On phase segregation in nonlocal two-particle Hartree systems
aa r X i v : . [ m a t h . A P ] S e p ON PHASE SEGREGATION IN NONLOCALTWO-PARTICLE HARTREE SYSTEMS
WALTER H. ASCHBACHER AND MARCO SQUASSINA
Abstract.
We prove the phase segregation phenomenon to occur in the groundstate solutions of an interacting system of two self-coupled repulsive Hartree equa-tions for large nonlinear and nonlocal interactions. A self-consistent numerical in-vestigation visualizes the approach to this segregated regime. Introduction
In this paper, we study the phase segregation phenomenon in the ground states ofthe eigenvalue system consisting of two interacting repulsive Hartree equations whoseinteraction, as well as the respective self-couplings, are nonlinear and nonlocal,(1.1) − ∆ φ + V ( x ) φ + ϑ ( V ∗ | φ | ) φ + κ ( V ∗ | φ | ) φ = µ φ in R N , − ∆ φ + V ( x ) φ + ϑ ( V ∗ | φ | ) φ + κ ( V ∗ | φ | ) φ = µ φ in R N , k φ k L = N , k φ k L = N . Here, the external potentials V and V are assumed to be nonnegative and confining,whereas the interaction potential V is, for example, of Coulomb type. Moreover, thesystem is purely repulsive, i.e. the self-coupling constants and the interaction strengthare nonnegative, ϑ , ϑ ≥ , κ ≥ . For fixed ϑ , ϑ ≥ N , N >
0, the phase segregation phenomenon inthe ground state ( φ , φ ) with ground state energy ( µ , µ ) of the system (1.1) ischaracterized by the decay to zero of the Coulomb energy functional D ( φ , φ ) = Z R N Z R N | φ ( x ) | V ( x − y ) | φ ( y ) | d x d y (1.2) Mathematics Subject Classification.
Key words and phrases.
Coupled Hartree equations, Quantum many-body problem, Hartree ap-proximation, ground states solutions, phase segregation, finite elements, self-consistent iteration.Zentrum Mathematik, Technische Universit¨at M¨unchen, Boltzmannstrasse 3, 85747 Garching, Ger-many. E-mail: [email protected] .Department of Computer Science, University of Verona, C`a Vignal 2, Strada Le Grazie 15, 37134Verona, Italy. E-mail: [email protected] .The research of the second author was partially supported by the 2007 MIUR national researchproject:
Variational and Topological Methods in the Study of Nonlinear Phenomena . in the regime of large interaction strength κ , i.e. by D ( φ , φ ) = o (1) for κ → ∞ . This study is not only of independent mathematical interest but it can also be mo-tivated by various physical applications like, e.g., electromagnetic waves in a Kerrmedium in nonlinear optics, surface gravity waves in hydrodynamics, and groundstates in Bose-Einstein condensed bosonic quantum mechanical many-body systems(see also [1]). The latter domain has been a subject of great interest since many years,both on the experimental and the theoretical side, starting off from a series of suc-cessful experimental realizations of Bose-Einstein condensation for atomic gases, firstachieved in 1995 for a single condensate (see e.g. [2]), then, in 1997, for a mixture oftwo interacting atomic species with equal masses (see e.g. [12]), and, finally, in 2003 fortriplet species states (see e.g. [16]). On the theoretical side, the standard scenario oftwo interacting Bose-Einstein condensates for a very dilute system of repulsive bosonsuses the description based on a system of two coupled Gross-Pitaevskii equations (seee.g. [9, 10, 13, 14]). These equations are formally contained in (1.1) for the case of thezero range interaction potential V = δ , i.e. in the case of local nonlinearities. For acomplete survey paper, we also refer the reader to [7] and references therein. One mayargue that, for higher density regimes, it is sensible to capture more of the boson-bosoninteraction by allowing for its nonlocal and, hence, less coarse grained resolution byuse of a potential V = δ (see e.g. [3]). The phase segregation phenomenon has beenstudied e.g. in [13, 15, 17] for Gross-Pitaevskii equations, and it has been given a gen-eral variational framework in [6]. Recently, the second author, jointly with M. Caliari,has investigated both numerically and analytically the behavior of ground state solu-tions highlighting their location and the occurring phase segregation phenomena inthe highly interacting regime (see [5]). In the present paper, we extend the analysis to the nonlocal system (1.1) and give aproof of the phase segregation phenomenon in the variational calculus setup. Moreover,in contradistinction to [5], we adopt a classical self-consistent numerical approach tothe solution of the ground state of (1.1) in order to compute the phase segregatedstates and to monitor the decay of the Coulomb interaction (1.2).As we aim at keeping the paper self-contained and easily readable also for thosereaders who are more acquainted with the physical or the numerical side, we willprovide rather detailed mathematical arguments throughout the paper.2.
Strong interaction and phase segregation
Throughout this section we shall denote by C a generic positive constant which canvary from line to line inside the proofs. For the case of a single condensate, the stationary and dynamical Gross-Pitavskii equation hasbeen rigorously derived from the many-body bosonic Schr¨odinger equation in the weak coupling limit,see e.g. [11] and [8], respectively. And, in some particular cases, also excited state solutions. With respect to the off-centering of the confining potentials V i . For a complete numerical study of ground states for vector like nonlinear Schr¨odinger systemswith cubic coupling, we also refer to [4].
N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 3
Functional setting.
As described in the introduction, we are interested in thecase of nonnegative confining external potentials V and V . More precisely, we makethe following assumption. Assumption 2.1.
The external potentials V i are nonnegative, continuous, and con-fining, i.e. for i = 1 ,
2, we have V i ∈ C ( R N , R +0 ) withlim | x |→∞ V i ( x ) = ∞ . The functional setting we want to apply makes use of the following Hilbert space.
Definition 2.2.
Let the external potentials V i satisfy Assumption 2.1 and let H be theHilbert subspace of H ( R N ) × H ( R N ) defined by H = n ( φ , φ ) ∈ H ( R N ) × H ( R N ) : Z R N V i ( x ) | φ i ( x ) | d x < ∞ , i = 1 , o , (2.1) where the scalar product of φ = ( φ , φ ) ∈ H with ψ = ( ψ , ψ ) ∈ H is given by h φ, ψ i H = X i =1 (cid:16) Z R N ∇ φ i ( x ) · ∇ ψ i ( x ) d x + Z R N V i ( x ) φ i ( x ) ψ i ( x ) d x (cid:17) . (2.2)This functional setting is the natural framework for the study of bound states of sys-tems (1.1) in external potentials as it allows (together with Lemma 2.5) the associatedenergy functional (see (2.9)) to be well-defined and finite. Lemma 2.3.
Under Assumption 2.1, for any ≤ N ≤ , the embedding H ֒ → L NN +2 ( R N ) ⊕ L NN +2 ( R N ) is compact, H being the Hilbert space (2.1) equipped with the norm (2.2) .Proof. Let ( φ h , φ h ) be a bounded sequence in H , say k ( φ h , φ h ) k H ≤ C for all h ∈ N .Up to a subsequence, it converges weakly in H to some ( φ , φ ) ∈ H . Moreover, bythe Rellich-Kondrachov compactness theorem, up to a further subsequence, ( φ h , φ h )converges strongly to ( φ , φ ) in L ( B R ) ⊕ L ( B R ) for any R >
0, where B R denotesthe open ball in R N of radius R , centered at the origin. Let now M >
R > V i ( x ) ≥ M for all x ∈ R N \ B R and any i = 1 ,
2. Hence, we can write Z R N | φ hi ( x ) − φ i ( x ) | d x = Z B R | φ hi ( x ) − φ i ( x ) | d x + Z R N \ B R | φ hi ( x ) − φ i ( x ) | d x ≤ Z B R | φ hi ( x ) − φ i ( x ) | d x + 1 M Z R N \ B R V i ( x ) | φ hi ( x ) − φ i ( x ) | d x ≤ Z B R | φ hi ( x ) − φ i ( x ) | d x + 4 C M .
Let now ε > M > C /M < ε/
2. Then, as thecorresponding radius R > h ≥ R B R | φ hi ( x ) − φ i ( x ) | d x < W.H. ASCHBACHER AND M. SQUASSINA ε/ h ≥ h . This yields k φ hi − φ i k L → h → ∞ . Moreover, by theGagliardo-Nirenberg inequality and the boundedness in H , we have k φ hi − φ i k L NN +2 ≤ C k φ hi − φ i k − NL k φ hi − φ i k N − H ≤ C k φ hi − φ i k − NL , from which it follows that lim h →∞ k φ hi − φ i k L NN +2 = 0 . This completes the proof. (cid:3)
The interaction between the components φ and φ is described by the followingCoulomb energy functional which is well-known from classical Hartree theory. Definition 2.4.
The Coulomb energy functional D : H ( R N ) × H ( R N ) → R is defined by D ( φ , φ ) = Z R N Z R N | φ ( x ) | V ( x − y ) | φ ( y ) | d x d y, (2.3) where the interaction potential V is the Coulomb potential in R N for N ≥ , V ( x ) = 1 | x | N − . (2.4)Due to the following lemma, for any 3 ≤ N ≤
6, the Coulomb energy functionalwith potential (2.4) is well-defined.
Lemma 2.5.
Let ≤ N ≤ and let φ i ∈ H ( R N ) with k φ i k L = N i > for i = 1 , .Then, there exists a constant C such that D ( φ , φ ) ≤ C ( N N ) − N k φ k N − H k φ k N − H . Proof.
Due to Schwarz’ inequality, we have D ( φ , φ ) ≤ D ( φ , φ ) D ( φ , φ ) . (2.5)Hence, by the Hardy-Littlewood-Sobolev inequality (for N ≥
3) and the Gagliardo-Nirenberg inequality (for 2 ≤ N ≤ D ( φ i , φ i ) ≤ C k φ i k L NN +2 ≤ C k φ i k − NL k φ i k N − H = CN − N i k φ i k N − H , which yields the assertion. (cid:3) Remark 2.6. If N ≥ V λ ( x ) = 1 | x | λ for some 0 < λ < min { , N } ,(2.6)we can again estimate the energy functional D λ ( φ , φ ) := Z R N Z R N | φ ( x ) | V λ ( x − y ) | φ ( y ) | d x d y (2.7) Also called direct term in the Hartree (-Fock) theory.
N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 5 by virtue of the Hardy-Littlewood-Sobolev inequality (for any 0 < λ < N ) and theGagliardo-Nirenberg inequality (for any 0 < λ ≤
4) as D λ ( φ , φ ) ≤ C k φ k L N N − λ k φ k L N N − λ ≤ C ( N N ) − λ k φ k λ/ H k φ k λ/ H . In particular, if N = 2 and λ = 1, we have D ( φ , φ ) ≤ C ( N N ) / k φ k / H k φ k / H . If λ = N − N ≥ D N − = D and one recovers the estimate of the previousLemma 2.5.2.2. Existence of a minimizer.
Let us consider the following two component Hartreeeigenvalue system in R N for 3 ≤ N ≤ V from (2.4) and N , N > − ∆ φ + V ( x ) φ + ϑ ( V ∗ | φ | ) φ + κ ( V ∗ | φ | ) φ = µ φ , − ∆ φ + V ( x ) φ + ϑ ( V ∗ | φ | ) φ + κ ( V ∗ | φ | ) φ = µ φ , k φ k L = N , k φ k L = N . Since we are interested in the phase segregation phenomenon in the case of a purelyrepulsive Hartree system, we make the following assumption.
Assumption 2.7.
The self-coupling constants ϑ , ϑ and the interaction strength κ are nonnegative, ϑ , ϑ ≥ , κ ≥ . Remark 2.8.
In the case of coupled Bose-Einstein condensates discussed in the in-troduction (where the nonlinearities are local, i.e. V = δ ), the self-coupling constants ϑ , ϑ as well as the interaction strength κ are explicitly related to the scattering lengthsand the masses of the atomic species in the condensates (see e.g. [9]).In order to study the nonlinear ground states of the Hartree system (2.8), we makeuse of the following energy functional. Definition 2.9.
The Hartree energy functional E κ : H → [0 , ∞ ) defined by (2.9) E κ ( φ , φ ) = E ∞ ( φ , φ ) + κ D ( φ , φ ) , where the decoupled energy functional E ∞ : H → [0 , ∞ ) consists of the sum of the twosingle particle energies E i : H → [0 , ∞ ) , E ∞ ( φ , φ ) = X i =1 E i ( φ i ) , (2.10) E i ( φ i ) = Z R N |∇ φ i ( x ) | d x + Z R N V i ( x ) | φ i ( x ) | d x + ϑ i D ( φ i , φ i ) . (2.11) From here on, since ϑ , ϑ ≥ N , N > κ only. W.H. ASCHBACHER AND M. SQUASSINA
In view of Lemma 2.5, the functional E κ is well-defined for 3 ≤ N ≤
6. Moreover, itis readily seen that E κ is a C smooth functional and that its critical points constrainedto the set { ( φ , φ ) ∈ H : k φ i k L = N i for i = 1 , } are weak solutions of (2.8). Remark 2.10.
The case κ = 0 corresponds to a noninteracting Hartree system (2.8)consisting of two independent Hartree equations, each describing a repulsive singleparticle self-coupling. Definition 2.11.
The ground state energy E κ ≥ of the Hartree functional (2.9) atinteraction strength κ ∈ [0 , ∞ ) is defined by (2.12) E κ = inf ( φ ,φ ) ∈S E κ ( φ , φ ) , where the infimum is taken over the set S = { ( φ , φ ) ∈ H : k φ i k L = N i for i = 1 , } . (2.13) Moreover, the segregated ground state energy E ∞ ≥ is defined by E ∞ = inf ( φ ,φ ) ∈S ∞ E ∞ ( φ , φ ) , where now the infimum is taken over the set S ∞ = { ( φ , φ ) ∈ S : D ( φ , φ ) = 0 } . Let us now prove that the Hartree functional (2.9) admits a real and positive mini-mizer for any positive interaction strength κ . Proposition 2.12.
Let κ ∈ (0 , ∞ ) and ≤ N ≤ . Then, there exists a positiveminimizer ( φ κ , φ κ ) ∈ S of the Hartree functional (2.9) with ground state energy E κ given in (2.12) .Proof. In order to prove the assertion, we make use of the direct method in the calculusof variations. Hence, we verify the three standard assumptions implying the existenceof a minimizer. First, since by Lemma 2.3, the normed space H from (2.1) with thenorm from (2.2) is compactly embedded in L ( R N ) ⊕ L ( R N ), it follows that the set S from (2.13) is weakly closed in H . Second, since k ( φ , φ ) k H ≤ E κ ( φ , φ )(2.14) = k ( φ , φ ) k H + X i =1 ϑ i D ( φ i , φ i ) + κ D ( φ , φ ) , the set { ( φ , φ ) ∈ S : E κ ( φ , φ ) ≤ a } is a bounded nonempty subset of S for anypositive number a . Third, we have to show that the functional E κ is weakly lowersemicontinuous on S . For this purpose, consider a sequence of elements ( φ h , φ h ) ∈ S which converges for h → ∞ weakly in H to some ( φ , φ ) ∈ S . Since, for any i, j = 1 , | φ hi ( x ) | | φ hj ( y ) | | x − y | N − → | φ i ( x ) | | φ j ( y ) | | x − y | N − for a.e. ( x, y ) ∈ R N , Fatou’s Lemma implies(2.15) D ( φ , φ ) ≤ lim inf h →∞ D ( φ h , φ h ) , D ( φ i , φ i ) ≤ lim inf h →∞ D ( φ hi , φ hi ) . N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 7
Therefore, due to (2.14), the fact that the norm k · k H from (2.2) on H is weakly lowersemicontinuous on S , and (2.15), the Hartree functional E κ is indeed weakly lowersemicontinuous on S . Hence, the three assumptions are verified and the existence ofa minimizer is proven. Moreover, due to the convexity inequality for gradients, Z R N |∇| φ i | ( x ) | d x ≤ Z R N |∇ φ i ( x ) | d x, the Hartree functional E κ satisfies the following inequality for any ( φ , φ ) ∈ S , E κ ( | φ | , | φ | ) ≤ E κ ( φ , φ ) . Consequently, with no loss of generality, we can assume that any minimizer of E κ ispositive. (cid:3) Remark 2.13.
Note that, for 3 ≤ N ≤
5, the Coulomb energy functional D is notonly weakly lower semicontinuous as given in (2.15), but even weakly continuous over H , i.e. for any sequence of elements ( φ h , φ h ) ∈ S which converges for h → ∞ weaklyin H to some ( φ , φ ) ∈ S , we have(2.16) lim h →∞ D ( φ h , φ h ) = D ( φ , φ ) , lim h →∞ D ( φ hi , φ hi ) = D ( φ i , φ i ) . In order to prove this claim, we make use of Lemma 2.3, which states that the embed-ding H ֒ → L NN +2 ( R N ) ⊕ L NN +2 ( R N ) is compact. Hence, up to a subsequence, it followsthat, for i = 1 , h →∞ k φ hi − φ i k L NN +2 = 0 . Using (2.17), we want to show that D ( φ hi , φ hi ) → D ( φ i , φ i ) as h → ∞ . To this end, weuse that the Coulomb potential V from (2.4) is even and write | D ( φ hi , φ hi ) − D ( φ i , φ i ) | ≤ D ( || φ hi | − | φ i | | / , ( | φ hi | + | φ i | ) / ) . By inequality (2.5), the Hardy-Littlewood-Sobolev inequality, and H¨older’s inequality,it follows that there exist a constant C with | D ( φ hi , φ hi ) − D ( φ i , φ i ) | ≤ C k || φ hi | − | φ i | | / k L NN +2 k ( | φ hi | + | φ i | ) / k L NN +2 ≤ C k φ hi − φ i k L NN +2 . This implies, via (2.17), the desired convergence of D ( φ hi , φ hi ) to D ( φ i , φ i ). Hence,it follows that all the terms in E κ containing the Coulomb energy functional D areweakly continuous on S . Notice that, as a consequence of (2.16), the weak lowersemicontinuity of E κ over S also holds in the case of attractive self-coupling or attractiveinteraction, i.e. for ϑ , ϑ ≤ κ ≤
0. In fact, this case amounts to the replacementof some (or all) D terms (with positive coupling) in the Hartree functional by − D . The convergence D ( φ h , φ h ) → D ( φ , φ ) as h → ∞ can be proved in a similar fashion. W.H. ASCHBACHER AND M. SQUASSINA
Phase segregation.
As pointed out in the introduction, we are interested in thesituation where the values of the self-coupling constants ϑ , ϑ (and N , N ) remainfixed whereas the interaction strength κ becomes very large. Definition 2.14.
A sequence of minimizers ( φ κ , φ κ ) ∈ S of the Hartree energy func-tional E κ from (2.9) is said to be phase segregating if D ( φ κ , φ κ ) = o (1) for κ → ∞ . Remark 2.15.
If the phase segregating sequence ( φ κ , φ κ ) is convergent in H , then thelimiting configuration ( φ ∞ , φ ∞ ) satisfies D ( φ ∞ , φ ∞ ) = 0.Let us now state our main assertion. Theorem 2.16.
Let ≤ N ≤ and let D be the Coulomb energy functional from (2.3) . Then, for κ ∈ (0 , ∞ ) , any sequence of minimizers ( φ κ , φ κ ) ∈ S of the Hartreeenergy functional E κ from (2.9) is phase segregating for κ → ∞ , and D ( φ κ , φ κ ) = o ( κ − ) . In addition, such a sequence converges in the H norm to a minimizer ( φ ∞ , φ ∞ ) ∈ S ∞ of the decoupled functional E ∞ from (2.10) and (2.11) . Corollary 2.17.
Under the assumptions of Theorem 2.16, the limiting configurationsatisfies the following set of uncoupled variational inequalities, (2.18) − ∆ φ ∞ i + V i ( x ) φ ∞ i + ϑ i (cid:0) V ∗ | φ ∞ i | (cid:1) φ ∞ i ≤ µ ∞ i φ ∞ i , where N i µ ∞ i = E i ( φ ∞ i ) + ϑ i D ( φ ∞ i , φ ∞ i ) and i = 1 , . Remark 2.18.
Although we stated Theorem 2.16 for minimizers of the Hartree func-tional in R N with 3 ≤ N ≤ D , it also holdsfor minimizers of the Hartree functional in R N with 0 < λ < min { , N } containinginstead D λ from (2.7). This corresponds to the system(2.19) − ∆ φ + V ( x ) φ + ϑ ( V λ ∗ | φ | ) φ + κ ( V λ ∗ | φ | ) φ = µ φ , − ∆ φ + V ( x ) φ + ϑ ( V λ ∗ | φ | ) φ + κ ( V λ ∗ | φ | ) φ = µ φ , k φ k L = N , k φ k L = N , where V λ stems from (2.6). In particular, in view of the numerical setup, the case N = 2 and λ = 1 is covered. Proof.
Consider a sequence of minimizers ( φ κ , φ κ ) ∈ S for κ → ∞ whose existenceis assured by Proposition 2.12. Note first that, in the light of Definition 2.11, thesequence of corresponding ground state energies ( E κ ) is uniformly bounded because E κ = inf ( φ ,φ ) ∈ S E κ ( φ , φ ) ≤ inf ( φ ,φ ) ∈ S ∞ E κ ( φ , φ ) = inf ( φ ,φ ) ∈ S ∞ E ∞ ( φ , φ ) = E ∞ . (2.20) N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 9
In particular, due to (2.14) and the definition of a minimizer, the sequence ( φ κ , φ κ ) isuniformly bounded in H with respect to κ , k ( φ κ , φ κ ) k H ≤ E κ ( φ κ , φ κ ) = E κ ≤ E ∞ . Hence, since H is weakly sequentially compact, there exists a pair ( φ ∞ , φ ∞ ) ∈ H and asubsequence of ( φ κ , φ κ ), again denoted by ( φ κ , φ κ ) which, for κ → ∞ , converges weaklyin H to ( φ ∞ , φ ∞ ). Next, we want to show that ( φ ∞ , φ ∞ ) ∈ S ∞ . Since ( φ κ , φ κ ) ∈ S and the embedding H ֒ → L ( R N ) ⊕ L ( R N ) is compact, we have, for i = 1 , k φ ∞ i k L = N i . Hence, ( φ ∞ , φ ∞ ) ∈ S . Moreover, again due (2.14) and (2.20), we have κ D ( φ κ , φ κ ) ≤ E κ ( φ κ , φ κ ) ≤ E ∞ , (2.21)which implies that the sequence ( φ κ , φ κ ) is phase segregating for κ → ∞ , D ( φ κ , φ κ ) = O ( κ − ) . Also, since we know that the Coulomb energy D is weakly continuous on S , we get D ( φ ∞ , φ ∞ ) = 0 , and, therefore, ( φ ∞ , φ ∞ ) ∈ S ∞ . In order to prove that ( φ ∞ , φ ∞ ) is a minimizer of E ∞ and that ( φ κ , φ κ ) converges strongly in H to ( φ ∞ , φ ∞ ), we next show that D ( φ κ , φ κ ) = o ( κ − ). To this end, consider the sequence κ D ( φ κ , φ κ ) which is bounded due to (2.21),and pick a convergent subsequence, denoted by κ n D ( φ κ n , φ κ n ). Then, using that( φ ∞ , φ ∞ ) ∈ S ∞ , the weak lower semicontinuity of the decoupled energy functional E ∞ on S , and (2.20), we get E ∞ ( φ ∞ , φ ∞ ) + lim n →∞ κ n D ( φ κ n , φ κ n ) ≤ lim inf n →∞ E κ n ( φ κ n , φ κ n )(2.22) ≤ E ∞ ≤ E ∞ ( φ ∞ , φ ∞ ) , with the consequence that κ n D ( φ κ n , φ κ n ) = 0 as n → ∞ . Therefore, since this holdsfor all convergent subsequences of κ D ( φ κ , φ κ ), we arrive at D ( φ κ , φ κ ) = o ( κ − ) . (2.23)This implies, on one hand, that ( φ κ , φ κ ) converges strongly in H to ( φ ∞ , φ ∞ ) sincefrom E k ≤ E ∞ ( φ ∞ , φ ∞ ), (2.23), and the weak continuity of D ( φ κi , φ κi ), we getlim sup κ →∞ k ( φ κ , φ κ ) k H ≤ k ( φ ∞ , φ ∞ ) k H . On the other hand, using (2.22) and (2.23), we see that ( φ ∞ , φ ∞ ) is a minimizer of E ∞ , that is E ∞ = E ∞ ( φ ∞ , φ ∞ ). Finally, we note that, again due to (2.22), we have E κ → E ∞ as κ → ∞ . This brings the proof of Theorem 2.16 to an end. (cid:3) Finally, we prove the assertion of Corollary 2.17.
Proof.
Observe that, by virtue of(2.24) µ κi = 1 N i n E i ( φ κi ) + ϑ i D ( φ κi , φ κi ) + κ D ( φ κ , φ κ ) o , the inequality (2.21) and E i ( φ κi ) ≤ E ∞ , we getsup κ ≥ µ κi < ∞ , where µ κi denotes the nonlinear eigenvalue of the minimizer φ κi as the weak nonlinearground state in the corresponding nonlinear eigenvalue system (2.8). Then, up to asubsequence, µ κi → µ ∞ i as κ → ∞ . Testing the equations of (2.8) with arbitrarynonnegative functions η of compact support, we get, recalling that φ i ≥ Z R N ∇ φ κi ( x ) · ∇ η ( x ) d x + Z R N V i ( x ) φ κi ( x ) η ( x ) d x + ϑ i Z R N Z R N φ κi ( y ) φ κi ( x ) η ( x ) | x − y | N − d x d y ≤ µ κi Z R N φ κi ( x ) η ( x ) d x. Hence, letting κ → ∞ , it turns out that φ ∞ i satisfies the variational inequality (2.18).Finally, the strong convergence and (2.24) yields, for i = 1 , N i µ ∞ i = E i ( φ ∞ i ) + ϑ i D ( φ ∞ i , φ ∞ i ) . This ends the proof of Corollary 2.17. (cid:3) Numerical approach
Galerkin approximation of the nonlinear eigenvalue system.
In order tocarry out the numerical simulation, we treat the Hartree system in the plane from(2.19) with N = 2 and λ = 1 in the framework of the following finite element approx-imation. As physical subdomain of R , we choose the open squareΩ = (0 , D ) × (3.1)with D > m − congruent closed subsquaresgenerated by dividing each side of Ω equidistantly into m − M = ( m − the total number of interior vertices of this lattice and by h = D/ ( m − Moreover, let us choose the Galerkin space S h to be spanned bythe bilinear Lagrange rectangle finite elements ϕ j ∈ C (Ω). Hence, with this choice, we have S h ⊂ C (Ω) ∩ H (Ω) As bijection from the one-dimensional to the two-dimensional lattice numbering, we may use themapping τ : { , ..., m − } × → { , ..., m − } with j = τ ( m , m ) := m + m m . The reference basis function ϕ : Ω → [0 , ∞ ) is defined on its support [0 , h ] × by ϕ ( x, y ) := 1 h xy, if ( x, y ) ∈ [0 , h ] × , (2 h − x ) y, if ( x, y ) ∈ [ h, h ] × [0 , h ] , (2 h − x )(2 h − y ) , if ( x, y ) ∈ [ h, h ] × ,x (2 h − y ) , if ( x, y ) ∈ [0 , h ] × [ h, h ] , (3.2)see Figure 1. The functions ϕ j are then defined to be of the form (3.2) having their support translatedby ( m h, m h ) with m , m = 0 , ..., m − N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 11
Figure 1. ϕ ( x, y ) on its support [0 , h ] × with maximum at vertex ( h, h ).and dim S h = ( m − . The Hartree system (2.19) in its weak finite element approxi-mation form reads, for all ϕ ∈ S h , (3.3) ( ∇ ϕ, ∇ φ ) + ( ϕ, V φ ) + ( ϕ, ( V ∗ [ ϑ | φ | + κ | φ | ]) φ ) = µ , ( ϕ, φ ) , ( ∇ ϕ, ∇ φ ) + ( ϕ, V φ ) + ( ϕ, ( V ∗ [ ϑ | φ | + κ | φ | ]) φ ) = µ , ( ϕ, φ ) , k φ k L = N , k φ k L = N . If we expand φ α ∈ S h with expansion coefficients z α = ( z α, , ..., z α,M ) ∈ C M w.r.t. thefinite element basis { ϕ j } Mj =1 of S h , φ α = M X j =1 z α,j ϕ j , (3.4)and if we plug this expansion together with ϕ = ϕ i , i = 1 , ..., M , into (3.3), we get thefollowing coupled matrix system on C M , (3.5) Q [ z , z ] z = µ , z ,Q [ z , z ] z = µ , z , | z | = N , | z | = N . Here, the matrix-valued mappings Q , Q : C M × C M → C M × M are defined by Q [ z , z ] := A − ( B + Y + ϑ G [ z ] + κ G [ z ]) ,Q [ z , z ] := A − ( B + Y + ϑ G [ z ] + κ G [ z ]) , In this section, ( · , · ) and k · k stand for the L (Ω)-scalar product and L (Ω)-norm, respectively.Moreover, the convolution on the finite domain Ω is defined, for any ( x, y ) ∈ Ω, by ( V ∗ φ )( x, y ) := R Ω V ( x − x ′ , y − y ′ ) φ ( x ′ , y ′ ) d x ′ d y ′ . | z | := h z, z i denotes the L -norm on C M , where h z, w i := P Mj =1 z j w j and h z, w i := h z, Aw i are the Euclidean and the L -scalar product on C M , respectively. For φ α from (3.4), we have k φ α k = h z α , Az α i = | z α | = N α for α = 1 , and A ∈ C M × M is the mass matrix, B ∈ C M × M the stiffness matrix, and Y α ∈ C M × M the matrices generated by the external potentials V α , A ij := ( ϕ i , ϕ j ) , B ij := ( ∇ ϕ i , ∇ ϕ j ) , ( Y α ) ij := ( ϕ i , V α ϕ j ) . (3.6)Moreover, the matrix-valued mapping G : C M → C M × M is defined on w = ( w , ..., w M ) ∈ C M by G [ w ] ij := ϕ i , g (cid:2) M X k =1 w k ϕ k (cid:3) ϕ j ! = M X k,l =1 w k w l V iklj , where the function g and the Hartree convolution term V iklj are defined by g [ φ ] := V ∗ | φ | , V iklj := ( ϕ i , V ∗ ( ϕ k ϕ l ) ϕ j ) . (3.7) Remark 3.1.
We avoid the inversion of the mass matrix A and simplify the evaluationof the double integral in the Hartree convolution term (3.7) by approximating theintegrals over Ω by the standard mass lumping quadrature procedure.In order to simplify the eigenvalue system (3.5) with the help of Remark 3.1, let usintroduce the mappings H , H : C M × C M → C M × M defined by H [ z , z ] := 1 h ( B + Y + ϑ diag( G [ z ]) + κ diag( G [ z ])) ,H [ z , z ] := 1 h ( B + Y + ϑ diag( G [ z ]) + κ diag( G [ z ])) , where diag : C M → C M × M is defined to be the matrix-valued mapping on w =( w , ..., w M ) ∈ C M defined by diag( w ) ij := δ ij w j for all i, j = 1 , ..., M , and G : C M → C M is defined by G [ w ] i := h M X j =1 | w j | V ( h ( τ − ( i ) − τ − ( j ))) , where τ is the grid numbering bijection from footnote 8. Hence, Remark 3.1 amountsto the replacement G [ z ] diag( G [ z ]) and we get approximated Hartree system(3.8) H [ z , z ] z = µ , z ,H [ z , z ] z = µ , z , | z | = N , | z | = N . Algorithms.
In order to solve the nonlinear coupled eigenvalue system (3.8), wemake use of the method of successive substitution whose fixed-point map is con-structed with the help of the power method used for the solution of the correspondinglinearized problem. In the following, we briefly describe the basic ideas of these algo-rithms. Also called nonlinear Richardson iteration or Picard iteration.
N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 13
Method of successive substitution (MSS)
Let M be the compact set M := { [ z , z ] ∈ C M × C M | | z | = N , | z | = N } . The MSS is an iterative method of the form[ z ( n +1)1 , z ( n +1)2 ] = F [ z ( n )1 , z ( n )2 ] , (3.9)where the fixed point map F : M → M is constructed as follows. Given an approx-imate nonlinear system ground state [ z ( n )1 , z ( n )2 ] ∈ M at iteration level n ∈ N , theapproximate nonlinear system ground state [ z ( n +1)1 , z ( n +1)2 ] ∈ M at iteration level n + 1is defined to be the linear system ground state of the linearized eigenvalue system(3.10) H [ z ( n )1 , z ( n )2 ] z ( n +1)1 = ǫ ( n +1)0 , z ( n +1)1 ,H [ z ( n )1 , z ( n )2 ] z ( n +1)2 = ǫ ( n +1)0 , z ( n +1)2 , | z ( n +1)1 | = N , | z ( n +1)2 | = N . Remark 3.2.
Here and in the following, we make the assumption that H α [ z ( n )1 , z ( n )2 ]has a unique linear ground state. E.g., using perturbation theory in the regime ofsmall nonlinear couplings, this holds as soon as the linear operator H α [0 ,
0] has anondegenerate ground state energy.
Remark 3.3.
We can write the fixed point map F α [ z ( n )1 , z ( n )2 ] from (3.9) with the helpof the linear ground state projection P α [ z , z ] = − π i I Γ α [ z ,z ] ( H α [ z , z ] − ζ ) − d ζ , where Γ α [ z , z ] is a path which encircles the linear ground state energy of H α [ z , z ]in the positive direction and no other point of the spectrum of H α [ z , z ]. The map F can now be written as F α [ z ( n )1 , z ( n )2 ] = p N α P α [ z ( n )1 , z ( n )2 ] z ( n ) α | P α [ z ( n )1 , z ( n )2 ] z ( n ) α | . (3.11)Since H α [ · , · ] is Lipschitz continuous on M , the map F has a not necessarily uniquefixed point due to Schauder’s fixed point theorem.The system (3.10) being not only linearized but also decoupled, we can solve thetwo linear eigenvalue problems separately. In order to approximately determine theground states of the linear eigenvalue problems, we make use of the power methodwhich works as follows. Power method (PM)
The PM computes the eigenvector of H α [ z ( n )1 , z ( n )2 ] whose eigenvalue has largest modu-lus amongst all the eigenvalues whose eigenvectors appear in the eigenvector expansion of the starting approximation. To access the ground state of H α [ z ( n )1 , z ( n )2 ], we applythe following spectral shift s ( n ) α := | H α [ z ( n )1 , z ( n )2 ] | + 1 . Moreover, we define the shifted operator by b H α [ z ( n )1 , z ( n )2 ] := H α [ z ( n )1 , z ( n )2 ] − s ( n ) α . Now, the p -th iterate of the PM iteration is defined by z ( n +1) ,pα := b H α [ z ( n )1 , z ( n )2 ] p z ( n ) α | b H α [ z ( n )1 , z ( n )2 ] p z ( n ) α | . (3.12) Remark 3.4.
Since b H α [ z ( n )1 , z ( n )2 ] is real symmetric, the spectral theorem implies theexistence of an orthonormal basis of C M of eigenvectors { w α,k } M − k =0 of b H α [ z ( n )1 , z ( n )2 ]. Moreover, since the spectral radius of b H α [ z ( n )1 , z ( n )2 ] is smaller than s ( n ) α , we have for alleigenvalues of the shifted operator ˆ ǫ α,k ∈ spec( H α [ z ( n )1 , z ( n )2 ]) − s ( n ) α that − s ( n ) α < ˆ ǫ α, < ˆ ǫ α, ≤ ... ≤ ˆ ǫ α,M − < . Let us expand z ( n ) α w.r.t. the orthonormal basis { w α,k } M − k =0 as z ( n ) α = M − X k =0 ξ α,k w α,k , use that b H α [ z ( n )1 , z ( n )2 ] w α,k = ˆ ǫ α,k w α,k , and divide the numerator and the denominatorin (3.12) by | ˆ ǫ α, | p . Moreover, let us assume that ξ α, = 0. Then, in the large p limit,( − p z ( n +1) ,pα converges to a multiple of the ground state of H α [ z ( n )1 , z ( n )2 ], z ( n +1) ,pα = ( − p ξ ,α (cid:12)(cid:12) ξ ,α (cid:12)(cid:12) w ,α + o (1) . (3.13) Remark 3.5.
Using formula (3.13), the fixed point map F from (3.9), (3.11) can alsobe written as F α [ z ( n )1 , z ( n )2 ] = p N α lim p →∞ ( − p z ( n +1) ,pα | z ( n +1) ,pα | . Stopping criteria
Both for the inner PM iteration and the outer MSS iteration, we use a relative errorstopping criterion in the numerical computation. For the PM iteration, let us definethe energy ˆ ǫ ( n +1) ,pα, := h z ( n +1) ,pα , b H α [ z ( n )1 , z ( n )2 ] z ( n +1) ,pα i . (3.14) For A = [ a ij ] ∈ C M × M , we define the ℓ - matrix norm by | A | := P Mi,j =1 | a ij | . | z | := h z, z i / denotes the Euclidean norm of z ∈ C M . We suppress the superscript n in the eigenvectors, eigenvalues, and in the expansion coefficients,since the PM iteration acts at a fixed n . Moreover, the numbering starts at 0 being the index of theground state. N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 15
Then, for suitably chosen accuracy tolerance δ PM >
0, we stop the PM iteration foreach component α = 1 , | ( b H α [ z ( n )1 , z ( n )2 ] − ˆ ǫ ( n +1) ,pα, ) z ( n +1) ,pα || ˆ ǫ ( n +1) ,pα, + s ( n ) α | ≤ δ PM . (3.15) Remark 3.6.
Note that the quotient (3.15) does not depend on the shift s ( n ) α , sincethe iterates z ( n +1) ,pα are normalized w.r.t. the Euclidean norm on C M . Remark 3.7.
Clearly, the stopping criterion (3.15) is satisfied for any eigenvector of b H α [ z ( n )1 , z ( n )2 ]. But as soon as ξ α, = 0, e.g. due to finite precision arithmetic, the PMiterate z ( n +1) ,pα converges to a multiple of the ground state w α, . But note that thechosen accuracy may be reached before a nonvanishing ξ α, is generated.For the MSS iteration, we implement a similar stopping criterion. To this end, wedefine the approximate nonlinear ground state energies as µ ( n +1) α, := 1 N α h z ( n +1) α , H α [ z ( n +1)1 , z ( n +1)2 ] z ( n +1) α i , where, compared to (3.14), the Hartree energy H α depends on iteration level n + 1instead of level n . We stop the MSS iteration as soon as | ( H α [ z ( n +1)1 , z ( n +1)2 ] − µ ( n +1)0 ,α ) z ( n +1) α | | µ ( n +1)0 ,α | ≤ δ MSS , where δ MSS > Phase segregation.
As it has been defined above in Definition 2.14, a sequenceof nonlinear ground state solutions ( φ κ , φ κ ) is phase segregating if its Coulomb energyvanishes in the limit of large interaction strength κ , i.e. D ( φ κ , φ κ ) = ( φ κ , ( V ∗ | φ κ | ) φ κ ) → κ → ∞ . (3.16)Plugging the expansions (3.4) into (3.16), we get D (cid:16) M X i =1 z ,i ϕ i , M X j =1 z ,j ϕ j (cid:17) = h z , G [ z ] z i . Hence, making use of Remark 3.1, we define the approximated Coulomb energy D : C M × C M → R by D [ z , z ] := h z , diag( G [ z ]) z i . (3.17)Below, we will use this approximation in the numerical computation of the Coulombenergy. Figures
The numerical computations leading to the following figures visualize the qualita-tive picture of the approach to the segregated regime. First, we exhibit the densitiesof the wave functions φ and φ for increasing values of the interaction strength κ approaching the segregated regime. Second, we report on the decay of the Coulombenergy (3.17).We choose the external potentials V α for α = 1 , V α ( x, y ) = c α (cid:0) ( x − a α ) + ( y − b α ) (cid:1) , (4.1)and the interaction potential V to be a regularized Yukawa potential, V ( x, y ) = e − Γ √ x + y p x + y + γ . (4.2) Remark 4.1.
The potential (4.2) being the regularized three-dimensional Yukawapotential, it may be argued that we consider a physically three-dimensional systemconstrained to a two-dimensional submanifold of the three-dimensional configurationspace.The specification of the parameters used in the simulations below is summarized inthe following table (cf. (3.1), (3.3), (4.1), and (4.2)). N N a b c a b c ϑ ϑ κ Γ γ D/ D/ D/ D/ − Remark 4.2.
All the qualitative features of the following simulations have been testedfor stability in different physical and numerical parameter ranges.4.1. κ = 0 . For the interaction strength κ = 0, the system is uncoupled and linear,and we find the ground state wave functions of the harmonic oscillator. The supportsare fully overlapping, see Figure 2. κ = 0 . . The wave functions φ and φ start to feel their respective repulsion.The support of φ is retracting whereas the one of φ gets pushed outwards. Thesupports are still heavily overlapping, see Figure 3. The code is part of our
Hartree package written in C++. All the figures have been produced with the help of gnuplot . N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 17
Figure 2.
The wave function densities | z α | and their contours with α = 1 above and α = 2 below for the interaction strength κ = 0. Figure 3.
The interaction strength is κ = 0 . κ = 10 . In the regime of large interaction strength κ , the segregation phenomenonoccurs: the supports get more and more disjoint, see Figure 4. Figure 4.
The interaction strength is κ = 10. Remark 4.3.
Up to the shape of the support of φ i , there is no qualitative change inthe picture if the two harmonic potentials are slightly dislocated with respect to eachother.4.4. Coulomb energy.
Finally, we monitor the decay of the Coulomb energy fromformula (3.17), see Figure 5.
Figure 5.
The decay of D [ z κ , z κ ] as a function of κ . N PHASE SEGREGATION IN NONLOCAL TWO-PARTICLE HARTREE SYSTEMS 19
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