

(√

a_`1f₁)²+ (√

a_`2f₂)²+ · · · + (√

a_`nf_n)²1/2

, and we get

|∇g`|<sup>2</sup> ≤ a`1|∇f1|²+ a`2|∇f2|<sup>2</sup>+ · · · + a`na`n|∇fn|<sup>2</sup>. The assertion is then obtained by summing over ` = 1, 2, . . . , m.

Remark 2.3. It should be clear why we use the word “redistribution”. On the other hand, we will consider mass redistributions of u ∈ A. The adjective “mass” is added since the square of u_j represents the mass density of the j-th component. Indeed, we might as well use the term “square redistribution”. For convenience, however, we shall later only say redistribution.

Let’s write AM and GM,q here. Redistribution provides a simple and concrete way to variate an element in AM into another element, in the same space or in another AM⁰. Indeed, if v is a redistribution of some u ∈ AM, then (1) of Proposition 2.5 implies N [v] = 1, and one needs only to take care of the value of M[v]. Also, it’s easy to compare E [v] with E [u]. Precisely, again from (1) we have

E_pot[v] = E_pot[u] and E₀[v] = E₀[u], (2.5)

and from (2) we have

E_kin[v] ≤ E_kin[u]. (2.6)

As will be seen, these features make it easy to deduce some facts by using redistribu- tion, which might otherwise be harder to obtain or need more elaboration.

The true merit of redistribution (in my opinion and for our purpose) exhibits in Chapter 5, when we use it to obtain simple inequalities satisfied by ground states. To be precise, let u ∈ GM,q for some M, q. Then for any redistribution v of u in the same class AM, the fact E [u] ≤ E[v] together with (2.5) imply

E_kin[u] + E₁[u] + E_Zee[u] ≤ E_kin[v] + E₁[v] + E_Zee[v].

And (2.6) further implies

E₁[u] + E_Zee[u] ≤ E₁[v] + E_Zee[v]. (2.7)

Inequality (2.7) is particularly simple in that it involves only algebraic expressions of u (v is practically also expressed in terms of u). This inequality, with suitable constructions of v, will be sufficient for our proof of the bifurcation phenomenon.

The “best” way to gain sharper inequalities from redistribution will be the topic of Section 6

Chapter 3

The Simplified Characterizations

In this chapter we assume q = 0, i.e. no external magnetic field. Thus E = E_kin+ E_pot+ E₀+ E₁ and H = H_kin+ H_pot+ H₀+ H₁.

In Section 3.1, we prove the SMA. And in Section 3.2, we consider the phenomenon u₀ ≡ 0 for u ∈ G. A direct consequence of these results is that we can characterize elements in G (and hence ground states) by systems with fewer (one or two) compo- nents.

It’s interesting that, though these two phenomena look quite different, they can be proved in essentially the same way. To explain the idea, let P denote the property (SMA or u0 ≡ 0) to be justified. We will prove that, for every u ∈ A, there corresponds a redistributioneu of u which also lies in A, such that

(a) u has the property P, ande (b) E [u] ≤ E [u].e

From (b), we have E [u] = E [u] provided u ∈ G, from which we shall prove u ise exactlyu, and hence u has the property P.e

3.1 The single-mode approximation

In this section we assume β1 < 0. Let

A1 = {u ∈ A | u = (γ1f, γ₀f, γ−1f ) for some constants γ_j and some function f } .

The goal is to prove G ⊂ A1.

Now given any u ∈ A. It’s easy to see that a redistribution of u in A1 can be expressed as γ|u|, where γ = (γ₁, γ₀, γ−1) is any triple of nonnegative constants sat- isfying







γ₁²+ γ₀²+ γ₋₁² = 1 γ₁²− γ₋₁² = M.

(3.1)

Let Γ denote the set containing all such γ:

Γ := (γ₁, γ₀, γ−1) ∈ R³

 γ_j ≥ 0 for each j, γ satisfies (3.1) . (3.2) Then for each γ ∈ Γ, since γ|u| is a redistribution of u, we have

Hpot(γ|u|) ≡ Hpot(u) and H0(γ|u|) ≡ H0(u). (3.3)

Also,

H_kin(γ|u|) = |∇|u||² ≤X

j

|∇u_j|² = H_kin(u). (3.4)

On the other hand,

H₁(γ|u|) = β₁P (γ)|u|⁴, where

P (γ) = 2γ₀²(γ₁ + γ−1)²+ M².

Since β₁ < 0, to make E[γ|u|] ≤ E[u] as possible as we can, we compute the max- imum of P (γ) for γ ∈ Γ. It’s easy to check that there is a unique γ^? ∈ Γ such that

max

γ∈Γ P (γ) = P (γ^?) = 1.

Indeed the maximizer γ^? = (γ₁^?, γ₀^?, γ₋₁^? ) is given by

γ₁^? = 1

2(1 + M ) , γ₀^? = r1

2(1 − M²), γ₋₁^? = 1

2(1 − M ) . We can now state our main theorem in this section.

Theorem 3.1. Assume q = 0 and β1 < 0. If u ∈ G, then u = γ^?|u|.

Proof. By direct calculation we have H₁(u) − H₁(γ^?|u|) = −β₁n

|u|⁴ −2u²₀(u₁+ u−1)²+ (u²₁− u²₋₁)²o

= −β₁(u²₀− 2u₁u−1)² ≥ 0.

(3.5)

By (3.3), (3.4) and (3.5), we have H(u) ≥ H(γ^?|u|) for every u ∈ A. And hence u ∈ G implies E[u] = E[γ^?|u|], and the equality holds if and only if the inequalities in (3.4) and (3.5) are equalities. That is

X

j

|∇u_j|²− |∇|u||² = 0, (3.6)

and

u²₀− 2u1u−1 = 0. (3.7)

From (8.1), the equality (3.6) holds iff

u_j∇u_k− u_k∇u_j = 0 for j 6= k. (3.8) Since u is not identically zero (by the assumption N [u] = 1), at least one component of u is strictly positive everywhere. Assume u₁ > 0. Then from (3.8) we have

∇ u₀ u₁



= ∇ u₋₁ u₁



= 0,

which implies u0 and u−1 are both constant multiples of u1. This shows u ∈ A1. That u must be γ^?|u| then follows either by (3.7) or by the fact that γ^? is the unique maximizer of P over Γ. The case u₀ > 0 or u−1 > 0 can be proved similarly.

Remark 3.1. Since |u| is bounded away from zero, we can also conclude from (3.6) and Corollary 8.3 that u ∈ A1.

The above theorem implies that searching for ground states of a ferromagnetic spin-1 BEC can be reduced to an one-component minimization problem. Precisely, let

A^s = {|u| | u ∈ A} =f ∈ H¹∩ L⁴∩ L²_V 

f ≥ 0 and R f² = 1 , (3.9)

and define E^s : A^s→ R by E^s[f ] =

Z n

|∇f |²+ V f²+ (β₀+ β₁)f⁴o .

Then E [γ^?f ] = E^s[f ] for f ∈ A^s. Also let

G^s=n

f ∈ A^s  

E^s[f ] = min

g∈A^sE^s[g]o .

Then if u ∈ G, by Theorem 3.1 we have for every f ∈ A^s E^s[|u|] = E [γ^?|u|] ≤ E[γ^?f ] = E^s[f ].

Thus |u| ∈ G^s. Conversely if f ∈ G^s, then for every u ∈ A we have E[γ^?f ] = E^s[f ] ≤ E^s[|u|] = E [γ^?|u|] ≤ E[u].

Hence γ^?f ∈ G. We thus obtain the following one-component characterization of G.

Corollary 3.2. G = {γ^?f | f ∈ G^s}.

3.2 The vanishing of u

We assume β₁ > 0 in this section. Recall the definition of A^twoin Section 2.2 We want to show that G ⊂ A^two. Now, similarly, for every u ∈ A we want to find an appropriate redistribution u = (e ue₁, 0,eu−1) ∈ A^two so that E [u] ≤ E [u]. This time, however, thee assumption eu ∈ A^two alone doesn’t give us an obvious candidate of eu. In view that such eu satisfies |eu| = |u| and hence N [eu] = 1, as a guess, we try just imposing the additional assumption thatu also satisfiese

ue²₁ −ue²₋₁ = u²₁− u²₋₁,

to make M[u] = M automatically. This results in only one possibility, that ise

ue_j = r

u²_j +u²₀

2 for j = 1, −1. (3.10)

It’s fortunate that it works, and we obtain our main theorem of this section as follows.

Theorem 3.3. Assume β1 > 0 and M > 0, then u ∈ G implies u0 ≡ 0.

Proof. For u ∈ A, defineu ∈ Ae ^two by (3.10). Again sinceeu is a redistribution of u, we have

H_pot(eu) = H_pot(eu) and H₀(u) = He ₀(u),e

and

Hkin(u) ≤ He kin(u).

Also obviously

H₁(u) − H₁(eu) = 2β₁u²₀(u₁− u−1)² ≥ 0.

Thus for u ∈ G we have E[u] = E[eu], and

u²₀(u₁− u−1)² ≡ 0.

From this equality, we have either u0 ≡ 0 or u1 ≡ u−1. However, since we assume M > 0, we cannot have u₁ ≡ u₋₁, and hence u0 ≡ 0.

Remark 3.2. From Theorem 3.3 and Theorem 2.4, z is then the unique element in G when 0 < M ≤ 1 and q = 0. From Theorem 3.4 below, z is also an element in G when M = q = 0, but is not the unique one.

3.3 Some degenerate situations

The requirement M > 0 in Theorem 3.3 is necessary. In fact, for M = 0, ground states are not unique, and u0 ≡ 0 corresponds to only one possible state. Moreover, the SMA is again valid. Precisely, consider the following variational problem:

f ∈Amin^s Z n

|∇f |²+ V f²+ β₀f⁴o

, (3.11)

where A^sis defined by (3.9). We have the following characterization.

Theorem 3.4. Assume β1 > 0 and M = 0, then

G = n

t,√

1 − 2t², t
f 

 0 ≤ t ≤ 1/√

2, f is a minimizer of (3.11)o .

Proof. Since M = 0, γ ∈ Γ (defined by (3.2)) implies

γ = t,√

1 − 2t², t

for some t ∈0, 1/√ 2.

Now it’s easy to see that for any u ∈ A and γ ∈ Γ we have

H(γ|u|) = |∇|u||²+ V |u|²+ β₀|u|⁴,

which is independent of γ. Obviously, H(γ|u|) ≤ H(u). It remains to show that u ∈ G (and hence E[γ|u|] = E[u]) implies u = γ|u| for some γ ∈ Γ. The proof is almost the same as before and we omit it.

In contrast to the above result, the following corollary of Theorem 3.3 shows that SMA is almost never the case when M > 0.

Corollary 3.5. Assume β₁ > 0 and 0 < M < 1, then u ∈ G ∩ A1impliesu₁ andu−1

are constants. Moreover, suchu exists only if V is a constant.

Proof. By Theorem 3.3, the Euler-Lagrange system (2.4) is reduced to the following two-component system:







(µ + λ)u₁ = Lu₁+ 2β₁u₁(u²₁ − u²₋₁) (µ − λ)u−1 = Lu−1+ 2β₁u−1(u²₋₁− u²₁),

(3.12)

where L = −∆ + V + 2β₀(u²₁+ u²₋₁).

Since 0 < M < 1, for j = 1, −1, uj > 0. So u ∈ A¹implies u−1 = κu1 for some constant 0 < κ < 1. The system (3.12) then gives the following two equations for u1:

(µ + λ)u₁ = −∆u₁+ V u₁+ 2β₀(1 + κ²)u³₁+ 2β₁(1 − κ²)u³₁, (A) (µ − λ)u1 = −∆u1+ V u1+ 2β0(1 + κ²)u³₁+ 2β1(κ²− 1)u³₁. (B)

Now

1

2((A) − (B)) =⇒ λu₁ = 2β₁(1 − κ²)u³₁. Since u₁ > 0, we get

u1 = s

λ 2β₁(1 − κ²) .

In particular u₁ and u−1 = κu₁are constants. Hence ∆u₁ = 0. Then, 1

2((A) + (B)) =⇒ µu₁ = V u₁ + 2β₀(1 + κ²)u³₁, from which we get

V = µ − 2β₀(1 + κ²)u²₁. And hence V is also a constant.

Chapter 4

Some Further Properties

Now that we have proved the two simplified characterizations in the situation without external magnetic field, in the remaining of this thesis (except Chapter 8) we shall, more or less, focus on the bifurcation phenomenon. For convenience, we will thus only consider β₁ > 0, despite the fact that some assertions hold also for β₁ < 0. Also, we will, of course, not always assume q = 0.

In this chapter, we use the notations AM, GM,q and E_g(M, q) to specify the values of M and q. We will establish some more results for elements in GM,q. Most of the results are directly relevant to the proof of the bifurcation phenomenon. Some of them however are just given for completeness or serving as illustrations of using the redistribution technique.

4.1 Continuity and monotonicity of E

(M, q)

In this section we prove that E_g(M, q) is continuous and increasing in each variable.

Since the two variables are of quite different natures, we treat them separately.

4.1.1 E _g as a function of M

In this subsection we fix a q ∈ [0, ∞) and consider E_g(·, q). The proof of continuity will rely on the monotonicity, and hence we prove the latter first. For this we need the following lemma.

Lemma 4.1. E is bounded on ∪_{0≤M ≤1}GM,q.

Proof. The assertion is equivalent to say that we can choose for every M ∈ [0, 1]

an f^M ∈ AM, such that E [f^M] is uniformly bounded in M . This is easy to do. For example, let f be any nonnegative function in H¹∩ L²_V ∩ L⁴ such thatR f² = 1. Then for each M ∈ [0, 1], let f^M = ((^1+M₂ )^1/2f, 0, (^1−M₂ )^1/2f ). We have f^M ∈ AM and

E[f^M] = Z n

|∇f |²+ V f²+ β₀f⁴+ β₁M²f⁴+ qo ,

which is bounded above by the finite number E [f¹].

Proposition 4.2. Eg(·, q) is strictly increasing on [0, 1].

Proof. Let u ∈ GM,q. We first consider 0 < M ≤ 1. For small δ ≥ 0, let u(δ) be the redistribution of u defined by















u₁(δ)² = (1 − δ)u²₁

u₀(δ)² = u²₀+ δu²₁+ δu²₋₁ u−1(δ)² = (1 − δ)u²₋₁.

Then u(δ) ∈ A(1−2δ)M. Since u(δ) is a redistribution of u, E_kin[u(δ)] ≤ E_kin[u]. One can also check by direct computation that

E_Zee[u] − E_Zee[u(δ)] = qδ Z

(u²₁+ u²₋₁) ≥ 0,

and

E₁[u] − E₁[u(δ)] = β₁δ Z

(u₁− u−1)²2u²₀+ 4u₁u−1+ δ(u₁− u−1)² ≥ 0. (4.1)

Moreover, if δ > 0, strict inequality holds in (4.1). To see this, for 0 < M < 1, note that u1u−1 > 0 (Corollary 2.3) and that (u1 − u−1)² can not be identically zero (otherwise M = 0). While for M = 1, only u1 > 0, and the positivity of (4.1) is obvious. Thus we obtain

E_g((1 − 2δ)M, q) ≤ E [u(δ)] < E [u] = E_g(M, q)

for each small δ > 0, which shows E_g(·, q) is strictly increasing on (0, 1].

It remains to show that Eg(·, q) is strictly increasing at 0. Let {M_n} be a sequence in (0, 1) such that Mn→ 0⁺. And let uⁿ ∈ GMn,qfor each n. By Lemma 4.1, E [uⁿ] is uniformly bounded, and hence Lemma 2.2 implies there is a subsequence {u^n(k)} of {uⁿ} such that u^n(k) * u^∞weakly in B for some u^∞ ∈ A0. Moreover,

E_g(0, q) ≤ E [u^∞] ≤ lim inf

k→∞ E[u^n(k)] = lim inf

k→∞ E_g(M_n(k), q) = inf

0<M ≤1E_g(M, q).

The last equality is due to the just proved monotonicity of Eg(·, q) on (0, 1]. Thus Eg(0, q) ≤ Eg(M, q) for every M ∈ (0, 1]. To see why strict inequality must hold, as- sume Eg(0, q) = E_g(M, q) for some M > 0. Then since E_g(·, q) is strictly increasing on (0, 1], we have Eg(M/2, q) < E_g(0, q), a contradiction.

Proposition 4.3. E_g(·, q) is continuous on [0, 1].

Proof. The ideas of proving the left continuity and the right continuity are different.

We first prove the right continuity. Let u ∈ GM,q for some 0 ≤ M < 1. For small δ ≥ 0, let u(δ) be the redistribution of u defined by















u1(δ)² = u²₁+ δu²₀+ δu²₋₁ u₀(δ)² = (1 − δ)u²₀

u−1(δ)² = (1 − δ)u²₋₁.

Let’s use M_δ to denote M[u(δ)]. Then M_δ = M + δR (u²₀+ 2u²₋₁). Since M < 1, u₀ and u−1 cannot both vanish, and hence Mδ > M for δ > 0. Obviously Mδ → M⁺as δ → 0⁺. Now since Eg(·, q) is strictly increasing, we have

0 < E_g(M_δ, q) − E_g(M, q) (4.2)

for δ > 0. On the other hand, since u ∈ GM,q while u(δ) need not lie in GMδ,q, we have E_g(M_δ, q) − E_g(M, q) ≤ E [u(δ)] − E [u]. Thus

E_g(M_δ, q) − E_g(M, q) ≤ (E₁[u(δ)] − E₁[u]) + (E_Zee[u(δ)] − E_Zee[u]) (4.3)

from (2.5) and (2.6). It’s easy to check that the right-hand side of (4.3) tends to zero as δ → 0⁺, and hence we obtain

lim sup

δ→0⁺

(E_g(M_δ, q) − E_g(M, q)) ≤ 0.

This together with (4.2) imply the right continuity of E_g(·, q) on [0, 1).

For the left-continuity on (0, 1], we prove by contradiction. Let M ∈ (0, 1]. As- sume there is a sequence {Mn} in (0, 1) such that Mn→ M⁻, and Eg(Mn, q) doesn’t converge to Eg(M, q). By choosing a suitable subsequence, we can assume without loss of generality that the sequence {Mn} itself satisfies

E_g(M, q) − E_g(M_n, q) > ε for each n, for some ε > 0.

Now for each n choose one uⁿ ∈ GMn,q. Lemma 2.2 implies that there is a subse- quence {u^n(k)}^∞_k=1 such that u^n(k) → u^∞for some u^∞∈ AM. Moreover, we have

Eg(M, q) ≤ E [u^∞] ≤ lim inf

k→∞ E[u^n(k)] = lim inf

k→∞ Eg(M_n(k), q) ≤ Eg(M, q) − ε, a contradiction.

Proposition 4.3 implies the following approximation result.

Corollary 4.4. For any M ∈ [0, 1], we can find a sequence uⁿ ∈ G^Mn,q such that M_n ∈ [0, 1], M_n→ M , M_n6= M for each n, and uⁿ → u^∞in the norm of B for some u^∞∈ GM,q.

Proof. Let {M_n} be a sequence in [0, 1] such that M_n → M and M_n 6= M for each n.

Then let uⁿ∈ GMn,q for each n. By definition we have N [uⁿ] = 1 and M[uⁿ] → M . Since E [uⁿ] = E_g(M_n, q), by continuity of E_g(·, q) we also have E[uⁿ] → E_g(M, q).

Thus by Lemma 2.2, {uⁿ} has a subsequence {u^n(k)}^∞_k=1 such that u^n(k) → u^∞ strongly in B for some u^∞ ∈ G^M,q. The sequence u^n(k) ∈ G^Mn(k),q thus satisfies the assertion to be proved.

Remark 4.1. Suppose we have uniqueness of element in G^M,q, and let u^M be the unique element in GM,q. Then Corollary 4.4 simply says the map M 7→ u^M is contin- uous from [0, 1] into B. In particular, let’s here write z^Mfor z to specify the dependence on M explicitly. Then we have the corollary that M 7→ z^M is continuous from [0, 1]

into B. (We’ll use this fact in §5.2.2.) To be rigorous, this is true since z^M is the unique element in GM,0for 0 < M ≤ 1, and as M → 0⁺, the limit of z^M in B, which should lie in G0,0by Corollary 4.4, must be z⁰. Of course, we might as well just prove the analogue of Corollary 4.4 for the “two-component world”, and the continuity of M 7→ z^M follows Theorem 2.4 directly.

4.1.2 E _g as a function of q

Now we consider the function Eg(M, ·) for fixed M ∈ [0, 1]. For u ∈ B+, let’s here write E [u, q] instead of E [u] to indicate the value of q. The proofs of monotonicity and continuity of Eg(M, ·) are much easier than those of Eg(·, q) above, and the proof of continuity doesn’t rely on the monotonicity. We put the assertions in a single proposi- tion.

Proposition 4.5. For fixed M ∈ [0, 1], E_g(M, ·) is an increasing and continuous func- tion on[0, ∞). Moreover, it’s strictly increasing if M > 0.

Proof. Let q₁ > q₂ ≥ 0 and u ∈ GM,q1. We have

E_g(M, q₁) − E_g(M, q₂) ≥ E [u, q₁] − E [u, q₂]

= (q₁− q₂) Z

(u²₁+ u²₋₁) ≥ 0,

(4.4)

which implies E_g(M, ·) is an increasing function on [0, ∞). If M > 0, we have u₁ > 0, and hence the last inequality in (4.4) is strict, which proves the strict monotonicity .

We next prove the continuity. Given any q1, q₂ ≥ 0, let u^k= (u^k₁, u^k₀, u^k₋₁) ∈ GM,qk

for k = 1, 2. Since E [u¹, q₁] = E_g(M, q₁) and E[u¹, q₂] ≥ E_g(M, q₂), we have (q₁− q₂)

Z

(u¹₁)²+ (u¹₋₁)²
= E[u¹, q₁] − E [u¹, q₂]

≤ E_g(M, q₁) − E_g(M, q₂).

(4.5)

Similarly,

E_g(M, q₁) − E_g(M, q₂) ≤ E [u², q₁] − E [u², q₂]

= (q₁− q₂) Z

(u²₁)²+ (u²₋₁)²
.

(4.6)

From (4.5) and (4.6), and the factR

(u^k₁)²+ (u^k₋₁)²
≤ N [u^k] = 1 for k = 1, 2, we find

|E_g(M, q₁) − E_g(M, q₂)| ≤ |q₁− q₂|, and hence E_g(M, ·) is continuous.

Remark 4.2. E_g(0, ·) is not strictly increasing. Indeed, by Proposition (4.7) below, for q > 0, u ∈ G0,q satisfies u1 = u₋₁ = 0. Such one-component ground state, as the two-component z, is unique and independent of q. This is easily obtained by imitating the proof of Theorem 2.4). Thus Eg(0, ·) is a constant function on (0, ∞), and hence on [0, ∞) by continuity.

With the continuity of E_g(M, ·), we can show the following analogue of Corollary 4.4. The proof is the same as that of Corollary 4.4 by changing the roles of M and q, and hence we omit it.

Corollary 4.6. For any q ∈ [0, ∞), there is a sequence uⁿ ∈ GM,qn such that q_n ∈ [0, ∞), q_n → q, q_n 6= q for each n, and uⁿ → u^∞ in the norm of B for some u^∞ ∈ G^M,q.

4.2 u

is no larger than u

The goal in this section is indicated by the title. The relevant assertions are Proposition 4.7 (and Remark 4.3 following it), Proposition 4.8, and Proposition 4.11.

Proposition 4.7. Suppose q > 0 and u ∈ G0,q(i.e.M = 0). We have u₁ = u−1 = 0.

Proof. Let v = (v₁, v₀, v−1) be the element in A0defined by







v²₁ = v²₋₁ = (u²₁+ u²₋₁)/2 v²₀ = u²₀.

Then E [u] − E[v] = (Ekin[u] − E_kin[v]) + E₁[u]. Since v is a redistribution of u, E_kin[u] − E_kin[v] ≥ 0. Also, E₁[u] ≥ 0, and hence E[u] − E[v] ≥ 0. Nevertheless, u ∈ G0,q, so we must have E [u] − E[v] = 0. Thus actually E_kin[u] − E_kin[v] = E₁[u] = 0.

In particular the term (u²₁ − u²₋₁)² in H₁(u) is zero, which implies u₁ = u−1. To see why they must vanish, note that now we have

E[u] = Z 

X

j

|∇u_j|²+ V (x)|u|²+ β₀|u|⁴+ q(u²₁+ u²₋₁)



≥ Z n

|∇|u||²+ V (x)|u|²+ β₀|u|⁴o

= E [(0, |u|, 0)].

(4.7)

Again since u ∈ G0,q and (0, |u|, 0) ∈ A0, we must have E [u] = E [(0, |u|, 0)]. Thus the inequality in (4.7) is equality, which implies u = (0, |u|, 0) since P

j|∇uj|² ≥

|∇|u||²and q > 0.

Remark 4.3. From Theorem 3.4, for M = q = 0, we also have u1 = u−1, while u₁ = u₋₁ = 0 corresponds to only one possibility. This together with Proposition 4.7 provide satisfactory descriptions of the degenerate situation M = 0. Also, on the other extreme M = 1, only u₁ > 0. Therefore, there is no need to consider the bifurcation phenomenon for M = 0, 1.

Proposition 4.8. For every 0 ≤ M ≤ 1 and q ≥ 0, u ∈ GM,q satisfiesu−1 ≤ u₁. Proof. Let v be defined by v1 = max(u₁, u₋₁), v₋₁ = min(u₁, u₋₁), and v₀ = u₀. Then we have E [v] = E [u]. To check this equality, for the kinetic part E_kinone can use the formula

v_j = 1

2(u_j + u−j + j|u_j− u−j|) for j = 1, −1. Then direct computation gives

|∇v₁|²+ |∇v−1|² = 1 2

n|∇u₁|²+ |∇u−1|²+ 2∇u₁· ∇u−1+ 2

∇|u₁− u−1| 

2o . And |∇v₁|²+ |∇v−1|² = |∇u₁|²+ |∇u−1|²is obtained by applying the fact

|∇|f ||² = |∇f |² a.e. for every f of class H¹.

The equalities of the other parts are obvious. Thus, we have E_g(M[v], q) ≤ E [v] = E [u] = E_g(M, q).

Since E_g(·, q) is strictly increasing, we thus obtain

M[v] ≤ M. (4.8)

On the other hand, it’s also obvious by definition that

v₁²− v₋₁² ≥ u²₁− u²₋₁. (4.9) (4.8) and (4.9) imply v₁²− v²₋₁ = u²₁− u²₋₁, that is v²₁− u²₁ = v₋₁² − u²₋₁, of which the left-hand side is nonnegative while the right-hand side is nonpositive by definition of v. Thus we really have v1 = u1 and v−1 = u−1, which means u−1 ≤ u1.

Proposition 4.8 can be used to improve itself. Precisely, we shall prove that strict inequality u−1 < u₁ holds when M > 0, by using the strong maximum principle. In doing so, the knowledge of the non-strict inequality itself is needed.

Lemma 4.9. Let f ∈ L¹(R³, R³) be such that the distributional divergence ∇ · f ∈ L¹(R³). ThenR ∇ · f = 0.

Proof. For R > 0, let ϕR: R³ → R be defined by

ϕR(x) =







1, |x| < R

R + 1 − |x|, R ≤ |x| < R + 1

0, R + 1 ≤ |x|.

Then it’s obvious that

lim

R→∞

Z

(∇ · f ) ϕ_R = Z

∇ · f .

On the other hand, Z

(∇ · f ) ϕ_R= − Z

R≤|x|<R+1

f (x) · n(x),

where n(x) = x/|x|. ThusR (∇ · f) ϕ_R → 0 as R → ∞, which proves the assertion.

Corollary 4.10. Let u ∈ G^M,q. If 0 < M < 1, the Lagrange multiplier λ in the GP system (2.4) is positive.

Proof. (2.4a) multiplied by u−1 minus (2.4c) multiplied by u₁ gives

2λu₁u−1 = ∇ · (−u−1∇u₁ + u₁∇u−1) + 2β₁(u²₁− u²₋₁)(u²₀+ 2u₁u−1).

By Lemma 4.9,R ∇ · (−u−1∇u1+ u1∇u−1) = 0, and hence λ

Z

u₁u−1 = β₁ Z

(u²₁− u²₋₁)(u²₀+ 2u₁u−1). (4.10) Now u₁u−1 > 0 by Corollary 2.3, and hence R u₁u−1 > 0. On the other hand, by Proposition 4.8 we have u²₁− u²₋₁ ≥ 0, which cannot be identically zero since M > 0.

Thus we also haveR (u²₁− u²₋₁)(u²₀+ 2u1u−1) > 0, and (4.10) implies λ > 0.

Proposition 4.11. For 0 < M ≤ 1 and q ≥ 0, u ∈ GM,q satisfiesu−1 < u₁.

Proof. If M = 1, we have u₁ > 0 ≡ u−1. For 0 < M < 1, let w = u₁− u−1. Then (2.4a) minus (2.4c) gives

∆w + Qw = −λ(u₁+ u−1) − µw, (4.11) where

Q = −V − 2β0|u|²− 2β12u²₀+ (u1+ u−1)² − q.

Since λ > 0 and w ≥ 0, by subtracting |µ|w from both sides of (4.11), we obtain

∆w + eQw ≤ 0, where eQ = Q − |µ| is locally bounded. By Corollary 8.11, either w > 0 everywhere or w ≡ 0. But w ≡ 0 means u₁ = u−1, contradicting to the assumption M > 0. Thus w > 0, which is what we want to show.

Remark 4.4. The subtraction of |µ|w in the proof above is indeed not necessary since we also have µ > 0 for 0 < M < 1. This is easy to obtain by using (2.4b) when u₀ > 0, and by using (2.4c) when u₀ ≡ 0. We omit the details.

Recall the definition of z from Theorem 2.4. Since z ∈ GM,0(for any 0 ≤ M ≤ 1), we have the following corollary.

Corollary 4.12. For 0 < M ≤ 1, z−1 < z₁.

Remark 4.5. Although z is independent of q, it’s dependent on M . To be precise we shall sometimes write z^M = (z₁^M, 0, z^M₋₁) to specify this dependence. For notational simplicity, we will however not do so when such explicitness is not really necessary.

4.3 Exponential decay of ground states

In this section we prove the exponential decay of ground states with the aid of Propo- sition 4.8, The approach of using the fundamental solution of Helmholtz equation is exactly taken from [14], Lemma A.5.

Proposition 4.13. Let u ∈ GM,q, for arbitrary0 ≤ M ≤ 1 and q ≥ 0. For any a > 0, there exist constantsU_j(a) (j = 1, 0, −1) such that u_j(x) ≤ U_j(a)e^−a|x|.

Proof. (2.4b) can be arranged as (−∆ + a²)u₀ = Q₀u₀, where

Q0 = a²+ µ − V − 2β0|u|²− 2β1(u1− u−1)². (4.12) Thus

u₀(x) = (Y_a∗ (Q₀u₀))(x) = Z

Y_a(x − y)Q₀(y)u₀(y)dy,

where Ya(x) = e^−a|x|/(4π|x|) is the fundamental solution of the operator −∆ + a². (Y_a is also referred to as the Yukawa potential. See [13], 6.23.) By the assumption (A1), Q0 < 0 outside a bounded set, say B(R0), the open ball centered at the origin with radius R0. Thus we obtain

u₀(x) ≤ Z

|y|<R₀

Y_a(x − y)Q₀(y)u₀(y)dy = e^−a|x|

Z

|y|<R₀

ea(|x|−|x−y|)

4π|x − y| Q₀(y)u₀(y)dy.

Thus u0(x) ≤ U₀(a)e^−a|x|, where (see also Lemma 4.14 below) U₀(a) = sup

x∈R³

Z

|y|<R0

ea(|x|−|x−y|)

4π|x − y| Q₀(y)u₀(y)dy < ∞. (4.13) For uj, j = 1, −1, we similarly have

(−∆ + a²)u_j = Q_ju_j − 2β₁u²₀(u_j − u−j)

from (2.4a) and (2.4c), where

Q_j = a²+ µ + jλ − V − 2β₀|u|²− 2β₁(u²_j − u²_−j) − q.

Now since u−1 ≤ u₁, Q₁is also negative outside B(R₁) for some radius R₁, and

u₁(x) = Z

Y_a(x − y)Q₁(y)u₁(y) − 2β₁u₀(y)²(u₁(y) − u−1(y))dy

≤ Z

Y_a(x − y)Q₁(y)u₁(y)dy

≤ Z

|y|<R1

Y_a(x − y)Q₁(y)u₁(y)dy.

As above we conclude that u₁(x) ≤ U₁(a)e^−a|x|, where U₁(a) is given by (4.13) with all the indices 0 replaced by 1. In contrast, the fact u−1 ≤ u1makes it difficult to apply the same argument to u−1. Nevertheless, also since u−1 ≤ u₁, at least we can choose U−1(a) = U₁(a).

For our next result, we give the following estimate of Uj(a).

Lemma 4.14. For j = 1 and 0,

U_j(a) ≤ e^aRj 4π sup

x∈R³

Z

|y|<Rj

Q_j(y)²

|x − y|²dy

!1/2

.

Proof. Since |x| − |x − y| ≤ |y|, we have for j = 1, 0 Z

|y|<Rj

ea(|x|−|x−y|)

4π|x − y| Q_j(y)u_j(y)dy ≤ Z

|y|<Rj

e^aRj

4π|x − y|Q_j(y)u_j(y)dy

= e^aRj 4π

Z

|y|<R_j

Q_j(y)

|x − y|uj(y)dy

≤ e^aRj 4π

Z

|y|<Rj

Q_j(y)²

|x − y|²dy

!1/2

,

where the last inequality is obtained by H¨older’s inequality and the fact Z

(uj)² ≤ Z

|u|² = 1.

We thus obtain the assertion of the lemma.