Abstract
We consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov’s approximation of the ground state energy and the ground state.
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1 Introduction
We consider a homogeneous system of N bosons on the unit torus \({{\mathbb {T}}}^d\), for any dimension \(d\ge 1\). The system is governed by the mean-field Hamiltonian
which acts on the bosonic Hilbert space
Here the kinetic operator \(-\Delta \) is the usual Laplacian (with periodic boundary conditions). The interaction potential w is a real-valued, even function. We assume that its Fourier transform is non–negative and integrable, namely
In particular, w is bounded. Since w is even, \(\widehat{w}\) is also even.
Under the above conditions, \(H_N\) is well defined on the core domain of smooth functions. Moreover, it is well-known that \(H_N\) is bounded from below and can be extended to be a self-adjoint operator by Friedrichs’ method. The self-adjoint extension, still denoted by \(H_N\), has a unique ground state \(\Psi _N\) (up to a complex phase) which solves the variational problem
Here \(\langle \cdot , \cdot \rangle \) is the inner product in \({\mathcal {H}}^N\).Footnote 1
In the present paper, we are interested in the asymptotic behavior of the ground state \(\Psi _N \in {\mathcal {H}}^N\) of \(H_N\) in the limit when \(N\rightarrow \infty \). More precisely, we will focus on the one-body density matrix \(\gamma _{\Psi _N}^{(1)}\) which is a trace class operator on \(L^2({{\mathbb {T}}}^d)\) with kernel
Note that \(\gamma _{\Psi _N}^{(1)}\ge 0\) and \(\mathrm{Tr}\gamma _{\Psi _N}^{(1)}=N\).
1.1 Main result
Our main theorem is
Theorem 1
(Ground state density matrix) Assume that \(0\le {\widehat{w}}\in \ell ^1( (2\pi {\mathbb {Z}})^d )\). Then the ground state \(\Psi _N\) of the Hamiltonian \(H_N\) in (1) satisfies
where
Here \(|u\rangle \langle u|\) is the orthogonal projection on u. We use the bra-ket notation, where \(|u\rangle =u\) is a vector in the Hilbert space \({\mathcal {H}}\) and \(\langle u|\) is an element in the dual space of \({\mathcal {H}}\) which maps any vector \(v\in {\mathcal {H}}\) to the inner product \(\langle u,v\rangle _{{\mathcal {H}}}\).
To the leading order, our result implies Bose–Einstein condensation, namely
in the trace norm. This result is well-known and it follows easily from Onsager’s inequality
(see [18]). The significance of Theorem 1 is that it gives the next order correction to \(\gamma ^{(1)}_{\Psi _N}\), thus justifying Bogoliubov’s approximation in a rather strong sense as we will explain.
1.2 Bogoliubov’s approximation
It is convenient to turn to the grand canonical setting. Let us introduce the Fock space
For any Fock space vector \(\Psi =(\Psi _n)_{n=0}^\infty \in {\mathcal {F}}\) with \(\Psi _n\in {\mathcal {H}}^n\), we define its norm by
and define the particle number expectation by
In particular, the vacuum state \(|0\rangle = (1,0,0,\ldots )\) is a normalized vector on Fock space which has the particle number expectation \(\langle 0| {\mathcal {N}}|0\rangle =0\).
For any \(f\in {\mathcal {H}}\), the creation operator \(a^*(f)\) on Fock space maps from \({\mathcal {H}}^n\) to \({\mathcal {H}}^{n+1}\) for every \(n\ge 0\) and satisfies
Its adjoint is the annihilation operator a(f), which maps from \({\mathcal {H}}^n\) to \({\mathcal {H}}^{n-1}\) for every \(n\ge 0\) (with convention \({\mathcal {H}}^{-1}=\{0\}\)) and satisfies
We will denote by \(a_p^*\) and \(a_p\) the creation and annihilation operators with momentum \(p\in 2\pi {\mathbb {Z}}^d\), namely
They satisfy the canonical commutation relation (CCR)
where \([X,Y]=XY-YX.\)
The creation and annihilation operators can be used to express several operators on Fock space. For example, the number operator can be written as
Similarly, the Hamiltonian \(H_N\) in (1) can be rewritten as
The right side of (4) is an operator on Fock space, which coincides with (1) when being restricted to \({\mathcal {H}}^N\). In the following we will only use the grand–canonical formula (4).
In 1947, Bogoliubov [4] suggested a heuristic argument to compute the low-lying spectrum of the operator \(H_N\) by using a perturbation around the condensation. Roughly speaking, he proposed to first substitute all operators \(a_0\) and \(a_0^{*}\) in (4) by the scalar number \(\sqrt{N}\) (c-number substitutionFootnote 2), and then ignore all interaction terms which are coupled with coefficients of order \(o(1)_{N\rightarrow \infty }\). All this leads to the formal expression
where
Note that the expression (5) is formal since \(H_N\) acts on the N-body Hilbert space \({\mathcal {H}}^N\) while the Bogoliubov Hamiltonian \({\mathbb {H}}_{\mathrm{Bog}}\) acts on the excited Fock space
where we have introduced the projections
In particular, unlike \(H_N\), the quadratic Hamiltonian \({\mathbb {H}}_{\mathrm{Bog}}\) does not preserve the number of particles. Nevertheless, \({\mathbb {H}}_{\mathrm{Bog}}\) can be diagonalized by the following unitary transformation on \({\mathcal {F}}_+\)
where the coefficients \(\beta _p>0\) are determined by
In fact, by using the CCR (3) it is straightforward to check that
where
Consequently,
where
Note that the assumption \(0\le \widehat{w} \in \ell ^1(2\pi {{\mathbb {Z}}}^d)\) ensures that \(E_{\mathrm{Bog}}\) is finite. Moreover we have the uniform bounds
Thus Bogoliubov’s approximation predicts that the ground state energy of \(H_N\) is
In 2011, Seiringer [18] gave the first rigorous proof of (11). He also proved that the low-lying spectrum of \(H_N\) is given approximately by the elementary excitation \(e_p\). These results have been extended to inhomogeneous trapped systems in [11], to more general interaction potentials in [12], to a large volume limit in [9], and to situations of multiple-condensation in [14, 17].
Let us recall the approach in [12] which also provides the convergence of the ground state of the mean-field Hamiltonian \(H_N\) in (1). Mathematically, the formal expression (5) can be made rigorous using the unitary operator introduced in [12]
which is defined by
Recall from [12, Proposition 4.2] that
where \({\mathcal {N}}_+\) is the number operator on the excited Fock space \({\mathcal {F}}_+\),
Thus \(U_N\) implements the c-number substitution in Bogoliubov’s argument because it replaces \(a_0\) by \(\sqrt{N-{\mathcal {N}}_+}\approx \sqrt{N}\) (we have \({\mathcal {N}}_+\ll N\) due to the condensation). Then the formal expression (5) can be reformulated as
which is rigorous since the operators on both sides act on the same excited Fock space. By justifying (14), the authors of [12] recovered the convergence of eigenvalues of \(H_N\) first obtained in [18], and also obtained the convergence of eigenfunctions of \(H_N\) to those of \({\mathbb {H}}_{\mathrm{Bog}}\). In particular, for the ground state, we have from [12, Theorem 2.2] that
where \(|0\rangle \) is the vacuum in Fock space. The convergence (15) holds strongly in norm of \({\mathcal {F}}_+\), and also strongly in the norm induced by the quadratic form of \({\mathbb {H}}_{\mathrm{Bog}}\) in \({\mathcal {F}}_+\). In particular, this implies the convergence of one-body density matrix
in trace class (see (68) for a detailed explanation). Since \(\mathrm{Tr}\gamma _{\Psi _N}^{(1)}=N\), (16) is equivalent to
Recall that \(P=|u_0\rangle \langle u_0|=1-Q\). The formula (17) looks similar to the result in Theorem 1, except that the cross term \(P \gamma _{\Psi _N}^{(1)}Q + Q \gamma _{\Psi _N}^{(1)}P\) is missing. Putting differently, to get the result in Theorem 1 we have to show that
As explained in [12, Eq. (2.19)], from (16) and the Cauchy–Schwarz inequality one only obtains that the left side of (18) is of order \(O(\sqrt{N})\). Moreover, (18) implies that
thus answering an open question in [13]. As explained in [13, Section 5], (19) would follow if we could replace \(U_N \Psi _N\) by \(U_B |0\rangle \) (which is a quasi-free state, and thus satisfies Wick’s Theorem [19, Chapter 10]). However, the norm convergence (15) is not strong enough to justify (19).
1.3 Outline of the proof
To prove Theorem 1 we have to extract some information going beyond Bogoliubov’s approximation. Roughly speaking, we will refine (14) by computing exactly the term of order \(O(N^{-1})\). Our proof consists of three main steps.
Step 1 (Excitation Hamiltonian). After implementing the c-number substitution, instead of ignoring all terms with coefficients of order \(o(1)_{N\rightarrow \infty }\), we will keep all terms of order \(O(N^{-1})\). More precisely, in Lemma 7 below we show that
in an appropriate sense, where
The formula (20) is obtained by a direct computation using the actions of \(U_N\) as in [12], plus an expansion of \(\sqrt{N-{\mathcal {N}}_+}\) and \(\sqrt{(N-{\mathcal {N}}_+)(N-{\mathcal {N}}_+-1)}\) in the regime \({\mathcal {N}}_+ \ll N\). The advantage of using \({\mathcal {G}}_N\) is that it is well-defined on the full Fock space \({\mathcal {F}}_+\). This idea has been used to study the norm approximation for the many-body quantum dynamics in [8].
Step 2 (Quadratic transformation). Then we conjugate the operator on the right side of (20) by the Bogoliubov transformation \(U_B\) in (7). In Lemma 8 we prove that
where
and \(R_2\) is an error term whose expectation against the ground state is of order \(O(N^{-3/2})\).
Note that in \({\mathcal {C}}_N\) we keep only cubic terms with three creation operators or three annihilation operators. These are the most problematic terms. All other cubic terms, as well as all quartic terms, are of lower order and can be estimated by the Cauchy–Schwarz inequality (the quartic terms always come with a factor \(N^{-1}\) instead of \(N^{-1/2}\) and this helps).
As we will see, the energy contribution of the cubic term \({\mathcal {C}}_N\) is of order \(O(N^{-1})\). Thus (21) implies that
which improves (11). Moreover, for the ground state we have
which in turn implies the norm approximation (up to an appropriate choice of the phase factor for \(\Psi _N\))
and the following bound on the one-body density matrix
Unfortunately the latter bound is still weaker than (18). Thus the desired result (18) cannot be obtained within Bogoliubov’s theory.
Step 3 (Cubic transformation). To factor out the energy contribution of the cubic term \({\mathcal {C}}_N\) in (21), we will use a cubic transformation. It is given by
where
From the assumption \({\widehat{w}} \in \ell ^1(2\pi {{\mathbb {Z}}}^d)\) and the bounds (10) we have the summability
Here we insert the cut-off \({\mathbb {1}}^{\le N}\) in the definition of \(U_S\) to make sure that it does not change the particle number operator \({\mathcal {N}}_+\) too much; see Lemma 5 for details.
The choice of the cubic transformation above can be deduced on an abstract level. Consider an operator of the form
where X stands for some perturbation. Then, in principle, we can remove X by conjugating A with \(e^{S}\) provided that
and that \([S,[S,A_0]]=-[S,X]\) is small in an appropriate sense. This can be seen by the simple expansions
and
In our situation, \(A_0= \sum _{p\ne 0} e(p) a_p^* a_p\) and \(X={\mathcal {C}}_N\), allowing to find S explicitly in (25).
In Lemma 9 we prove that
with an error term \(R_3\) whose expectation against the ground state is of order \(O(N^{-3/2})\). This allows us to obtain the following improvements of (22), (23) and (24).
Theorem 2
(Refined ground state estimates) Assume that \(0\le {\widehat{w}}\in \ell ^1( (2\pi {\mathbb {Z}})^d )\). Then the ground state energy of the Hamiltonian \(H_N\) in (1) satisfies
Moreover, if \(\Psi _N\) is the ground state of \(H_N\), then \(\Phi = U_S U_B U_N \Psi _N\) satisfies
Consequently, we have the norm approximation (up to an appropriate choice of the phase factor for \(\Psi _N\))
As we will explain, Theorem 2 implies (18) and thus justifies Theorem 1.
The idea of using cubic transformations has been developed to handle dilute Bose gases in [1,2,3, 16, 20], where the interaction potential has a much shorter range but the interaction strength is much larger in its range. In this case, the contribution of the cubic terms is much bigger, and Bogoliubov’s approximation has to be modified appropriately to capture the short-range scattering effect. Results similar to (11) have been proved recently for the Gross-Pitaevskii limit [2] and for the thermodynamic limit [10, 20]. It is unclear to us how to extend Theorem 1 to the dilute regime.
Our work shows that in the mean-field regime, in contrast to the dilute regime, the cubic terms are smaller, and they actually contribute only to the next order correction to Bogoliubov’s approximation (there are also some quadratic and quartic terms which contribute to the same order of the cubic term). On the other hand, it is interesting that the contribution of the cubic terms is not visible in the expansion of the one-body density matrix in Theorem 1; putting differently the approximation in Theorem 1 can be guessed using only Bogoliubov’s theory (although its proof requires more information).
There have been also remarkable works concerning higher order expansions in powers of \(N^{-1}\) in the mean-field regime; see [15] for a study of the ground state, [7] for the low-energy spectrum, and [5, 6] for the quantum dynamics. These works are based on perturbative approaches which are very different from ours. Note that the method of Bossmann, Petrat and Seiringer in [7] also gives access to the higher order expansion of the reduced density matrices (see [7, Eq. (3.15)] for a comparison). We hope that our rather explicit strategy complements the previous analysis in [5,6,7, 15] concerning the correction to Bogoliubov’s theory in the mean-field regime.
1.4 Organization of the paper
In Sect. 2 we will derive some useful estimates for the particle number operator \({\mathcal {N}}_+\). Then we analyze the actions of the transformations \(U_N\), \(U_B\), \(U_S\) in Sects. 3, 4 and 5, respectively. Finally, we prove Theorem 2 in Sect. 7 and conclude Theorem 1 in Sect. 7.
2 Moment estimates for the particle number operator
In this section we justify the Bose–Einstein condensation by showing that the ground state has a bounded number of excited particles. As explained in [18], the uniform bound on the expectation of \({\mathcal {N}}_+\) follows easily from Onsager’s inequality (2). For our purpose, we will need uniform bounds for higher moments of \({\mathcal {N}}_+\). The following lemma is an extension of [13, Lemma 5].
Lemma 3
(Number of excited particles) If \(\Psi _N\) is the ground state of \(H_{N}\), then
Proof
As in [13, Lemma 5], from the operator inequality
we obtain
for \(s=1,2,3\). Let us assume that \(s\in {\mathbb {N}}\) is even. We will show that \(\langle \Psi _N, {\mathcal {N}}_+^{s+1} \Psi _N \rangle \le C\).
Since \(\Psi _N\) is a ground state of \(H_N\), it solves the Schrödinger equation
Consequently, we get the identity
The left side of (30) can be estimated using (28) and (29) as
For the right side of (30), since
using (4) and the CCR (3) we write
Now we take the expectation against \(\Psi _N\) and estimate. For the first term on the right side of (32), by the Cauchy–Schwarz inequality, we get for a given j
Here we have used that \(a_0 a_0\) commutes with \({\mathcal {N}}_+\), that \(a_0^*a_0 \le N\) on \({\mathcal {H}}^N\) and that \(\sum |{\widehat{w}}(\ell )|^2=\Vert w\Vert _{L^2}^2<\infty \). Similarly, for the second term, we have
as before. For the third term, we can bound
and proceed similarly for other terms. Thus in summary, from (32) we get
Inserting (31) and (33) into (30), we obtain
By the Cauchy–Schwarz inequality
we get
which implies
We can now use
and obtain
Telescoping this inequality and using [13, Lemma 5] we arrive at a bound on \(\left\langle \Psi _N, {\mathcal {N}}_+^{s+1} \Psi _N \right\rangle \) that is uniform in N. This gives the desired result for odd powers of \({\mathcal {N}}_+\). Finally, using (34), we obtain the bound for any \(s\in {\mathbb {N}}\) and this ends the proof. \(\square \)
In order to put Lemma 3 in a good use, we will also need the fact that the moments of \({\mathcal {N}}_+\) are essentially stable under the actions of the Bogoliubov transformation and the cubic transformation.
Lemma 4
Let \(U_B\) be given in (7). Then
Lemma 5
Let \(U_S=e^S\) be given in (25). Then for all \(t\in [-1,1]\) and \(k\in {\mathbb {N}}\),
The results in Lemma 4 and Lemma 5 are well-known. For the completeness, let us quickly explain the proof of Lemma 5, following the strategy in [2, Proposition 4.2] (the proof of Lemma 4 is similar and simpler).
Proof of Lemma 5
Take a normalized vector \(\Phi \in {\mathcal {F}}_+\) and define
Then
with
Here we have used
It is obvious that \(|\Theta _k({\mathcal {N}}_+)| \le C_k ({\mathcal {N}}_+ +1)^{k-1}\). Combining with the summability (27) and the Cauchy–Schwarz inequality we obtain
Thanks to the cut-off, we can bound
Thus (37) implies that
From Grönwall’s lemma, it follows that
Since the latter bound is uniform in \(\Phi \), we get the desired operator inequality. \(\square \)
We will also need the following refinement of Lemma 5.
Lemma 6
Let \(U_S=e^S\) be given in (25). Then for all \(t\in [-1,1]\) and \(k\in {\mathbb {N}}\),
Proof
Take a normalized vector \(\Phi \in {\mathcal {F}}_+\) and define
Then proceeding similarly to (37), we have
with f(t) being defined in the proof of Lemma 5. Using (38) and the Cauchy–Schwarz inequality we obtain
From Grönwall’s lemma, it follows that
The latter bound is uniform in \(\Phi \) and it implies the desired conclusion. \(\square \)
3 Excitation Hamiltonian
In this section, we study the action of the transformation \(U_N\) in (12). By conjugating \(H_N\) with \(U_N\), we can factor out the contribution of the condensation. More precisely, we have
Lemma 7
We have the operator identity on \({\mathcal {F}}_{+}^{\le N}\)
where
and the error term \(R_1\) satisfies the quadratic form estimate
Moreover, we have the operator inequality on \({\mathcal {F}}_{+}\)
Proof
A straightforward computation using the relations (12) shows that
This operator identity holds on \({\mathcal {F}}_+^{\le N}\). For further analysis, we will expand \(\sqrt{N-{\mathcal {N}}_+}\) and \(\sqrt{(N-{\mathcal {N}}_+)(N-{\mathcal {N}}_+ -1)}\), making the effective expressions well-defined on the whole Fock space \({\mathcal {F}}_+\). This idea has been used before in [8]. Here it suffices to use
and
The operator inequalities (40) and (41) hold on \({\mathcal {F}}_+^{\le N}\). Thus we can write
with \( {\mathcal {G}}_N\) given in the statement of Lemma 7 and with the error term \(R_1=R_{1a}+R_{1b}\) where
By the Cauchy–Schwarz inequality, we have the quadratic form estimates
and
This completes the first part of Lemma 7.
Now let us turn to the operator inequality on the Fock space \({\mathcal {F}}_+\). We have proved that
Let us compare the right side of (42) with the corresponding version without the cut-off \({\mathbb {1}}^{\le N}\). First, consider the terms commuting with \({\mathcal {N}}_+\). Since
this operator is not smaller than its product with the cut-off \({\mathbb {1}}^{\le N}\). Moreover, using
we have
Finally, consider
By the Cauchy–Schwarz inequality \(\pm (Y^*Z+ Z^* Y) \le Y^* Y + Z^* Z\) we can bound
Moreover, since X changes the number of particles by at most 2, we have
Hence, combining with the above bound on \(\pm X\) we find that
This completes the proof of the operator inequality on \({\mathcal {F}}_+\) in Lemma 7. \(\square \)
4 Quadratic transformation
Recall that the Bogoliubov transformation \(U_B\) in (7) diagonalizes \({\mathbb {H}}_{\mathrm{Bog}}\) as in (9). In this section, we will study the action of \(U_N\) on the operator \({\mathcal {G}}_N\). We have
Lemma 8
Let \({\mathcal {G}}_N\) be given in Lemma 7. Then we have the operator identity on \({\mathcal {F}}_{+}\)
where
and the error term \(R_2\) satisfies
Proof
Let us decompose
where
Non-cubic terms. Let us prove that \(U_B {\widetilde{{\mathcal {D}}}}_N U_B^*- \langle 0| U_B {\widetilde{{\mathcal {D}}}}_N U_B^* |0\rangle \) contains only the terms of the form
with \(1\le s+t \le 4\) and the coefficients \(A_{m_1, \ldots ,m_s, n_1, \ldots ,n_t}\) satisfy
Let us start with the quadratic terms involving \(a_p^* a_{-p}^*\). Using (8) and the CCR (3) we have
Obviously the constant in (45) satisfies
Moreover, the other terms in (45) can be rewritten as
All of the sums in (46), (47), (48) are of the general form (43)–(44), thanks to the uniform bounds (10). The quadratic terms involving \(a_p^* a_{p}\) can be treated similarly.
Next, consider
It is straightforward to see that, except the constant
all other terms in (49) can be written as in (43), with the corresponding bound (44) following from (10). By the same argument, we can show that the terms involving \(a_p^* a_{p}{\mathcal {N}}_+\), \(a^*_{p+\ell } a^*_{q-\ell } a_p a_q\) and \({\mathcal {N}}_+ ({\mathcal {N}}_+-1)\) are of the general form (43) and (44).
Next, let us bound the terms of the general form (43) and (44). We consider the case \(s\ge t\) (the other case is treated similarly). By the Cauchy–Schwarz inequality
we have
for all \(\varepsilon >0\). Note that if \(\min (t,s) \ge 1\), then on the right side of (50) we can replace \(({\mathcal {N}}_++1)^{t+s-1}\) by \({\mathcal {N}}_+^{t+s-1}\).
In particular, for the non-cubic term \({\widetilde{{\mathcal {D}}}}_N\), using (50) with \(\varepsilon =N^{-1/2}\) and \(t+s\le 4\) we get
Cubic terms. By using (8) we have
By using (10), we can write the last sum of (52) as
with
Using (50) with \(\varepsilon =1\), \(t=1,s=2\), we get
Here \({\mathcal {N}}_+\le {\mathcal {N}}_+^2\) since the spectrum of \({\mathcal {N}}_+\) is \(\{0,1,2, \ldots \}\). The second sum on the right side of (52) can be treated by the same way. Thus from (52) and its adjoint, we have
which is equivalent to
In particular, (54) implies that
Therefore, from (9), (51) and (54) we obtain the desired conclusion of Lemma 8. \(\square \)
5 Cubic transformation
To factor out the cubic term \({\mathcal {C}}_N\) in Lemma 8, we will use a cubic renormalization. We will prove
Lemma 9
Let \({\mathcal {C}}_N\) be the cubic term in Lemma 8 and let \( U_S\) be given in (25). Then we have the operator identity on Fock spacee \({\mathcal {F}}_+\)
where
Proof
Recall that from Lemma 8 we have
with
Thanks to Lemmas 5 and 6, we find that
Thus this error term is part of \(R_3\).
For the main term, we use \(U_S=e^{S}\) and the Duhamel formula
we can write
Controlling \({\mathcal {C}}_N + [S,\mathrm{d}\Gamma (\xi )]\). Since \(\mathrm{d}\Gamma (\xi )\) commutes with \({\mathcal {N}}_+\) and
we find that
which is equivalent to
where \({\mathbb {1}}^{>N}={\mathbb {1}}-{\mathbb {1}}^{\le N}={\mathbb {1}}({\mathcal {N}}_+ >N)\). Thanks to the summability (10), we can use the Cauchy–Schwarz inequality similarly to (50) (with \(\varepsilon =1\)) to get
Combining with Lemmas 5 and 6 we obtain
Controlling \([S,{\mathcal {C}}_N]\). Let us decompose \(S= {\widetilde{S}} - S^{>}\) where
The main contribution comes from
By using the CCR (3) as in (49), we can write
where
By the Cauchy–Schwarz inequality as in (50), we get
It remains to bound \([S^>, {\mathcal {C}}_N]\). From the explicit form of \(S^{>}\) and \({\mathcal {C}}_N\), it is straightforward to check that
On the other hand, we observe that
and that \({\mathcal {C}}_N\) does not change the number of particles more than 3. Therefore,
Moreover, it is obvious that
for \(N\ge 10\). Thus from (59) and (60) we obtain
Combining with Lemma 5 we conclude that
Conclusion. Inserting (58) and (61) in (57) we find that
Combining with (55) we deduce that
Taking the expectation of the latter bound again the vacuum, we find that
Thus we obtain the desired conclusion
This completes the proof of Lemma 9. \(\square \)
6 Proof of Theorem 2
Proof
We will prove the ground state energy estimate
Upper bound. We use the following N-body trial state
Then by the variational principle and the operator inequality on \({\mathcal {F}}_+\) in Lemma 7 we have
By Lemmas 4 and 5 we know that
Consequently,
Combining with Lemma 9 we find that
In the last estimate, we have also used the simple upper bound
which will be justified below.
Lower bound. Let \(\Psi _N\) be the ground state of \(H_{N}\) and denote \( \Phi := U_S U_B U_N \Psi _N \in {\mathcal {F}}_+.\) By Lemmas 3, 35 and 36, we have
Then from the operator identity on \({\mathcal {F}}_+^{\le N}\) in Lemma 7 it follows that
Next, using Lemma 9 together with two simple estimates:
and
for \(\varepsilon >0\) small (but independent of N), we obtain
Therefore,
Thus
From (63), since \(\Big \langle \Phi , {\mathcal {N}}_+ \Phi \Big \rangle \ge 0\) we obtain the desired energy lower bound
This and the obvious upper bound \(E_N \le {\widehat{w}}(0) (N/2)\) imply the simple estimate (62). Thus the matching energy upper bound is valid, and hence we conclude that
Ground state estimates. By comparing the ground state energy expansion (64) with the lower bound (63) we deduce that
Let us write \(\Phi =(\Phi _j)_{j=0}^\infty \) with \(\Phi _j\in {\mathcal {H}}_+^j\). We can choose a phase factor for \(\Psi _N\) such that \(\Phi _0 \ge 0\). Then
Putting back the definition \(\Phi = U_S U_B U_N \Psi _N\) we obtain the norm approximation
This completes the proof of Theorem 2. \(\square \)
7 Proof of Theorem 1
Proof
Let \(\Psi _N\) be the ground state for \(H_N\). As explained in the introduction, we will decompose
Diagonal terms. For \(Q \gamma _{\Psi _N}^{(1)} Q\), recall from [12, Theorem 2.2 (iii)] that
strongly in the quadratic form of \({\mathbb {H}}_{\mathrm{Bog}}\) on \({\mathcal {F}}_+\). Moreover, it is easy to see that
(see e.g. [13, Proof of Theorem 1]). Therefore, in the limit \(N\rightarrow \infty \),
Here we have used Bogoliubov’s transformation (8). Similarly, for any \(p,q\ne 0\) we have
From (66) and (67), we conclude that
strongly in trace class. Consequently,
and hence
Off-diagonal terms. Let us prove that
By using \(P=|u_0\rangle \langle u_0|\) and the Cauchy–Schwarz inequality, it suffices to show that
Since \(\{u_p\}_{p\ne 0}\) is an orthonormal basis for \({\mathcal {H}}_+\), we have
Using the excitation map \(U_N\) and the relations (12) we can decompose
Therefore, by the Cauchy–Schwarz inequality
For the second sum in (70), using the Cauchy–Schwarz inequality, the simple bound
and Lemma 3, we find that
To control the first sum in (70), we will use the bound from Theorem 2:
Also, from Lemmas 3, 4 and 5 it follows that
Using the action of Bogoliubov’s transformation in (8) and the uniform bounds (10) we obtain
To estimate further the right side of (74), we use the Duhamel formula
and the Cauchy–Schwarz inequality to get
Thanks to (72) we can bound
It remains to handle the term involving the commutator \([S,a_p]\) in (75). Using the CCR (3) and the identity \([a_p, {\mathbb {1}}^{\le N}] = -{\mathbb {1}}({\mathcal {N}}_+=N) a_p\) we can decompose
Hence, by the Cauchy–Schwarz inequality we have for all \(t\in [0,1]\),
The right side of (77) can be bounded using the Cauchy–Schwarz inequality, the summability (27), Lemmas 5, 6, (72) and (73). For the terms involving \(I_1(p)\), we have
Similarly, the terms involving \(I_2(p)\) are bounded by
Finally for the terms involving \(I_3(p)\), using
we have
Thus we conclude from (77) that
Consequently,
Inserting (79) and (76) in (75) and (74) we obtain
Using the latter bound and (71), we deduce from (70) that
This implies (69) and completes the proof of Theorem 1. \(\square \)
Notes
We use the convention that \(\langle \cdot , \cdot \rangle \) is linear in the second argument and anti-linear in the first.
Strictly speaking, for \(a_0^*a_0^*a_0a_0\) we should rewrite it as \((a_0^*a_0)^2-a_0^*a_0\) before doing the substitution
References
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Acknowledgements
We thank Robert Seiringer and Nicolas Rougerie for helpful discussions. The research is funded by the Polish-German Beethoven Classic 3 project “Mathematics of many-body quantum systems”. PTN acknowledges the support from the Deutsche Forschungsgemeinschaft (DFG Project Nr. 426365943). MN acknowledges the support from the National Science Centre (NCN Project Nr. 2018/31/G/ST1/01166).
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