On the Infimum of the Excitation Spectrum of a Homogeneous Bose Gas

Transcription

1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria On the Infimum of the Excitation Spectrum of a Homogeneous Bose Gas H.D. Cornean J. Derezińsi P. Ziń ienna, Preprint ESI November 1, 005 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via

2 ON THE INFIMUM OF THE EXCITATION SPECTRUM OF A HOMOGENEOUS BOSE GAS H.D. CORNEAN, J. DEREZIŃSKI, AND P. ZIŃ Abstract. We consider a second quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature, and in the grand canonical setting the chemical potential µ is fixed, the number of particles can vary. The interaction potential has a positive Fourier transform. We define the infimum of the excitation spectrum as the infimum of the grandcanonical Hamiltonian restricted to the subspace of total momentum. We investigate its upper bound given by the expectation value at squeezed states. Our method can be viewed as a rigorous version of the usual Bogoliubov approach. Contents 1. Introduction 1. Original Bogoliubov approach 4 3. Grand-canonical version of the original Bogoliubov approach 6 4. Infimum of the excitation spectrum a rigorous definition 8 5. Infimum of the excitation spectrum in the squeezed states approximation A rigorous version of the Bogoliubov approximation Bogoliubov translation of the Hamiltonian 1 8. Bogoliubov rotation of the Hamiltonian 1 9. Conditions arising from minimization of B 14 Appendix A. Bogoliubov transformations 17 Appendix B. Computations of the Bogoliubov rotation 17 References Introduction The main subject of our paper is a rigorous study of the infimum of the excitation spectrum abbreviated IES of the Bose gas of density ρ in the thermodynamic limit. One can distinguish two possible approaches to this subject: canonical and grand-canonical. Further on in our paper we will concentrate on the latter setting. Nevertheless, in the introduction, let us stic to the canonical approach. Suppose that the -body potential of an interacting Bose gas is described by a real function v defined on R d, satisfying vx = v x. We assume that the Fourier transform of v, given by ˆv := vxe ix d, 1.1 is nonnegative. In order to define the quantities of interest to us at nonzero density, we start with a system of n bosonic particles in Λ = [0, L] d the d-dimensional cubic box of side length L. We assume that the number of particles equals n = ρ, where Date: December 15,

3 = L d is the volume of the box. Following the accepted although somewhat unphysical tradition we replace the potential by v L x = 1 π L Zd e i x ˆv, 1. where π L Zd is the discrete momentum variable. Note that v L is periodic with respect to the domain Λ, and v L x vx as L. The system is described by the Hamiltonian H L,n = 1 i + v L x i x j 1.3 i=1 1 i<j n acting on the space L sλ n symmetric square integrable functions on Λ n. We assume that the Laplacian has periodic boundary conditions. The total momentum is given by the operator n P L,n := i xi. i=1 P L,n commutes with H L,n and both have purely discrete spectrum, hence we can find a joint orthonormal system of eigenfunctions. Thus we can fix an eigenvalue of P L,n and study the restriction of H L,n to the spectral subspace associated to this eigenvalue. The IES at finite volume in the zero temperature canonical ensemble, denoted ǫ L, is defined as the minimal energy needed to go up from the ground state energy of H L,n to its lowest eigenvalue from the sector of total momentum. The IES in the thermodynamic limit, denoted ǫ, is defined by an appropriate limiting procedure as L, as we will explain further on. It is easy to see that the ground state energy of H L,n is attained at the zero momentum. Therefore, the IES both at finite volume and in thermodynamic limit is zero for = 0. It is believed that in the thermodynamic limit, at a fixed positive density, the IES as a function of the total momentum has the following features: ǫ is continuous in. In particular, the continuity at zero means that there is no gap in the spectrum there are excited states with an arbitrarily small positive energy. There exists c 1 > 0 such that ǫ lies above c 1. This implies that as long as the system travels with velocity less than c 1, the motion should be frictionless which is a manifestation of superfluidity. Again, this property does not hold in the noninteracting case. It has a linear cusp at its bottom, more precisely, it behaves lie c near = 0, where c > 0. Thus low-lying excitations propagate with a well defined sound velocity c. This differs from the noninteracting Bose gas, whose IES is proportional to the square of ; These properties of the Bose gas were predicted by Landau in the 40 s. Shortly thereafter, they were derived by a somewhat heuristic argument by Bogoliubov []. They seem to have been confirmed experimentally see e.g. [10]. Since Bogoliubov, many theoreticians have wored on this subject. Nevertheless, to our nowledge, no satisfactory rigorous analysis of the excitation spectrum exists. Only in the one-dimensional case, when the interaction v is a repulsive delta-function, the excitation spectrum can be described fairly explicitly see [8, 7] and at least one can prove that there is no gap in it.

4 Almost all theoretical wors on the excitation spectrum of the Bose gas instead of the correct Hamiltonian H L,n or its second-quantized version H L and the grandcanonical version Kµ L considered its modifications. They either replaced the zero mode by a c-number or dropped some of the terms, or did both modifications [1, 6, 4, 5, 1, 13]. These Hamiltonians have no independent physical justification apart from being approximations to H L in some uncontrolled way. In our paper we are not interested in any such modified Hamiltonians: all our statements will be related to the grand-canonical Hamiltonian Kµ L the natural second quantization of H L,n µn. Note however, that some of the quantities we study are not quite the ones we are interested in: instead of the bottom of the spectrum, most of the time we consider its upper bounds given by the expectation values between some restricted classes of vectors. In many papers on the Bose gas one does not obtain a cusp at the bottom of the excitation spectrum. One usually obtains a small gap between the ground state energy and the lowest excitation. There is no such gap in the calculations of Bogoliubov, as crude as they are. However, when one tries to improve on Bogoliubov s calculations, the gap usually appears. Yet, there are arguments that this gap is an artifact of these approximations, and in the complete treatment it should not appear [6, 4, 3]. In our approach, we expect that there is no gap if we consider the true IES defined mathematically by the joint spectrum of the grand-canonical Hamiltonian and the momentum in the thermodynamic limit. On the other hand, if we consider one of the approximations to the IES obtained not by changing the Hamiltonian, but choosing an appropriate family of test vectors, the gap seems to appear. One should remar that there exists a large rigorous literature on the Bose gas, see e.g. recent lecture notes of Lieb, Seiringer, Solovej and Yngvason [9] and references therein. This literature, however, is devoted mostly to the study of the ground state energy and not to the dependence of the excitation spectrum on momentum. In our paper we describe two methods that can be used to give upper bounds on the infimum of the joint spectrum of the Hamiltonian and the momentum that is, the sum of the ground state energy and the IES. The first, which we call the Squeezed States Approximation SSA, uses squeezed states and 1-particle excitations over squeezed states to obtain a variational estimate. The second, which we call the Improved Bogoliubov Approximation IBA is a modification of the SSA, which gives a less precise estimate, but is more convenient computationally. The method IBA is also well suited as the first step to a systematic perturbative treatment of the Bose gas. It leads to the construction of an effective Hamiltonian, which can serve as the main part of the full Hamiltonian Kµ L, whereas the reminder can be treated as a perturbation. By squeezed states we mean states obtained from the vacuum by a Bogoliubov translation and rotation. Note that the requirement of the translation symmetry restricts the choice of these transformations. In particular, only a translation of the zero mode is allowed and only pairs of particles of opposite momenta can be created. To our nowledge, in the context of the Bose gas, the idea of using squeezed states to bound the ground state energy first appeared in the paper of Robinson [11]. Robinson considered a slightly more general class of states quasi-free states. He noticed, however, that in the case he looed at it is sufficient to restrict to pure quasi-free states which coincide with squeezed states. Only an upper bound to the ground state energy is considered in [11]. We go one step further: we show how this method can be extended to obtain upper bounds on 3

5 the ground state energy plus the IES by using one-particle excitations over squeezed states. The structure of our paper is as follows: In Section we recall the original Bogoliubov calculations. They use the canonical approach. In Section 3 we describe the original Bogoliubov calculations adapted to the grand-canonical approach. The grand-canonical approach is used throughout the later part of the paper as well. Sections and 3 should be viewed as an extension of the introduction, included for a historical perspective. In Section 4 we describe and discuss our definitions of the excitation spectrum. We have little to say rigorously about these quantities. In Sections 5 and 6 we describe the Squeezed States Approximation and the Improved Bogoliubov Approximation. In Sections 7 and 8 we derive basic formulas for the Improved Bogoliubov Approximation. Acnowledgments. The research of all authors was partly supported by the EU Postdoctoral Training Program HPRN-CT The research of J.D. was also supported by the Polish grants SPUB17 and P03A 07 5 and was partly done during his visit to the Erwin Schrödinger Institute as a Senior Research Fellow. A part of the wor of H.C. was also done during his visit to the Erwin Schrödinger Institute. H.C. and J.D. also acnowledge support from the Statens Naturvidensabelige Forsningsråd grant Topics in Rigorous Mathematical Physics. The research of P.Z. was also supported by the grant P03B435.. Original Bogoliubov approach As realized by Bogoliubov, even if one is interested in properties of the Bose gas with a fixed but large number of particles, it is convenient to pass to the second quantized description of the system, allowing an arbitrary number of particles. To this end one introduces the second quantized form of 1.3, that is H L = a 1 x xa x dx + 1 a x a y vl x ya y a x dxdy,.1 acting on the symmetric Foc space Γ s L Λ. It is convenient to pass to the momentum representation: H L = 1 a a. + 1 δ ˆv 3 a 1 a a 3 a 4, 1,, 3, 4 where we used 1. to replace v L x with the Fourier coefficients ˆv. Note that ˆv = ˆv, and a x = 1/ eix a. Recall that the number and momentum operator are defined as N L := a a, P L := a a..3 H L,n and P L,n coincide with the operators H L and P L restricted to the eigenspace of N L with the eigenvalue n. We define the IES at the momentum as the infimum 4

6 of H L,n restricted to the eigensubspace of P L at after subtracting the ground state energy of H L. Let us proceed following the original method of Bogoliubov. We mae an ansatz consisting in replacing the operators a 0, a 0 with c-numbers: a 0a 0 α, a 0 α, a 0 α..4 We drop higher order terms. We will write for 0. H L ˆvα Then we replace α with ρ, setting ˆv0 + ˆv α a a a a + ˆvα a a + ˆv0 α 4..5 α + a a = ρ, and again we drop the higher order terms: H L 1 + ˆvρ a a + ˆvρ α α a a + α α a a + ˆv0 ρ..6 We perform the Bogoliubov rotation to separate the Hamiltonian into normal modes: a = c b s b, a = c b s b, where We obtain c = 1 1/ 1/ ρˆv , + ρˆv s = 1/ α ρˆv 1 1 α 1 + ρˆv 1/. H L ωbg b b + E bg,.7 where the excitation spectrum is ω bg = and the ground state energy equals Now, the IES is given by E bg = ˆv0 ρ ˆvρ,.8 ω bg 1 + ρˆv..9 ǫ bg = inf{ω bg ω bg n : n =, n = 1,,...}. 5

7 Note that under quite general assumptions on the potential, ǫ bg and ω bg coincide. In any case, they coincide for small. Thus, the IES has all the properties predicted by Landau, in particular, for small, it behaves as We can also compute that for small s Therefore, if Ψ denotes the ground state of.9, then The density of particles ρ equals ǫ ρˆv0..10 α α ρˆv0 1/4 1/..11 Ψ a a Ψ = s ρˆv01/. α + 1 s.1 We expect that for large L,.1 converges to ρ π d s d..13 But 1 is integrable only in dimension d > 1. Therefore, if we eep the density ρ fixed as L, then the asymptotics.10 and.11 cannot be satisfied for d = 1. To our nowledge,.11 and the above described problem of the Bogoliubov approximation in d = 1 was first noticed in [4]. 3. Grand-canonical version of the original Bogoliubov approach It was noted already by Beliaev [1], Hugenholz - Pines [6] and others that instead of studying the Bose gas in the canonical formalism, fixing the density, it is more convenient to use the grand-canonical formalism and fix the chemical potential. Then one can pass from the chemical potential to the density by the Legendre transformation. Physically, it corresponds to allowing the total number of particles to vary, eeping the chemical potential fixed. There are perhaps some physical situations, where it is more natural to eep the number of particles fixed. Nevertheless, mathematically, the chemical potential seems to be a more convenient parameter. More precisely, for a given chemical potential µ > 0, we define the grandcanonical Hamiltonian K L µ := H L µn L 3.1 = 1 µa a + 1 1,, 3, 4 δ ˆv 3 a 1 a a 3 a 4. If we determine the ground state energy E L µ of K L µ, then one can get the corresponding density by µ Eµ L µ = ρ. 3. 6

8 Again, following e.g. Zagrebnov-Bru [13], we mae the replacement.4 and drop higher order terms: K L µ + 1 µ + ˆvα +ˆv0 α 4 µ α. We perform the Bogoliubov rotation and obtain ˆv0 + ˆv α a a a a + ˆvα a a K L µ ωb b + E, 3.3 where the excitation spectrum is 1 ω = µ + α ˆv0 1 ˆv0 + ˆv µ + α and the ground state energy equals 3.4 E = ˆv0 α 4 µ α + 1 ω 1 ˆv0 + ˆv µ + α. 3.5 Note that ω 1 ˆv0 + ˆv µ + α 1 1 µ + α ˆv0 + ˆv 1 α 4 ˆv. Hence the term with in 3.5 is second order in the interaction. Therefore, eeping only first order terms we obtain E ˆv0 α 4 µ α. To find the lowest ground state energy for a given µ we compute which gives µ ˆv0 α. Thus 0 = α E ˆv0 α µ, E ˆv0 α 4 = µ ˆv0. Applying 3. we obtain µ = ρˆv0. If we insert this into 3.5 and 3.4, we obtain expressions identical to.9 and.8. Needless to say that both derivations of the IES for interacting Bose gas presented above are non-rigorous. 7

9 4. Infimum of the excitation spectrum a rigorous definition In this section we will try to introduce mathematical quantities related to the Bose gas, that in our opinion are relevant for its physical properties and which deserve to be called the infimum of the excitation spectrum IES. We will mostly use the grand-canonical formalism. We now that if we fix the number of particles to be n > 1, the operator H L,n from 1.3 has purely discrete spectrum and a non-degenerate ground state. Let us denote by E L,n the ground state energy in the box for n particles E L,n := inf sph L,n, where spk denotes the spectrum of an operator K. Then the ground state energy of E L µ is going to be E L µ = inf spk L µ = inf n 1 EL,n µn. 4.1 Both E L,n and E L are finite for a large class of potentials, which follows from a simple rigorous result, which we state and prove below. Theorem 4.1. Suppose that ˆv 0, ˆv0 > 0 and v0 <. Then H L,n and K L µ are bounded from below and E L,n ˆv0 n v0 n, 4. E L µ 1 v0 + µ. 4.3 ˆv0 Proof. Let us drop the subscript L and denote the self-interacting term in the Hamiltonian by := 1 ˆva a a a 4.4,, Define the operator N := a a. 4.5 Then, by a simple commutation, using that N 0 = N and that ˆv is non-negative, we obtain H = 1 ˆvN N v0 N ˆv0 N v0 N, which proves 4.. Moreover, which proves 4.3. K µ ˆv0 N = 1 ˆv0 v0 + µ N N 1 v0 + µ ˆv0 1 v0 + µ, ˆv0 Note that the operators K L µ and P L commute. Therefore we can define their joint spectrum spk L µ, P L. For L π L Zd which is the spectrum of P L, we define the IES in the box ǫ L L := inf{s : s, L spk L µ, P L } E L

10 In the limit L, the momentum lattice π L Zd converges in some sense to the continuous space R d. It is not obvious how to define ǫ in this limit. We propose to define it as follows: For R d we define the IES at the thermodynamic limit ǫ := sup δ>0 lim inf L inf ǫ L L L π L Zd, L <δ. 4.7 Thus first we consider a window in the momentum space of diameter δ. We obtain the quantity lim inf inf L L <δ ǫl L, which bounds from below the IES in this window, and increases as δ becomes smaller. Then tae the supremum or, equivalently, the limit of the above quantity as δ ց 0. Another natural definition would be: ǫ := inf δ>0 lim sup L sup L π L Zd, L <δ ǫ L L. 4.8 This bounds from above the IES in the box in every window of diameter δ. Clearly, ǫ ǫ. Although it is not obvious that ǫ and ǫ are actually equal, we only consider ǫ in this paper. Now let us formulate some results and conjectures on the behavior of the IES. Proposition 4.. At zero total momentum, the excitation spectrum has a global minimum where it equals zero: ǫ L 0 = ǫ0 = 0. Proof. Under the conditions imposed on our potential v, each E L,n is a nondegenerate eigenvalue of H L,n, and H L,n commutes with the total momentum and space inversion. Thus each E L,n corresponds to zero total momentum, and so does the minimum over n 1 in 4.1. Hence ǫ L 0 = 0. Moreover, for 0 L < δ we have ǫ L L 0, therefore the infimum in 4.7 is always 0, for all δ and L. Let us now formulate some conjectures about ǫ: Conjecture 4.3. We expect the following statements to hold true: 1 The map R d ǫ R + is continuous. Let R d. If {L s } s 1 R diverges to +, then for every sequence s π L s Z d which obeys s, we have that ǫ Ls s ǫ. 3 There exists c 1 > 0 such that ǫ > c 1 at all. 4 There exists c > 0 such that ǫ c for small. Statements 1 and can be interpreted as some ind of a spectral thermodynamic limit. Note that if 1 and are true around = 0, then we can say that there is no gap in the excitation spectrum Proposition 4. stated that ǫ0 = 0. The third statement implies the superfluidity of the Bose gas. The fourth statement claims that low energy excitations have a well defined speed. Throughout most of our paper, the chemical potential µ is considered to be the natural parameter of our problem. The IES, as we have defined it, should be written as ǫ, µ. If we want to pass to canonical conditions fixed density, then we need to prove that E L µ eµ := lim L L d exists and defines a smooth, concave function of µ. Denote by µ ρ the unique solution to the equation µ eµ ρ = ρ. Then ǫ, µ ρ will give the IES in the thermodynamic limit and canonical conditions. 9

11 5. Infimum of the excitation spectrum in the squeezed states approximation In this section we describe an approximate method, which will give a rigorous upper bound on E and E + ǫ. The main idea of this method is the use of the so-called squeezed states. See Appendix A for a brief summary of basic properties of squeezed states. Let α C and π L Zd θ C be a square summable sequence with θ = θ. Set W α := e αa 0 +αa0, U θ := e 1 θa a + 1 θaa Then U α,θ := W α U θ is the general form of a Bogoliubov transformation commuting with P. Let Ω denote the vacuum vector. Note that Ψ α,θ := U α,θω is the general form of a squeezed vector of zero momentum. ectors of the form Ψ α,θ, := U α,θa Ω are normalized and have momentum, that means P L Ψ α,θ, = 0. By the Squeezed States Approximation SSA we will mean applying the variational method using only vectors lie Ψ α,θ and Ψ α,θ,. We define the SSA ground state energy in the box E L ssa := inf α,θ Ψ α,θ K L µψ α,θ. For L π L Zd we define the SSA IES in the box ǫ L ssa L := inf α,θ Ψ α,θ, L K L µ Ψ α,θ, L E L ssa and for R d we define the SSA IES in the thermodynamic limit ǫ ssa := sup δ>0 lim inf L Clearly, from the mini-max principle we obtain: inf ǫ L ssa L L π L Zd, L <δ E L E L ssa, E L + ǫ L L E L ssa + ǫ L ssa L. Conjecture 5.1. We believe the following statements to hold true: 1 ǫ L ssa L > 0 for all L π L Zd. ǫ ssa > 0, for all R d. 3 The map R d ǫ ssa R + is continuous. 4 Let R d. If {L s } s 1 R diverges to +, then for every sequence s π L s Z d which obeys s, we have that ǫ Ls ssa s ǫ ssa.. Thus we conjecture that the Squeezed States Approximation does not capture the behavior at the bottom of the IES predicted by Landau, in particular ǫ ssa 0 > 0. 10

12 6. A rigorous version of the Bogoliubov approximation The method of SSA is still not very convenient computationally. In this section we describe a less precise method of finding an upper bound to E + ǫ, which seems, however, more convenient in practical calculations. This method resembles closely the original Bogoliubov method. It consists of similar steps: it introduces a c-number α for the zero mode, it involves a Bogoliubov rotation and dropping terms of higher order in creation and annihilation operators. In contrast to the original Bogoliubov approach, our approach leads to a rigorous upper bound. Therefore, we call it the Improved Bogoliubov Approximation IBA. For any L large enough we assume that the infimum of Ψ α,θ Kµ L Ψ α,θ is attained. Therefore, for any large enough L we can fix α L, θ L, where α L C and π L Zd θ L C, such that Essa L = Ψ αl,θ L KL µψ αl,θ L. We define for L π L Zd For R d we set Clearly, ǫ L iba L := Ψ α L,θ L, L Ω K L µ Ψ α L,θ L, L E L ssa. ǫ iba := sup δ>0 lim inf L ǫ L ssa ǫ L iba, inf L π L Zd, L <δ ǫ L iba L ǫ ssa ǫ iba. Again, we have a conjecture similar to the conjecture about the Squeezed States Approximation: Conjecture 6.1. We believe the following statements to hold true: 1 ǫ L iba L > 0 for all L π L Zd. ǫ iba > 0 for all R d. 3 The map R d ǫ iba R + is continuous. 4 Let R d. If {L s } s 1 R diverges to +, then for every sequence s π L s Z d which obeys s, we have that ǫ Ls iba s ǫ iba.. In order to implement the above method, let us see that Ψ α,θ K L µ Ψ α,θ = Ω U α,θ K L µ U α,θω, Ψ α,θ,l K L µψ α,θ,l = a L Ω U α,θ K L µ U α,θa L Ω. The Hamiltonian after the Bogoliubov transformation can be Wic ordered: U α,θ Kµ L Uα,θ = B L + C L a 0 + C L a O L a a + 1 O L a a + D L a a + terms higher order in a s. 6.1 Then we easily see that Ψ α,θ K L µ Ψ α,θ = B L, Ψ α,θ,l K L µ Ψ α,θ,l = B L + D L L. If we require that B L attains its minimum, then we will later on show that automatically C L and O L L vanish for all L. Besides, we get that B L = E L ssa and D L = ǫ L iba L. 11

13 Note that it is natural to introduce the effective Hamiltonian K L iba := E L ssa + ǫ L ibab b, 6. where b = U α,θ a Uα,θ. One-particle states of the Hamiltonian KL iba have the excitation spectrum coinciding with IES of the full Hamiltonian K L in the Improved Bogoliubov Approximation. The method IBA seems especially convenient, not only as a means of obtaining an upper bound, but also as the first step to a systematic perturbative treatment of the Bose gas. We can use the effective Hamiltonian 6. as the main part of the full Hamiltonian Kµ L, treating the higher order terms dropped when defining 6. as a perturbation. 7. Bogoliubov translation of the Hamiltonian In the sequel we drop the superscript L. We apply the Bogoliubov transformation in two steps. First we apply W α := e αa 0 +αa0. We have W α a 0 W α = a 0+α. Hence it suffices to displace the zero mode and our Hamiltonian taes the form: W α K µ Wα = µ α + ˆv0 α ˆv0 + α µ αa 0 + αa ˆv0 + ˆv µ + α a a + ˆv α a a + a a α +, ˆv αa + a a + αa a a + + δ ˆv 3 4 a 1 a a 3 a 4. 1,, 3, 4 Before the displacement of the zero mode the Hamiltonian possessed global phase symmetry. After the displacement, the Hamiltonian will still have it if we also rotate the c-number α with the same phase as each operator a. 8. Bogoliubov rotation of the Hamiltonian Next we perform the Bogoliubov rotation by U θ. We set Note that c := cosh θ, s := θ θ sinh θ. U θ a U θ = c a s a, U θ a U θ = c a s a 8.1 We have s = s and c = c = 1 + s s. 1

14 The result of the rotation is: B = µ α + ˆv0 α 4 + ˆv + ˆv0 µ + α s ˆv α s c + α s c C = + ˆv c s c s, + ˆv0 + ˆv s s ;, ˆv0 α µ + ˆv0 + ˆv s αc 0 αs 0 + ˆv αs 0c s αc 0 c s ; O = D = + + ˆv ˆv0 + ˆv µ + α + α ˆv s c c α ˆv s c s ; ˆv µ + ˆv0 + ˆv ˆv + ˆv0 s c s α + ˆv0 + ˆv s c + s α ˆv s c c s α ˆv s c s c. ˆv ˆv The main intermediate step of the calculations leading to the above result is described in Appendix B. 13

15 9. Conditions arising from minimization of B We demand that B attains a minimum. To this end we first compute the derivatives with respect to α and α: The first derivatives with respect to α and α are: α B = µ + ˆv0 α + ˆv0 + ˆv s α Note that: α B = so that the condition: ˆv s c α, µ + ˆv0 α + ˆv s c α. C = c 0 α B s 0 α B ˆv0 + ˆv s α α B = α B = entails C = 0. Computing the derivative with respect to s, s we can use s c = s c, s c = s c. s B = s B = ˆv0 + ˆv s s + ˆv α + ˆv c s c + s c + ˆv α + ˆv s c s c ˆv0 + ˆv µ + α + ˆv0 + ˆv s s ˆv0 + ˆv µ + α + + ˆv α + ˆv c s c + s c + ˆv α + ˆv s c s. c One can calculate that: O = c + s s B s s B c Thus s B = s B = 0 entails O = 0. c 14

16 All the above conditions can be rewritten using: f : = g : = Note that f is real. Then: and: This implies that Note also that: µ without minimizing The condition 9.1 yields ˆv0 + ˆv + α + ˆv + ˆv0 s α ˆv ˆv s c. 9. s B = f s g s B = f s g c + s c c + s c s g 4c g s 4c. s s B s s B = c g s g s g s = g s 9.3 D = f c + s c s g + s g 9.4 O = c s f + s g + c g 9.5 µ = ˆv0 α + ˆv0 + ˆv s α ˆv α s c. 9.6 This allows to eliminate µ from the expression for f : f : = + α ˆv + ˆv ˆv s + α ˆv α s c. 9.7 We will eep α instead of µ as the parameter of the theory, hoping that one can later on express µ in terms of α. The condition O = 0 together with 9.3 imply Therefore, 1 + s s f = 1 + s g. 4 s 4 f g + 4 s f g g = 0. If we neglect the solution s f = g = 0, we obtain f g s = f f g f g. Then we can compute s together with the phase: g 1 s = sgnf 1 g 1/ 1/ g 15 f

17 9.8, together with the definition of g, f given in 9. and 9.7 can be viewed as the fixed point equation for s depending on the parameter α. Note that it seems that in the thermodynamic limit one should tae α = ρ 0 e iφ0, for some fixed parameter ρ 0 having the interpretation of the density of the condensate and a fixed phase φ 0. Then one could expect that s will converge to a function depending on R d in a reasonable class and we can replace 1 1 by π d. d Let us compute the ground state energy in the Squeezed States Approximation. Inserting 9.6 to the expression for B we obtain B = ˆv s α + s + ˆv ˆv v s s ˆv α α s c s ˆv c s c s. Using 9.6 again to eliminate α in favor of µ, and then computing the derivative with respect to µ we obtain µ B = α + s. Therefore, the grand-canonical density is given by ρ = α + s. 9.9 Let us now compute the IES in the Improved Bogoliubov Approximation: D = sgnf f g hence f is positive and we can drop sgnf. We also compute: ˆv0 D0 = ˆv α s c Thus it seems to imply D0 > 0, which would mean that we have a gap in this approximation. Concluding remars. The analysis that we performed suggests a number of physically relevant open problems, which could be studied by rigorous methods. The most interesting among them are perhaps those related to Conjecture 4.3 about the true IES. Nevertheless, we believe that the problems about the methods SSA and IBA described in Conjectures 5.1 and 6.1 are also physically relevant and worth studying. It is liely that they are more accessible to rigorous investigations. A number of mathematical open problems is suggested by the analysis of the previous section. In particular it should be possible to study mathematically the fixed point equation 9.8 and to determine whether it has a solution and whether this solution has a limit as L in some reasonable function space. 16

18 Appendix A. Bogoliubov transformations In this appendix we recall the well-nown properties of Bogoliubov transformations and squeezed vectors. For simplicity we restrict ourselves to one degree of freedom. Let a, a are creation and annihilation operators and Ω the vacuum vector. Recall that [a, a ] = 1 and aω = 0. Here are the basic identities for Bogoliubov translations and coherent vectors. Let W := e αa +αa. Then WaW = a + α, Wa W = a + α, W Ω = e α e αa Ω. Here are the basic identities for Bogoliubov rotations and squeezed vectors. Let U := e θ a a + θ aa. Then UaU Ua U = cosh θ a + θ θ sinh θ a, = cosh θ a + θ sinh θ a, θ U Ω = 1 + tanh θ 1 4 e θ θ tanh θ a a Ω. ectors obtained by acting with both Bogoliubov translation and rotation will be also called squeezed vectors. Appendix B. Computations of the Bogoliubov rotation In this appendix we give the computations of the rotated terms in the Hamiltonian used in Section 8. U θ a a U θ = s U θ a a U θ = s c +c a a c s a a c s a a + s a a, +c a a c s a a c s a a + s a a ; U θ a a Uθ = s c +c a a c s a a c s a a + s a a ; U θ a + a a U θ = c 0 s δ + s δ + s 0 c s δ + a 0 s 0 s δ + s δ + c 0 c s δ + a 0 +higher order terms; U θ a + a a U θ = c 0 s δ + s δ + s 0 c s δ + a 0 s 0 s δ + s δ + c 0 c s δ + a 0 +higher order terms; 17

19 δ U θ a 1 a a 3 a 4 U θ = c 1 s 1 c 3 s 3 δ 1 + δ s 1 s δ 1 3 δ 4 + δ 1 4 δ 3 + s 1 c 1 s 3 c 3 a 3 a 3 s 3 a 3 a 3 c 3 a 3 a 3 + s 3 c 3 a 3 a 3 +s 3 c 3 s 1 c 1 a 1 a 1 s 1 a 1 a 1 c 1 a 1 a 1 + s 1 c 1 a 1 a 1 + δ 1 + δ s c 1 a 1 a 1 c 1 s 1 a 1 a 1 c 1 s 1 a 1 a 1 + s 1 a 1 a 1 + s 1 c a a c s a a c s a a + s a a δ 1 3 δ 4 + δ 1 4 δ 3 +higher order terms; References [1] Beliaev, S.T.: Energy spectrum of a non-ideal Bose gas, JETP-USSR 7 no [] Bogoliubov, N. N.: J. Phys. USSR ; J. Phys. USSR , reprinted in D. Pines The Many-Body Problem New Yor, W.A. Benjamin 196 [3] Bogoliubov, N. N.: Quasi-averages in problems of statistical mechanics, in colection of papers, ol. 3, Nauova Duma, Kiev 1971, [4] Gavoret, J., Nozières, P.: Structure of the perturbation expansion for the Bose liquid at zero temperature Ann. of Phys [5] Girardeau, M., Arnowitt, R.: Theory of Many-Boson Systems: Pair Theory, Phys. Rev [6] Hugenholtz, N. M., Pines, D.: Ground state energy and excitation spectrum of a system of interacting bosons, Phys. Rev [7] Lieb, E.H.: Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Phys. Rev. 130 no [8] Lieb, E.H., Liniger, W.: Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Phys. Rev. 130 no [9] Lieb, E. L., Seiringer, R., Solovej, J. P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation Oberwolfach Seminars, Birhuser erlag, 005 [10] C. Raman, M. Kohl, R. Onofrio, D.S. Durfee, C.E. Kulewicz, Z Hadzibabic and W. Ketterle, Phys. Rev. Lett. 83, ; Evidence for a Critical elocity in a Bose Einstein Condensed Gas. [11] Robinson, D. W.: On the ground state energy of the Bose gas, Comm. Math. Phys [1] Taano, F.: Excitation spectrum in many-boson systems, Phys. Rev [13] Zagrebnov,.A., Bru, J.B.: The Bogoliubov model of wealy imperfect Bose gas, Phys. Rep. 350 no H.D. Cornean Dept. of Math., Aalborg University, Fredri Bajers ej 7G, 90 Aalborg, Denmar address: cornean@math.aau.d J. Derezińsi Dept. of Math. Methods in Phys., University of Warsaw, Hoza 74, Warszawa, Poland address: Jan.Derezinsi@fuw.edu.pl P. Ziń Institute of Theoretical Physics, Uniwersity of Warsaw, Hoza 69, Warszawa, Poland address: Pawel.Zin@fuw.edu.pl 18