Persistence of translational symmetry in the BCS model with radial pair interaction
Andreas Deuchert, Alissa Geisinger, Christian Hainzl, Michael Loss
aa r X i v : . [ m a t h - ph ] M a y PERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCSMODEL WITH RADIAL PAIR INTERACTION
A. DEUCHERT, A. GEISINGER, C. HAINZL, AND M. LOSS
Abstract.
We consider the two-dimensional BCS functional with a radial pairinteraction. We show that the translational symmetry is not broken in a certaintemperature interval below the critical temperature. In the case of vanishingangular momentum our results carry over to the three-dimensional case. Introduction
In 1957 Bardeen, Cooper, and Schrieffer published their famous paper with the ti-tle "Theory of Superconductivity", which contained the first, generally accepted, mi-croscopic theory of superconductivity. In recognition of this work they were awardedthe Nobel prize in 1972. Originally introduced to describe the phase transition fromthe normal to the superconducting state in metals and alloys, BCS theory can alsobe applied to describe the phase transition to the superfluid state in cold fermionicgases. In this situation, one has to replace the usual non-local phonon-induced inter-action in the gap equation by a local pair potential. Apart from being a paradigmaticmodel in solid state physics and in the field of cold quantum gases, the BCS theoryof superconductivity, that is, the gap equation and the BCS functional show a richmathematical structure, which has been well recognized. See [22, 2, 23, 24, 21, 25] forworks on the gap equation with interaction kernels suitable to describe the physicsof conduction electrons in solids and [12, 6, 15, 16, 3, 11, 10] for works that treat thetranslation-invariant BCS functional with a local pair interaction. The gap equationand the BCS functional are related in the way that the former is the Euler-Lagrangeequation of the latter. One main question in the study of BCS theory is whether thegap equation ∆( p ) = − π ) d/ Z R d ˆ V ( p − q ) tanh ( E ( q ) / T ) E ( q ) ∆( q ) d q, (1.1)with E ( q ) = q ( q − µ ) + | ∆( q ) | has a non-trivial solution, that is, one with ∆ = 0 .If this is so, the system is said to be in a superconducting/superfluid state. The func-tion ∆ has the interpretation of a spectral gap of an effective mean-field Hamiltonianthat is present only in the superconducting/superfluid phase, see the Appendix in[12] for further explanations. In [12] it has been demonstrated that, although the gapequation is highly non-linear, the question whether there exists a non-trivial solutioncan be decided with the help of a linear criterion. To be more precise, it was shownthat the existence of a non-trivial solution of the gap equation is equivalent to thefact that a certain linear operator has a negative eigenvalue. Based on a character-ization of the critical temperature in terms of this linear operator, its behavior hasbeen investigated in the limit of small couplings and in the low-density limit, see[6, 16] and [14], respectively. Recently, there has also been considerable interest inthe BCS functional with external fields, and in particular, in its connection to theGinzburg-Landau theory of superconductivity, see [17, 4, 18, 7, 9, 5, 8, 20]. The gap equation in the form stated in Eq. (1.1) and the related BCS functionalcan be heuristically derived from Quantum Mechanics by a variational procedureunder several simplifying assumptions, see [12] and the discussion in Section 2 be-low. One of these assumptions is that states used in this variational procedure aretranslation-invariant which leads to a strong simplification of the model. Whilethis approximation is presumably valid in the case of cold fermionic gases with arotationally-invariant pair interaction and is of great importance when it comes tonumerical computations, it is in general hard to justify its validity. See [1] for exam-ples in the context of solid state physics where this approximation is not valid. Froma mathematical point of view one is faced with a functional that is invariant undertranslations in the sense that spatial translations do not change the energy of a state.Due to the non-linear nature of the functional, minimizers need not be translation-invariant, however. If they are not one says that the translational symmetry of thesystem is broken. The aim of this work is to prove the absence of translational sym-metry breaking in two situations: We start by considering the two-dimensional BCSfunctional with a radial pair interaction and show that there exists a certain temper-ature interval below the critical temperature, in which the translational symmetry ofthe system persists. Afterwards, we realize that our analysis directly carries over tothe three-dimensional case if the Cooper-pairs are in an s-wave state. Prior to thiswork, such a result was known only in the case of ˆ V ≤ and not identically zero, see[19]. 2. Main Results
We consider a sample of fermionic atoms in a cold gas in d -dimensional space( d = 2 , ) within the framework of BCS theory. It is convenient to think of thesample as infinite and periodic, since this setting avoids having to deal with boundaryconditions at the boundary of the sample. To describe the periodicity we introducethe lattice Z d with the unit cell [0 , d = Ω . The special form of the lattice does notplay any role for us and the proof carries over to an arbitrary Bravais lattice. Tonot artificially complicate the presentation, we therefore opt for the simplest choice.BCS states are most conveniently described by their generalized one-particle densitymatrix, that is, by a self-adjoint operator Γ on L ( R d ) ⊕ L ( R d ) of the form Γ = (cid:18) γ αα − γ (cid:19) , (2.1)with ≤ Γ ≤ . Here γ and α denote the one-particle density matrix and theCooper-pair wave function of the state Γ , respectively. Both of them are representedby periodic operators with period one. In terms of kernels, the latter means that γ ( x + u, y + u ) = γ ( x, y ) and α ( x + u, y + u ) = α ( x, y ) for all u ∈ Z d and all x, y ∈ R d . In (2.1), α = CαC , where C denotes complex conjugation. Note that, inparticular, α ( x, y ) = α ( y, x ) for all x, y ∈ R d , due to the self-adjointness of Γ . In thissetting, it is natural to consider energies per unit volume. Accordingly, we definefor a periodic operator A , the trace per unit volume Tr Ω by Tr Ω [ A ] = Tr [ χ Ω Aχ Ω ] ,where χ Ω denotes the characteristic function of Ω . We call Γ of the form (2.1) an admissible BCS state if Tr Ω ( −∇ + 1) γ < ∞ and denote the set of admissible BCSstates by D . We will, by a slight abuse of notation, write ( γ, α ) ∈ D , meaning thatthe BCS state Γ given by (2.1) is admissible. ERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCS MODEL 3
The BCS functional at temperature T ≥ , with chemical potential µ ∈ R , inter-action potential V ∈ L ( R d ) and entropy S (Γ) = −
12 Tr Ω [Γ log Γ + (1 − Γ) log (1 − Γ)] , is then given by F (Γ) = Tr Ω (cid:2)(cid:0) −∇ − µ (cid:1) γ (cid:3) + Z Ω × R d V ( x − y ) | α ( x, y ) | d( x, y ) − T S (Γ) . (2.2)Note that the same functional has been considered in [7], where the periodicity was in-troduced for ease of comparison with the translation-invariant functional. As alreadymentioned above, the BCS functional can be heuristically derived from Quantum Me-chanics by a variational procedure. To that end, one considers the full free energyfunctional of the system and restricts attention to quasi-free states only. Due to theWick rule, the energy and the entropy can then be expressed solely in terms of thegeneralized one-particle density matrix of the quasi-free state under consideration,see [1]. If one assumes additionally SU (2) -invariance as well as the above periodicityof the state and neglects the direct and the exchange term in the energy, one arrivesat Eq. (2.2). For more details see the Appendix of [12].The translation-invariant BCS functional F ti is obtained from F by restrictingthe set of admissible states to the translation-invariant ones. That is, the kernelsof γ and α take the form γ ( x, y ) = γ ( x − y ) and α ( x, y ) = α ( x − y ) , respectively.We describe translation-invariant BCS states via their momentum representationsby × matrices of the form ˆΓ( p ) = (cid:18) ˆ γ ( p ) ˆ α ( p )ˆ α ( p ) 1 − ˆ γ ( − p ) (cid:19) , (2.3)for p ∈ R d , where the bar denotes complex conjugation and the hats indicate thatthose objects are Fourier transforms of integral kernels that depend only on x − y .Obviously, ˆΓ( p ) satisfies ≤ ˆΓ( p ) ≤ for all p ∈ R d . The latter translates to | ˆ α ( p ) | ≤ ˆ γ ( p )(1 − ˆ γ ( p )) for p ∈ R d in terms of ˆ γ and ˆ α . Note that the fact that Γ is self-adjoint implies that ˆ α is an even function and that ˆ γ is real-valued. Atranslation-invariant BCS state Γ is admissible if and only if ˆ γ ∈ L ( R d , (1 + p ) d p ) and α ∈ H ( R d , d x ) . By D ti we denote the set of all admissible translation-invariantBCS states. For T ≥ the translation-invariant BCS functional with chemicalpotential µ ∈ R , interaction potential V ∈ L ( R d ) and entropy S , which we can nowwrite as S (Γ) = − Z R d tr C h ˆΓ( p ) log ˆΓ( p ) + (cid:16) − ˆΓ( p ) (cid:17) log (cid:16) − ˆΓ( p ) (cid:17)i d p, takes the form F ti (Γ) = Z R d ( p − µ )ˆ γ ( p ) d p + Z R d V ( x ) | α ( x ) | d x − T S (Γ) . (2.4)Given a state Γ , we define the gap function ∆ of that state as the Fourier transformof V ( x ) α ( x ) . One can then show that the gap function of any minimizing BCSstate satisfies Eq. (1.1), see [12]. We note that F ti was studied in [12] without theconstraint that α is reflection symmetric. The results there hold equally if one worksonly in the subspace of reflection-symmetric functions in L ( R d ) , however. In thecase of V = 0 , the translation-invariant BCS functional F ti is minimized by thepair ( γ , where ˆ γ ( p ) = (1 + e β ( p − µ ) ) − . The same statement is true for the A. DEUCHERT, A. GEISINGER, C. HAINZL, AND M. LOSS periodic BCS functional F . The state ( γ , is called the normal state and describesa situation where superfluidity is absent.It was shown in [12, Theorem 1] that there exists a critical temperature T c ≥ such for T < T c , the minimizer of the translation-invariant BCS functional has a non-vanishing Cooper-pair wave function. On the other hand, for T ≥ T c , the normalstate is the unique minimizer. Additionally, there is a characterization of T c in termsof a linear operator. To make this statement more explicit, let us introduce thefunction K T : R d → R given by K T ( p ) = p − µ tanh(( p − µ ) / (2 T )) . Then, K T = K T ( − i ∇ ) defines an operator on L ( R d ) acting by multiplicationwith K T ( p ) in Fourier space. The critical temperature of the translation-invariantBCS functional is given by T c = inf { T ≥ | K T + V ≥ } . In other words, T c is the value of T such that the operator K T + V has zero aslowest eigenvalue. Observe that this definition makes sense because K T is monotoneincreasing in T . The characterization of T c in terms of a linear operator comesabout because a minimizer of the translation-invariant BCS functional F ti has anon-vanishing Cooper-pair wave function if and only if the normal state is unstableunder pair formation. That is, if and only if the second variation of F ti at ( γ , has a negative eigenvalue. The operator K T + V is exactly the second variation of F ti at the normal state in the direction of a perturbation with γ = 0 and α .In this paper, we treat the question whether there is translational symmetry break-ing in the BCS model with radial pair interaction V . More precisely, we study theminimization problem inf {F (Γ) | Γ ∈ D} and we are, in particular, concerned with the question whether the infimum of F isattained by the minimizers of the translation-invariant BCS functional. If ˆ V ≤ with ˆ V not identically zero this is already known to be the case, see [7, 19]. In order tostudy this question, we consider the BCS functional F ti ℓ on the sector of translation-invariant BCS states with Cooper-pair wave functions of angular momentum ℓ ∈ N ,that we will define in the next paragraph. Our strategy consists of showing that thereexists ℓ such that the minimizers of F ti ℓ and F coincide under certain assumptions.Let us now introduce the functionals F ti ℓ in the case d = 2 . They are obtainedfrom F ti by restricting the domain to Cooper-pair wave functions of the form ˆ α ℓ ( p ) = e iℓϕ σ ℓ ( p ) , (2.5)for some ℓ ∈ Z , where ϕ denotes the angle of p ∈ R in polar coordinates and σ ℓ isa radial function. Recall that α is an even function, which requires ℓ to be even. Aswe will see, the Euler-Lagrange equation of F ti implies that if ( γ, α ℓ ) is a minimizerof F ti , then ˆ γ has to be a radial function. Therefore, we define the BCS functionalon the sector of Cooper-pair wave functions of angular momentum ℓ as follows. Wemake an angular decomposition for ( p, q ) ˆ V ( p − q ) , that is ˆ V ( p − q ) = X ℓ ∈ Z ˆ V ℓ ( p, q ) e iℓϕ , ERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCS MODEL 5 where ϕ denotes the angle between p and q . In other words, this means that ˆ V ℓ ( p, q ) = 12 π Z π e − iℓϕ ˆ V ( p − q ) d ϕ. (2.6)Since ˆ V is a radial function, it only depends on the absolute value of its argument,that is, on | p − q | = p p + q − | p || q | cos( ϕ ) and we conclude that ˆ V ℓ is radial inboth arguments. Furthermore, observe that ˆ V ℓ = ˆ V − ℓ .Then, the BCS functional F ti ℓ on the sector of Cooper-pair wave functions of evenangular momentum ℓ ∈ N is given by F ti ℓ (Γ ℓ ) = Z R ( p − µ ) γ ℓ ( p ) d p + Z R Z R σ ℓ ( p ) σ ℓ ( q ) ˆ V ℓ ( p, q ) d p d q − T S (Γ ℓ ) , where V ℓ is given in (2.6) and Γ ℓ is determined by the pair ( γ ℓ , σ ℓ ) with radialfunctions γ ℓ and σ ℓ . To be more precise, the domain of F ti ℓ is given by D ℓ := (cid:8) ( γ ℓ , σ ℓ ) | γ ℓ , σ ℓ radial and ( γ ℓ , α ℓ ) ∈ D ti , ˆ α ℓ ( p ) = e iℓϕ σ ℓ ( p ) for p ∈ R (cid:9) . Equivalently, F ti ℓ can be understood as the restriction of F ti to pairs ( γ, α ) ∈ D ti with the property that γ is radial and that α is of the form given in (2.5). In Section 3we will show that F ti ℓ has a minimizer.Next, we characterize the critical temperature T c ( ℓ ) corresponding to the BCSfunctionals F ti ℓ on the sector of Cooper-pair wave functions of angular momentum ℓ ∈ N . For this purpose, let us introduce H = { f ∈ H ( R , d p ) | f radial } . Thenthe critical temperature T c ( ℓ ) of F ti ℓ is given by T c ( ℓ ) := inf (cid:8) T ≥ (cid:12)(cid:12) ( K T + V ℓ ) (cid:12)(cid:12) H ≥ (cid:9) . (2.7)The definition of V ℓ in Eq. (2.6) and the fact that K T + V commutes with rotations,implies that T c = max ℓ ∈ N T c ( ℓ ) holds.Let us now assume that T c = T c ( ℓ ) and that the lowest eigenvalue of K T c + V is atmost twice degenerate. In other words, we assume the lowest eigenvalue of K T c + V tobe exactly twice degenerate in the case ℓ = 0 and we assume it to be non-degeneratein the case ℓ = 0 . An exemplary situation satisfying this assumption is illustrated inFigure 1. The meaning of this schematic pictures is the following. Since T c = T c ( ℓ ) ,the lowest eigenvalue of K T + V lies in the sector with angular momentum ℓ . Ifwe decrease the temperature this eigenvalue becomes negative and the second/thirdeigenvalue (depending on the degeneracy) will approach zero at some temperature ˜ T < T c ( ℓ ) . For this eigenvalue, there are two possibilities: Either it also lies in thesector of angular momentum ℓ , which means that ˜ T ∈ ( T c ( ℓ ) , T c ) and this is thecase illustrated in Figure 1, or the next eigenvalue lies in the next sector of angularmomentum, which means that ˜ T = T c ( ℓ ) . The following theorem shows that thetranslational symmetry in the BCS model persists if T ∈ ( ˜ T , T c ) . In particular, if ℓ = 0 , the periodic (and the translation-invariant) BCS functional has a, up to aphase, unique radial minimizer ( γ , α ) for T ∈ ( ˜ T , T c ) . If ℓ = 0 , the periodic (andthe translation-invariant) BCS functional has two minimizers, namely ( γ ℓ , α ℓ ) and ( γ ℓ , α − ℓ ) , with γ ℓ radial and α ± ℓ of the form ˆ α ± ℓ ( p ) = e ± iℓ ϕ σ ℓ ( p ) . A. DEUCHERT, A. GEISINGER, C. HAINZL, AND M. LOSS T T c ( ℓ ) T c ( ℓ ) ˜ T Figure 1.
Schematic picture of the lowest eigenvalues of K T + V as a function of the temperature T . The lowest two lines representeigenvalues in the sector of angular momentum ℓ . The third linecorresponds to the lowest eigenvalue in the angular momentum ℓ sector. The red dots highlight the temperatures at which one of theeigenvalues crosses the T -axis. Theorem 1.
Let V ∈ L ( R ) with ˆ V ∈ L r ( R ) , where r ∈ [1 , , be radial and suchthat T c > . Suppose that T c = T c ( ℓ ) and that the lowest eigenvalue of K T c + V isat most twice degenerate. If ( γ ℓ , σ ℓ ) ∈ D ℓ minimizes F ti ℓ , then there exists ˜ T < T c such that ( γ ℓ , α ℓ ) and ( γ ℓ , α − ℓ ) ∈ D ti , where ˆ α ± ℓ ( p ) = e ± iℓ ϕ σ ℓ ( p ) , minimize the BCS functional F for T ∈ [ ˜ T , T c ) . For T ∈ ( ˜ T , T c ) these are the only minimizers of F up to phases in front of α ℓ and α − ℓ . Remark 2.1.
We want to emphasize that ˜ T is determined by the lowest nonzeroeigenvalue of K T c + V . More precisely, ˜ T is given as the value of T such thatthe second eigenvalue (counted without multiplicities) of K T + V is zero, which isillustrated in Figure 1. In particular, if in addition to the assumption above, thesecond eigenvalue of K T c + V lies in the sector of angular momentum ℓ = ℓ , onecan show that ˜ T = T c ( ℓ ) . Remark 2.2.
The assumptions V ∈ L ( R ) and ˆ V ∈ L r ( R ) with r ∈ [1 , inTheorem 1 are of technical nature and we expect the Theorem to hold as long as V ∈ L ǫ ( R ) for ǫ > . Note that this is the L p regularity for which V is relatively formbounded with respect to the Laplacian in two space dimensions. The assumptionon the Fourier transform of V is only needed in the proof of Proposition 4.3. In [10,Proposition 5.6] a similar result is proved in the case d = 3 under the assumption V ∈ L / ( R ) which guarantees form boundedness relative to the Laplacian in thiscase. Although we expect the strategy of that proof to carry over to d = 2 , ourargument is much simpler than the one given in this reference and so we prefer tokeep the additional assumption on ˆ V . ERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCS MODEL 7
Remark 2.3.
The Fourier transform preserves angular momentum sectors, andhence the inverse Fourier transforms of the minimizing Cooper-pair wave functions ˆ α ± ℓ ( p ) = e ± iℓ ϕ p σ ℓ ( p ) are of the form e ± iℓ ϕ x f ℓ ( x ) with f ℓ radial. That is, theCooper-pairs have definite angular momentum also in position space. Remark 2.4.
An important step in the proof of Theorem 1 is to compare theminimizers of the BCS functional F ti ℓ on the sector of Cooper-pair wave functionswith angular momentum ℓ with the minimizers of the periodic BCS functional F .The crucial tool for this comparison will be the relative entropy inequality, [7, Lemma5]. Remark 2.5.
It is shown in [10], amongst other things, that for every ℓ ∈ N one can find a radial potential such that the ground state of K T c + V has angularmomentum ℓ . This in particular implies T c = T c ( ℓ ) for such a potential. In thecase of weak coupling, that is for K T + λV , where λ ∈ R is small enough, themethods of [6, 16] can be applied to determine the angular momentum ℓ of theground state of K T c + V . An application of these methods reduces the problem offinding the eigenvalues of K T + λV , for λ small enough, to finding the eigenvaluesof a simple matrix, that only depends on the behavior of ˆ V on the Fermi sphere.This is easily solvable numerically. In particular, one sees, that the eigenvalues arein one-to-one correspondence to the eigenvalues of the matrix ( h ψ n , ˆ V ψ m i ) n,m ≥ ,where ψ n ( p ) = e inϕ . Moreover, if the lowest eigenvalue of this matrix is at mosttwice degenerate one is in the situation described in Remark 2.1, i.e. ˜ T = T c ( ℓ ) . Remark 2.6.
In the non-interacting case, that is, for V = 0 , the minimizer of theBCS function F is given by the normal state ˆΓ = (cid:18) ˆ γ
00 1 − ˆ γ (cid:19) , where ˆ γ = (1 + exp(( −∇ − µ ) /T )) − . Let us assume that we are in the situation ofRemark 2.1. Having in mind that the linear operator K T + V , which characterizes T c ,is related to the second variation of F at the normal state Γ in the direction of α by d d t F ( γ , tα ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 2 h α, ( K T + V ) α i , one can understand Theorem 1 as follows. We find T < T c such that K T + V hasexactly one negative eigenvalue λ . Hence the second variation is smallest (and, inparticular, negative) if α is an element of the eigenspace of λ and one could thereforehope to find a minimizer of F which lies approximately in this eigenspace. In fact,Theorem 1 states that the minimizers of F for temperatures T in a certain intervalbelow T c lie in exactly one specific sector of angular momentum ± ℓ . For T = T c ( ℓ ) the next eigenvalue λ and its eigenspace become important, since now also elementsof the eigenspace of λ are candidates to lower the energy.In the special case ℓ = 0 , Theorem 1 also holds in three dimensions. Theorem 2.
Let V ∈ L ( R ) with ˆ V ∈ L r ( R ) for some r ∈ [1 , / be radial andsuch that T c > . Assume that zero is a non-degenerate eigenvalue of K T c + V , thatis, the corresponding eigenfunction is radial. Then, there exists ˜ T < T c such thatthe minimizer of the BCS functional F for T ∈ [ ˜ T , T c ) is given by a pair ( γ , α ) , A. DEUCHERT, A. GEISINGER, C. HAINZL, AND M. LOSS where γ and α are radial functions. Moreover, ( γ , α ) is, up to phases, the onlyminimizer of F for T ∈ ( ˜ T , T c ) . Remark 2.7.
Note that ˆ V ≤ implies that the ground state of K T c + V is radialin all dimensions. Hence, the assumption that K T c + V has a non-degenerate lowesteigenvalue is always satisfied for interaction potentials V with this property. Remark 2.8.
As in the case of Theorem 1, we expect Theorem 2 to hold under theonly assumption that V is relatively form bounded with respect to the Laplacian,that is, if V ∈ L / ( R ) .We recall the gap function ∆( p ) = 2(2 π ) − d/ ˆ V ∗ ˆ α ( p ) with d = 2 , . The Cooper-pair wave function of any minimizer of the translation-invariant BCS functional F ti satisfies the Euler-Lagrange equation (cid:0) K ∆ T + V (cid:1) α = 0 . (2.8)Here K ∆ T is the operator defined by multiplication in Fourier space with the function K ∆ T ( p ) = E ( p )tanh ( E ( p ) / (2 T )) , where E ( p ) = p ( p − µ ) + | ∆( p ) | . The key ingredient to the proof of Theorem 1 and Theorem 2 is that in both situations K ∆ T + V ≥ holds. The following Proposition tells us that this already implies that | ˆ α ( p ) | is a radial function. Hence, our strategy of proof can only work if this is thecase. In particular, it tells us that we cannot extend our results to situations wherethe absolute value of the Fourier transform of the ground state of K T c + V is notradial. Proposition 2.9.
Let V be a radial function with V ∈ L ( R ) if d = 2 and V ∈ L / ( R ) if d = 3 . Assume that ( γ, α ) is a minimizer of the translation-invariantBCS functional F ti such that | ˆ α ( p ) | is not a radial function. Then there exists arotation R ∈ SO ( d ) such that (cid:10) U ( R ) α, (cid:0) K ∆ T + V (cid:1) U ( R ) α (cid:11) < , (2.9) where ( U ( R ) f ) ( p ) = f ( R − p ) . Preparations
The proof of Theorem 2 works similarly to the proof of Theorem 1. In order toprove Theorem 1 we will show that there exists ℓ ∈ N , such that the minimizersof F ti ℓ also minimize F . The following lemma lays the basis for this approach.In [12] it was shown that F ti is bounded from below and attains its infimumon D ti in three dimensions. The same results hold in two dimensions by analogousarguments, which provides a solution of the BCS gap equation in this case. Lemma 3.1.
The BCS functional F ti ℓ is bounded from below and attains its mini-mum.Proof. Boundedness from below of F ti ℓ follows from the fact that F ti is bounded frombelow. As in the proof of [12, Lemma 1] we find a minimizing sequence ( γ ( n ) ℓ , σ ( n ) ℓ ) in D ℓ that converges strongly in L p ( R ) × L ( R ) to ( γ, σ ) for some p ∈ (1 , ∞ ) , as n tends to infinity. It is an easy consequence that ( γ, σ ) ∈ D ℓ . (cid:3) ERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCS MODEL 9
The Euler-Lagrange equation of F ti ℓ takes the same form as the Euler-Lagrangeequation of F ti , which will play an important role in the proof. The derivation of theEuler-Lagrange equation of F ti given in [19, Proposition 3.1] translates to the caseof F ti ℓ . Therefore, we will not rewrite the proof here, but only give the Euler-Lagrangeequation of F ti ℓ in its various forms.Let us define the gap function ∆ ℓ related to the Cooper-pair wave function σ ℓ by ∆ ℓ ( p ) = 1 π Z R V ℓ ( p, q ) σ ℓ ( p )d q. (3.1)Since V ℓ ( p, q ) is radial in both arguments ∆ ℓ ( p ) is a radial function, too. Also define H ∆ ℓ ( p ) = (cid:18) k ( p ) ∆ ℓ ( p )∆ ℓ ( p ) − k ( p ) (cid:19) (3.2)with k ( p ) = p − µ . For T > , the Euler-Lagrange equation of the functional F ti ℓ , isgiven by Γ ℓ ( p ) = (cid:18) γ ℓ ( p ) σ ℓ ( p ) σ ℓ ( p ) 1 − γ ℓ ( p ) (cid:19) = 11 + e H ∆ ℓ ( p ) /T . (3.3)The right-hand side of Eq. (3.3) depends only on σ ℓ through ∆ ℓ but not on γ ℓ . Thatis, γ ℓ is determined by σ ℓ .Let us define E ℓ ( p ) = p ( p − µ ) + | ∆ ℓ ( p ) | and the function K ∆ ℓ T , which for T > is given by K ∆ ℓ T ( p ) = E ℓ ( p )tanh ( E ℓ ( p ) / (2 T )) . Then K ∆ ℓ T = K ∆ ℓ T ( − i ∇ ) defines an operator on L ( R ) acting by multiplicationwith K ∆ ℓ T ( p ) in Fourier space. Calculations given explicitly in [19] show that (3.3) isequivalent to γ ℓ ( p ) = 12 − p − µ K ∆ ℓ T ( p ) , (3.4) σ ℓ ( p ) = − ∆ ℓ ( p )2 K ∆ ℓ T ( p ) . (3.5)Using Eq. (3.1), we see that Eq. (3.5) can be written as (cid:16) K ∆ ℓ T + V ℓ (cid:17) σ ℓ = 0 . (3.6)We will also make use of this equation in the form (cid:16) K ∆ ℓ T + V (cid:17) α ℓ = 0 , (3.7)where α ℓ is of the form (2.5).4. Proof of Theorem 1 and Theorem 2
We begin with the proof of Theorem 1. Let ( γ ℓ , σ ℓ ) ∈ D ℓ be a minimizer of F ti ℓ and assume T c = T c ( ℓ ) . Let Γ ℓ be the BCS state given by the pair ( γ ℓ , α ℓ ) with ˆ α ℓ ( p ) = e iℓ ϕ σ ℓ ( p ) . Our aim is to show that the inequality F (Γ) − F (Γ ℓ ) ≥ holdsfor all Γ ∈ D . We will use a generalization of the trace per unite volume, which fora periodic operator A on L ( R , C ) is defined by Tr [ A ] = Tr Ω [ P AP + Q AQ ] with P = (cid:18) (cid:19) and Q = (cid:18) (cid:19) . Note that if A is locally trace class, then Tr [ A ] = Tr Ω [ A ] .We begin by calculating the difference F (Γ) − F (Γ ℓ ) , where Γ ℓ corresponds to aminimizer of F ti ℓ as described above. The state Γ is defined by the pair ( γ, α ) . Wefind F (Γ) − F (Γ ℓ ) (4.1) = Tr Ω (cid:2)(cid:0) −∇ − µ (cid:1) ( γ − γ ℓ ) (cid:3) + Z Ω × R V ( x − y ) (cid:0) | α ( x, y ) | − | α ℓ ( x, y ) | (cid:1) d( x, y ) − T ( S (Γ) − S (Γ ℓ )) . First, we complete the square in the difference of the interaction terms, which yields Z Ω × R V ( x − y ) (cid:0) | α ( x, y ) | − | α ℓ ( x, y ) | (cid:1) d( x, y )= Z Ω × R V ( x − y ) (cid:0) | α ( x, y ) − α ℓ ( x, y ) | (cid:1) d( x, y ) − Z Ω × R V ( x − y ) (cid:16) | α ℓ ( x, y ) | − Re (cid:16) α ( x, y ) α ℓ ( x, y ) (cid:17)(cid:17) d( x, y ) . Next, we combine the second term on the right hand side and the first term on theright hand side of (4.1). Let ˜∆ ℓ ( p ) = e iℓϕ ∆ ℓ ( p ) where ϕ denotes the angle of p ∈ R in polar coordinates and ∆ ℓ is given by Eq. (3.1). Inserting the equation ˆ α ℓ ( p ) = − ˜∆ ℓ ( p ) / (2 K ∆ ℓ T ( p )) which follows from Eq. (3.5), we see that Tr Ω (cid:2)(cid:0) −∇ − µ (cid:1) ( γ − γ ℓ ) (cid:3) + 2 Re Z Ω × R V ( x − y ) (cid:16) α ℓ ( x, y ) α ( x, y ) − | α ℓ ( x, y ) | (cid:17) d( x, y )= 12 Tr h H ˜∆ ℓ (Γ − Γ ℓ ) i . Here H ˜∆ ℓ is given as in Eq. (3.2) with ∆ ℓ replaced by ˜∆ ℓ .At this point, it turns out to be convenient to introduce the relative entropy H ,which for two BCS states Γ , ˜Γ ∈ D is given by H (Γ , ˜Γ) = Tr h Γ (cid:16) log Γ − log ˜Γ (cid:17) + (1 − Γ) (cid:16) log (1 − Γ) − log (cid:16) − ˜Γ (cid:17)(cid:17)i . The fact that H ˜∆ ℓ /T = log(1 − Γ ℓ ) − log Γ ℓ yields the following statement. Lemma 4.1.
Let ( γ ℓ , σ ℓ ) ∈ D ℓ be a minimizer of F ti ℓ and let Γ ℓ be given by the pair ( γ ℓ , α ℓ ) where α ℓ ( p ) = e iℓϕ σ ℓ ( p ) . Then F (Γ) − F (Γ ℓ ) = T H (Γ , Γ ℓ ) + Z Ω × R V ( x − y ) | α ( x, y ) − α ℓ ( x, y ) | d( x, y ) for all Γ ∈ D , where α = (Γ) . Based on this identity, we estimate F (Γ) − F (Γ ℓ ) from below by applying therelative entropy inequality [7, 13]. ERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCS MODEL 11
Proposition 4.2.
Let ( γ ℓ , σ ℓ ) ∈ D ℓ , be a minimizer of F ti ℓ , let Γ ℓ be as in Lemma 4.1and denote V y ( x ) = V ( x − y ) . Then, for all Γ ∈ D , with α = (Γ) , F (Γ) − F (Γ ℓ ) ≥ Z Ω D α, (cid:16) K ∆ ℓ T + V y ( x ) (cid:17) x α E L ( R , d x ) d y + Tr Ω K ∆ ℓ T (Γ − Γ ℓ ) . Here, we understand ( K ∆ ℓ T + V y ( x )) x as an operator acting on the x -coordinate of α ( x, y ) .Proof. The claimed estimate is a consequence of an inequality for the relative entropythat has been proven in [7, Lemma 5]. An application of this inequality yields F (Γ) − F (Γ ℓ ) ≥
12 Tr Ω (Γ − Γ ℓ ) H ˜∆ ℓ tanh (cid:16) H ˜∆ ℓ / (2 T ) (cid:17) (Γ − Γ ℓ ) + Z Ω × R V ( x − y ) | α ( x, y ) − α ℓ ( x, y ) | d( x, y ) . The fact that x x (tanh( x/ − is an even function and H ℓ ( p ) = I C E ℓ ( p ) is diagonal, implies the statement. (cid:3) Next, we show that the operator K ∆ ℓ T + V is nonnegative for T ∈ [ ˜ T , T c ) . Proposition 4.3.
Assume V ∈ L ( R ) and ˆ V ∈ L r ( R ) for some r ∈ [1 , . If thelowest eigenvalue of K T c + V is at most twice degenerate then there exists ˜ T < T c such that K ∆ ℓ T + V is nonnegative as an operator on L ( R ) for all T ∈ [ ˜ T , T c ) . The proof of Proposition 4.3 is based on spectral perturbation theory and relieson the fact that K ∆ ℓ T + V → K T c + V , while ∆ ℓ ( T ) → , in norm resolvent sensefor T → T c . We will derive this convergence from the following lemmas. In order tosimplify the notation we write a . b if there exists a constant c > such that a ≤ cb .Moreover, we denote by k · k the operator norm and by k · k r the L r ( R ) -norm. Lemma 4.4.
Let T ∈ (0 , T c ) . The operators K T c − K T and K ∆ ℓ T − K T are bounded.More precisely, k K T c − K T k . ( T c − T ) and k K ∆ ℓ T − K T k . k ∆ ℓ k ∞ . Moreover, K T c − K T ≥ and K ∆ ℓ T − K T ≥ .Proof. In the proof we abbreviate A T := K T c − K T and B T := K ∆ ℓ T − K T . Noticethat K ∆ ℓ T ( p ) = p k ( p ) + | ∆ ℓ ( p ) | tanh (cid:16)p k ( p ) + | ∆ ℓ ( p ) | / (2 T ) (cid:17) is an increasing function in T for fixed ∆ ℓ and vice versa. Hence A T ≥ and B T ≥ . Both, A T and B T are pseudo-differential operators and by a slight abuseof notation we denote by A T ( p ) and B T ( p ) the symbols of A T and B T , respectively.In the following we abbreviate T c − T = δT and I T = 1 T − T c . A simple calculation yields A T ( p ) = Z I T k ( p ) ( k ( p ) / (2 T c ) + tI T k ( p ) /
2) d t. Obviously, for large | p | the smooth function A : p A ( p ) and all its derivatives haveexponential decay. Moreover, | I T | . T c − T implies k A T k . T c − T . In order toderive an analogous representation for B T ( p ) we define f ( x ) := dd x x tanh( x/ (2 T )) = T sinh( x/T ) − x T sinh ( x/ (2 T )) (4.2)as well as δE ℓ ( p ) = q k ( p ) + | ∆ ℓ ( p ) | − | k ( p ) | . (4.3)A straightforward calculation shows that B T ( p ) = δE ℓ ( p ) Z f ( | k ( p ) | + tδE ℓ ( p )) d t. (4.4)Since the function f defined in (4.2) is bounded by , we find that | B T ( p ) | ≤ | δE ℓ ( p ) | for all p ∈ R . It can be seen directly from the definition of δE ℓ ( p ) , see (4.3),that | δE ℓ ( p ) | ≤ | ∆ ℓ ( p ) | for all p ∈ R , which implies | B T ( p ) | ≤ | ∆ ℓ ( p ) | for all p ∈ R . (cid:3) Lemma 4.5.
Let T ∈ (0 , T c ) . If α ℓ is a solution of the BCS gap equation in theform of Eq. (3.7) , then k (1 + p ) / ˆ α ℓ k . h α ℓ , ( K ∆ ℓ T − K T ) α ℓ i .Proof. We will make use of the following observation, which is implied by the factthat the function | ∆ ℓ | 7→ | ∆ ℓ | /K ∆ ℓ T is strictly increasing. Eq. (3.1) implies that k ∆ ℓ k ∞ ≤ k V k k ˆ α ℓ k . (4.5)We will abbreviate k V k k ˆ α ℓ k by c ( α ℓ ) in the following. Thus, together with (3.5),the just mentioned monotonicity of | ∆ ℓ | /K ∆ ℓ T implies that | ˆ α ℓ ( p ) | ≤ c ( α ℓ )2 K c ( α ℓ ) T ( p ) for all p ∈ R . By taking the square and integrating, we see that ≤ k V k Z R (cid:16) K c ( α ℓ ) T ( p ) (cid:17) − d p. Next, we use that tanh( x ) ≤ for all x , which leads to ≤ k V k Z R (cid:0) ( p − µ ) + k V k k ˆ α ℓ k (cid:1) − d p We may assume that k V k k α ℓ k ≥ µ and conclude that ≤ k V k Z R (cid:0) p / − µ + k V k k ˆ α ℓ k (cid:1) − d p. From this estimate one easily derives that k ˆ α ℓ k ≤ k V k π
32 + µ k V k . Making use of (4.5), we see that this directly implies that k ∆ ℓ k ∞ ≤ k V k π
32 + µ . (4.6)In other words, there exists a constant m > that only depends on V and µ , suchthat | ∆ ℓ ( p ) | < m for all p ∈ R . In particular, m does not depend on T . ERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCS MODEL 13
We have to estimate K ∆ ℓ T − K T from below. We recall that | ∆ ℓ | 7→ K ∆ ℓ T / | ∆ ℓ | is decreasing. Having in mind that K ∆ T − K T behaves like | ∆ | for small | ∆ | we thusestimate K ∆ ℓ T − K T | ∆ ℓ | | ∆ ℓ | & (cid:18) K mT − K T m (cid:19) | ∆ ℓ | . Abbreviating y t = p k ( p ) + tm / (2 T ) we find that K ∆ ℓ T ( p ) − K T ( p ) = 2 T Z dd t y t tanh ( y t ) d t = m T Z (cid:18) y t tanh( y t ) − ( y t ) (cid:19) d t. (4.7)As one easily sees, the function g ( y ) = 1 y y ) − ( y ) is decreasing, which implies K ∆ ℓ T ( p ) − K T ( p ) & m T (cid:18) y tanh( y ) − ( y ) (cid:19) . Moreover, g is bounded from below by g ( y ) ≥ / y ) − . Together with (4.7) thisshows that K ∆ ℓ T ( p ) − K T ( p ) & | ∆ ℓ ( p ) |
11 + p . (4.8)Next, we make use of the Euler-Lagrange equation of F ti ℓ , that is the relation | ∆ ℓ ( p ) | = 2 K ∆ ℓ T ( p ) | ˆ α ℓ ( p ) | . Inserting this identity in (4.8) we see that K ∆ ℓ T ( p ) − K T ( p ) & (cid:16) K ∆ ℓ T ( p ) (cid:17) | ˆ α ℓ ( p ) | p & (cid:0) p (cid:1) | ˆ α ℓ ( p ) | , which implies the statement. (cid:3) Lemma 4.6.
Let T ∈ (0 , T c ) . If α ℓ is a solution of the BCS gap equation in theform (3.7) , then k α ℓ k . ( T c − T ) / . In particular, k ∆ ℓ k ∞ . ( T c − T ) / .Proof. The gap equation, see (3.7), can be written as h α ℓ , ( K T c + V ) α ℓ i + h α ℓ , Bα ℓ i = h α ℓ , Aα ℓ i , where we use the notation introduced in the proof of Lemma 4.4 but drop the sub-script, i.e. A = A T and B = B T for brevity. Lemma 4.4 and the definition of T c imply that h α ℓ , Bα ℓ i ≤ h α ℓ , Aα ℓ i . ( T c − T ) k α ℓ k . (4.9)From the combination of Lemma 4.5 and (4.9) we deduce that k (cid:0) p (cid:1) / ˆ α ℓ k . ( T c − T ) k α ℓ k . On the other hand, the L r ( R ) -norm of ˆ α is bounded from above by k ˆ α ℓ k r ≤ k (cid:0) p (cid:1) − / k s k (cid:0) p (cid:1) / ˆ α ℓ k , where r > , due to the fact that we have to choose s > . Thus, k ˆ α ℓ k r . ( T c − T ) k ˆ α ℓ k . (4.10) Furthermore, we conclude from the relation between ∆ ℓ and α ℓ given by Eq. (3.1)that k ∆ ℓ k ∞ . k ˆ V k t k ˆ α ℓ k r , (4.11)where we choose r > and t ∈ [1 , appropriately. Note that the gap equation inthe form (3.5) implies that k ˆ α ℓ k . k ∆ ℓ k ∞ . Together with (4.10) and (4.11) thisfinally shows that k ˆ α ℓ k . ( T c − T ) / k ˆ α ℓ k / and hence proves the first part of the claim. In order to get the estimate on k ∆ ℓ k ∞ ,we go back to (4.10) and insert k α ℓ k . ( T c − T ) / . Together with (4.11) this yieldsthe statement. (cid:3) Let T ∈ (0 , T c ) and z ∈ C \ R . Taken together, Lemma 4.4 and Lemma 4.6 showthat (cid:13)(cid:13)(cid:13)(cid:13) ( z − ( K T c + V )) − − (cid:16) z − (cid:16) K ∆ ℓ T + V (cid:17)(cid:17) − (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ( z − ( K T c + V )) − (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) K ∆ ℓ T − K T c (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) z − (cid:16) K ∆ ℓ T + V (cid:17)(cid:17) − (cid:13)(cid:13)(cid:13)(cid:13) . | Im( z ) | − ( T c − T ) / . In other words, K ∆ ℓ T + V → K T c + V for T → T c in norm resolvent sense for anarbitrary z ∈ C \ R and consequently for all z ∈ ρ ( K T c + V ) .We are now prepared for the proof of Proposition 4.3. Proof of Proposition 4.3.
We consider the case ℓ = 0 . The proof for the case ℓ = 0 is analogous. As illustrated in Figure 1, we have by assumption that T c = T c ( ℓ ) andthat the lowest eigenvalue of K T c + V is exactly twice degenerate. Note that in thecase that ℓ = 0 the smallest eigenvalue is non-degenerate. From the convergenceof K ∆ ℓ T + V to K T c + V in norm resolvent sense one concludes that the lowesteigenvalue of K ∆ ℓ T + V is stable.In particular, this tells us that there exists ˜ T < T c such that K ∆ ℓ T + V with T ∈ ( ˜ T , T c ] has exactly two eigenvalues λ ( T ) , λ ( T ) ∈ { z ∈ C | | z | < r } for someradius r > . Combining this with the fact that the Euler-Lagrange equation (3.7)of F ti ℓ reads ( K ∆ ℓ T + V ) α = 0 , (4.12)we conclude that λ ( T ) = λ ( T ) = 0 . Having in mind that K ∆ ℓ T is an increasingfunction of T and of ∆ ℓ , what we have seen by this argument is that the effects ofthese monotonicity properties exactly correspond. In other words, we have shownthat there exists ˜ T < T c such that K ∆ ℓ T + V is nonnegative for all T ∈ [ ˜ T , T c ] . It isnot hard to see that ˜ T can be chosen as pointed out in Remark 2.1. (cid:3) Proof of Theorem 1.
We know from Lemma 3.1 that for ℓ determined by T c ( ℓ ) =max ℓ ∈ N T c ( ℓ ) the functional F ti ℓ has a minimizer ( γ ℓ , σ ℓ ) . Proposition 4.2 andProposition 4.3 show that for Γ ℓ given by ( γ ℓ , α ℓ ) , with α ℓ as in (2.5), F (Γ) − F (Γ ℓ ) ≥ , holds for all Γ ∈ D . Moreover, if F (Γ) −F (Γ ℓ ) = 0 , then γ = γ ℓ and α ∈ ker( K ∆ ℓ T + V y ) by Proposition 4.2. Consequently, α takes the form α = ψ α ℓ + ψ α − ℓ , where α ± ℓ ( p ) = e ± iℓϕ σ ℓ ( p ) and ψ and ψ denote complex constants. It remains to show ERSISTENCE OF TRANSLATIONAL SYMMETRY IN THE BCS MODEL 15 that either ψ = 0 and | ψ | = 1 or | ψ | = 1 and ψ = 0 . Observe that, in particular, ( γ ℓ , α ) ∈ D ti and as we know that F ti has a minimizer, we conclude that ( γ ℓ , α ) satisfies the Euler-Lagrange equation of F ti , that is γ ℓ ( p ) = 12 − p − µ K ∆ T ( p ) , where ∆ = π − ˆ V ∗ ˆ α . Hence | ∆ | is a radial function and consequently either ψ = 0 or ψ = 0 . In other words, ( γ ℓ , σ ℓ ) ∈ D ℓ . Thus, in order to find minimizers of F ,it is sufficient to find the minimizers of F ti ℓ . As we know that F ti ℓ has minimizers,the only thing left to show is that ( γ ℓ , σ ℓ ) is, up to a phase, the only minimizerof F ti ℓ . The fact that other possible minimizers ( γ ℓ , ψσ ℓ ) , for some ψ ∈ C , have tosatisfy the gap equation (3.6) of F ti ℓ reads (cid:16) K ψ ∆ ℓ T + V ℓ (cid:17) ( ψσ ℓ ) = 0 . Together with the monotonicity of K ψ ∆ ℓ T in ψ this implies that | ψ | = 1. (cid:3) The proof of Theorem 2 is analogous to the proof of Theorem 1 with one exception.
Proof of Theorem 2.
In case ℓ = 0 all given arguments also apply in the three-dimensional case. The only exception is Lemma 4.6, where we need to modify theassumptions slightly. One easily sees that ˆ V ∈ L r ( R ) with r ∈ [1 , / is a sufficientassumption in this case. (cid:3) Proof of Proposition 2.9.
We will carry out the proof for d = 3 and afterwards com-ment on the case d = 2 . The Cooper-pair wave function of any minimizer of thetranslation-invariant BCS functional satisfies ˆ α ( p ) = − ∆( p ) / (2 K ∆ T ( p )) which is im-plied by the Euler-Lagrange equation of F , see [12] or compare with Section 3. Hence, | ˆ α | is radial if and only if | ∆ | is radial. With Eq. (2.8) and the assumption that V isa radial function, one checks that it is sufficient to show (cid:10) U ( R ) α, K ∆ T U ( R ) α (cid:11) < (cid:10) α, K ∆ T α (cid:11) . (4.13)Using the above relation between ˆ α and ∆ , we write (cid:10) U ( R ) α, K ∆ T U ( R ) α (cid:11) = 14 Z R | ∆( p ) | K ∆ T ( p ) K ∆ T ( Rp ) d p = 14 Z ∞ Z Ω r | ∆( p ) | K ∆ T ( p ) K ∆ T ( Rp ) d ω ( p ) r d r, where Ω r denotes the sphere with radius r and d ω ( p ) denotes the uniform measureon Ω r . On Ω r , that is, for fixed radius r = | p | , we can understand | ∆( p ) | /K ∆ T ( p ) as a function f that depends only on | ∆( p ) | . There also exists a function g suchthat K ∆ T ( Rp ) = g ( | ∆( Rp ) | ) for all p ∈ Ω r . The functions f and g are both strictlyincreasing.Consider the expression M ( R ) := Z Ω r [ g (∆( Rp )) − g (∆( p ))][ f (∆( Rp )) − f (∆( p ))] dω ( p ) The functions f and g depend only on the magnitude of ∆( Rp ) resp. ∆( p ) . Since f and g are strictly increasing we have that M ( R ) > unless | ∆( Rp ) | = | ∆( p ) | for a.e. p . To see this assume that | ∆( Rp ) | and | ∆( p ) | differ on a set of positive measure. Now consider the set { p : | ∆( Rp ) | > | ∆( p ) |} and the set { p : | ∆( Rp ) | < | ∆( p ) |} Atleast one of them must have positive measure. Hence on the union of these sets [ g (∆( Rp )) − g (∆( p ))][ f (∆( Rp )) − f (∆( p ))] > since f and g are both strictly increasing. Using the rotation invariance of themeasure ω , we find < M ( R ) = 2 Z Ω r g (∆( p )) f (∆( p )) dω ( p ) − Z Ω r g (∆( p )) f (∆( Rp )) dω ( p ) − Z Ω r g (∆( Rp )) f (∆( p ))] dω ( p ) and hence one of the integrals Z Ω r g (∆( p )) f (∆( Rp )) dω ( p ) or Z Ω r g (∆( Rp )) f (∆( p )) dω ( p ) must be strictly below Z Ω r g (∆( p )) f (∆( p )) dω ( p ) . Accordingly, there exists a R ∈ SO (3) such that Z Ω r | ∆( p ) | K ∆ T ( p ) K ∆ T ( Rp ) d ω ( p ) < Z Ω r | ∆( p ) | K ∆ T ( p ) K ∆ T ( p ) d ω ( p ) . (4.14)To conclude that Eq. (4.13) holds, it suffices to note that ∆ is a continuous function,see the first paragraph in the proof of [12, Proposition 3], which implies that bothsides of Eq. (4.14) are continuous functions of r . If d = 2 the proof goes throughin the same way with the only difference that the continuity of ∆ is concluded from ∆( p ) = π − ˆ V ∗ ˆ α ( p ) , the assumption that V ∈ L ( R ) and the Riemann-LebesgueLemma. (cid:3) Acknowledgments.
The paper was partially supported by the GRK 1838 and theHumboldt foundation. M.L. was partially supported by NSF grant DMS-1600560.Partial financial support by the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovation programme (grant agreementNo 694227) is gratefully acknowledged (A.D.). We are grateful for the hospitalityat the Department of Mathematics at the University of Tübingen (M.L.) and at theGeorgia Tech School of Mathematics (A.G.).
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