Abstract
We consider a system of N bosons in the limit \(N \rightarrow \infty \), interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.
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1 Introduction
We consider a system of N bosons with Hamilton operator
acting on \(L^2_s( {\mathbb {R}}^{3N}) \), the subspace of \(L^2( {\mathbb {R}}^{3N})\) consisting of functions which are symmetric with respect to permutations. The N-dependent two-body interaction potential is given through
In the following, we assume \(V \ge 0\) to be smooth, spherically symmetric, and compactly supported. For \(\beta =0\), the Hamiltonian (1.1) describes the mean-field regime characterized by a large number of weak collisions, whereas for \(\beta >1/3\) the collisions of the particles are rare but strong. In the Gross–Pitaevskii regime (\(\beta =1\)), pair correlations play a crucial role. Here, we study intermediate regimes \(\beta \in (0,1)\) in the limit \(N \rightarrow \infty \) where the particles interact through singular potentials.
The time evolution is governed by the Schrödinger equation
For \(\beta =0\) (mean-field regime), the solution of (1.2) can be approximated by products of solutions of the Hartree equation
with initial data \(\varphi _0 \in L^2( {\mathbb {R}}^3)\). See, for example, [1,2,3,4,5, 10, 16, 20,21,22, 25, 26, 34]. For \(0 < \beta \le 1\), on the other hand, the solution \(\psi _{N,t}\) of (1.2) can be approximated by the nonlinear Schrödinger equation
with \(\sigma = {\widehat{V}}(0)\) if \(\beta <1\) and \(\sigma = 8 \pi {\mathfrak {a}}_0\) if \(\beta =1\) (Gross–Pitaevskii regime). Hereafter, \({\mathfrak {a}}_0\) denotes the scattering length associated with the potential V defined through the solution of the zero-energy scattering equation
with boundary condition \(f(x) \rightarrow 1\) as \(|x| \rightarrow \infty \). Then, outside the support of V, the solution f is given through
where \({\mathfrak {a}}_0\) is defined as the scattering length of the potential V. In [17,18,19], it is shown that if the one-particle reduced density \(\gamma _N\) associated with \(\psi _N\) satisfies
in the trace norm topology and
then the one-particle reduced density \(\gamma _{N,t}\) associated with the solution \(\psi _{N,t}\) of (1.2) obeys
where \(\varphi _t\) denotes the solution of (1.3). In fact, in [17] considering the case \(\beta <1\), the energy condition (1.6) on the initial data is not needed. For more results in the Gross–Pitaevskii regime, see [7, 12, 15, 30, 31]. An overview on the derivation of the nonlinear Schrödinger equation from many-body quantum dynamics is given in [8, 23, 33].
1.1 Norm approximation
Besides the convergence of the one-particle reduced density \(\gamma _{N,t}\) associated with \(\psi _{N,t}\), the norm approximation of \(\psi _{N,t}\) has been studied for different settings of \(\beta \in (0,1)\) in [11, 24, 28, 29]. Our result is based on the norm approximation obtained in [11] covering \(\beta <1\) whose ideas we explain in the following.
Truncated Fock space. As first step toward the norm approximation in [11], the contribution of the Bose–Einstein condensate is factored out. This is realized through the unitary \({\mathscr {U}}_{\varphi _{N,t}}: L_s \left( {\mathbb {R}}^{3N} \right) \rightarrow {\mathscr {F}}_{\perp \varphi _{N,t}}^{\le N}\). It maps the N-particle sector of the bosonic Fock space
into the truncated Fock space
defined over the orthogonal complement \(L^2_{\perp \varphi _{N,t}}\left( {\mathbb {R}}^3\right) \) of the subspace of \(L^2( {\mathbb {R}}^3)\) spanned by the condensate wave function \(\varphi _{N,t}\). This unitary has first been used in [27] in the mean-field regime. Its definition is based on the observation that every \(\psi _N \in L^2_s ( {\mathbb {R}}^{3N}) \) has a unique decomposition
where \(\alpha ^{(n)} \in L^2_{\perp \varphi _{N,t}} ( {\mathbb {R}}^3)^{\otimes _s n}\) for all \(n=1, \ldots , N\). Then,
This unitary satisfies the following properties proven in [27]
for all \(f,g \in L^2_{\perp \varphi _{N,t}} ( {\mathbb {R}}^3)\). Here \(a^*(f), a(f)\) denote the standard creation and annihilation operators on the bosonic Fock space \({\mathscr {F}}\). On the truncated Fock space, we define modified creation and annihilation operators
The modified creation operator \(b^*(f)\) excites one particle from the condensate into its complement, while b(f) annihilates an excitation into the condensate. We define the vector \( \xi _{N,t}:= {\mathscr {U}}_{\varphi _{N,t}} \psi _{N,t}\) representing the fluctuation outside the condensate and observe
with initial data \(\xi _{N,0} = {\mathscr {U}}_{\varphi _{N,0}} \psi _{N,0}\).
From the truncated Fock space to the bosonic Fock space. We approximate the generator \({\mathscr {L}}_{N,t}\) acting on the truncated Fock space only with a modified generator \(\widetilde{{\mathscr {L}}}_{N,t}\) defined on the whole bosonic Fock space. We consider regimes with a small number of excitations \({\mathscr {N}}_+(t)\). For this reason, we realize the approximation of \({\mathscr {L}}_{N,t}\) through \(\widetilde{{\mathscr {L}}}_{N,t}\) by replacing \(\sqrt{N-{\mathscr {N}}_+(t)}\) with \(\sqrt{N} G_M({\mathscr {N}}_+(t) /N)\), where
is the Mth Taylor polynom of \(\sqrt{1-\tau }\) expanded at the point \(\tau _0 =0\). For a precise definition, see [11, eq. (54)].
Correlation structure through Bogoliubov transformation. In the intermediate regime, correlations are important (at least if \(\beta >1/2\)). For their implementation, we consider for fixed \(\ell >0\) the ground state of the scattering equation
with Neumann boundary conditions on the ball \(B_\ell (0) \). We fix \(f_N(x) = 1\) for all \(|x| =\ell \) and extend \(f_N\) to \({\mathbb {R}}^3\) by setting \(f_N(x) = 1\) for all \(|x| \ge \ell \).
In [10], the nonlinear Schrödinger equation (1.3) is replaced by the N-dependent Hartree equation
with initial data \(\varphi _{N,0} = \varphi _0\) (the condensate wave function at time \(t=0\)) to approximate the time evolved condensate wave function. The well-posedness of (1.12) is shown in [10, Appendix B].
The correlation structure is implemented through the Bogoliubov transformation
where \(\eta _{N,t}\) denotes the Hilbert–Schmidt operator with integral kernel
Here, \(\omega _N = 1- f_N\) and \(\varphi _{N,t}\) are as defined in (1.11) resp. (1.12) and \(q_{N,t} = 1- | \varphi _{N,t} \rangle \langle \varphi _{N,t} |\). The Bogoliubov transformation acts on the creation and annihilation operators as
for all \(f \in L^2( {\mathbb {R}}^3)\). The operators \(\sinh _{\eta _{N,t}}\) and \(\cosh _{\eta _{N,t}}\) are defined through the absolutely convergent series of products of the operator \(\eta _{N,t}\)
Let \({\mathscr {G}}_{N,t}\) be the generator given through
In fact, the special choice of (1.11) and (1.12) allows crucial cancellations in the generator \({\mathscr {G}}_{N,t}\). Note that \({\mathscr {G}}_{N,t}\) consists of terms which are quadratic in creation and annihilation operators and of terms of higher order. Nevertheless, in [11, Lemma 5], it is shown that \({\mathscr {G}}_{N,t}\) can be approximated through the generator \({\mathscr {G}}_{2,N,t}\) containing quadratic terms only.
Limiting quadratic dynamics. We are interested in the limit \(N\rightarrow \infty \) of \({\mathscr {G}}_{2,N,t}\) defined in (1.16). In order to replace the Bogoliubov transformation \(T_{N,t}\) defined in (1.13) with a limiting one, we define the limiting kernel \(\omega _\infty \) of \(\omega _N\) through
for \(|x| \le \ell \) and \(\omega _\infty (x) =0\) otherwise. Here, we used the notation \({\mathfrak {b}}_0 = \int \text {d}x \; V(x) \).
Furthermore, the solution \(\varphi _{N,t}\) of the modified Hartree equation (1.12) with initial data \(\varphi _0 \in H^4 ( {\mathbb {R}}^3)\) can be approximated with the solution \(\varphi _t\) of (1.3) with \(\sigma = {\widehat{V}}(0)\) and with initial data \(\varphi _0\). To be more precise, [10, Proposition B.1] shows that there exists a constant \(C>0\) (depending on \(\Vert \varphi _0 \Vert _{H^4}\)) such that
with \(\gamma = \min \lbrace \beta , 1-\beta \rbrace \). Standard arguments (see, for example, [10, Proposition B.1]) imply that there exists a constant \(C>0\) such that
for all \(n \in {\mathbb {N}}\). The approximations (1.17) and (1.18) lead to a limiting kernel
We define the limiting Bogoliubov transformation
In fact, (1.17) and (1.18) yield that there exists a constant \(C>0\) such that
where \(\gamma = \min \lbrace \beta , 1- \beta \rbrace \).
In order to define the limiting dynamics, we introduce some more notation. We use the shorthand notation \(j_x( \cdot ) = j(\cdot , x)\) for any \(j \in L^2( {\mathbb {R}}^3 \times {\mathbb {R}}^3)\). Furthermore, we decompose \(\text {sh}_{\eta _t} = \eta _t + \text {r}_t \), \(\text {ch}_{\eta _t} = \mathbb {1} + \text {p}_t\) and
for all \(x,y \in {\mathbb {R}}^3\).
A slight modification of the arguments in [10, Appendix C] shows some properties of the kernels. For these, we consider initial data \(\varphi _0 \in H^4 ( {\mathbb {R}}^3)\) of (1.3). There exist a constant \(C>0\) (depending only on \(\Vert \varphi _{0}\Vert _{H^4( {\mathbb {R}}^3)}\) and on V) such that on the one hand
where \(\Vert \cdot \Vert \) denotes the operator norm. On the other hand, denoting with \(\nabla _1 k_t \) and \(\nabla _2 k_t\) the operator with the kernel \(\nabla _x k_{t} (x;y)\)
Furthermore, let \(\varDelta _1 \text {r}_t \) resp. \(\varDelta _2 \text {r}_t \) be the operator having the kernel \(\varDelta _x \text {r}_t(x;y)\) resp. \(\varDelta _y \text {r}_t (x;y)\), then for all \(i =1,2\)
In order to simplify notation, we write in the following \(\text {sh}_{\eta _{t}} = \text {sh}, \; \text {ch}_{\eta _t} = \text {ch}\) resp. \(\text {r}_t = \text {r}, \; \text {k}_t = \text {k}, \; \text {p}_t = \text {p}\).
Definition 1
We define the limiting dynamics\({\mathscr {U}}_2( t;s)\) satisfying
where \({\mathscr {G}}_{2,t}\) is given by
with
and
and
Here, we used the notation \({\mathscr {K}} = \int \text {d}x \; a_x^* \left( - \varDelta _x \right) a_x \) and \(K_{1,t} = q_{t} {\widetilde{K}}_{1,t} q_{t}\) and \( K_{2,t} = \left( q_{t} \otimes q_{t} \right) {\widetilde{K}}_{2,t}\) where \({\widetilde{K}}_{1,t}\) is the operator with integral kernel
and \(K_{2,t}\) is the function given through
Note that (1.19) implies \(K_{1,t}, K_{2,t} \in L^\infty ( {\mathbb {R}}^6) \cap L^2( {\mathbb {R}}^6 )\) with norms uniform in N.
Norm approximation. We consider the solution \(\psi _{N,t}\) of the Schrödinger equation (1.2) with initial data \(\psi _{N,0} = U_{\varphi _0}^* \mathbb {1}^{\le N} T_{N,0}^* \varOmega \). It is proven in [11, Theorem 2] that for all \(\alpha < \min \lbrace \beta /2, (1-\beta )/2 \rbrace \) there exists a constant \(C>0\) such that
for all N sufficiently large and all \(t \in {\mathbb {R}}\).
1.2 Bogoliubov transformation
The limiting dynamics \({\mathscr {U}}_{2} (t;s)\) defined in (1.26) is quadratic in creation and annihilation operators. As the following proposition shows, it gives rise to a Bogoliubov transformation defined in the following. For this, we first define
On the one hand,
Here, \(J: L^2\left( {\mathbb {R}}^3 \right) \rightarrow L^2 \left( {\mathbb {R}}^3 \right) \) denotes the anti-linear operator defined by \(J f = {\overline{f}}\) for all \(f \in L^2 \left( {\mathbb {R}}^3 \right) \). On the other hand, the commutation relations imply for \(f_1,f_2, g_1,g_2 \in L^2( {\mathbb {R}}^3)\)
Definition 2
A Bogoliubov transformation is a linear map \(\nu : L^2\left( {\mathbb {R}}^3 \right) \oplus L^2\left( {\mathbb {R}}^3 \right) \rightarrow L^2 \left( {\mathbb {R}}^3 \right) \oplus L^2 \left( {\mathbb {R}}^3 \right) \) preserving the relations (1.33) and (1.34), i.e. \(\nu ^* S \nu = S\) and \( {\mathscr {J}} \nu =\nu {\mathscr {J}}\).
It turns out that a Bogoliubov transformation \(\nu \) is of the form
for linear operators \(U,V: L^2 \left( {\mathbb {R}}^3 \right) \rightarrow L^2 \left( {\mathbb {R}}^3 \right) \) satisfying \(U^* U - V^*V =1\) and \(U^* JVJ - V^*JUJ =0\).
The following proposition is proven in Sect. 2.2.
Proposition 1
Let \({\mathscr {U}}_{2}(t;s)\) be the dynamics defined in (1.27). For every \(t,s \in {\mathbb {R}}\), there exists a bounded linear map
such that
for all \(f,g \in L^2( {\mathbb {R}}^3)\). The map \(\varTheta (t;s)\) satisfies
where \({\mathscr {J}}\) and S are defined in (1.33) resp. (1.34). The Bogoliubov transformation \(\varTheta (t;s)\) can be written as
for bounded linear maps \(U(t;s), V(t;s): L^2( {\mathbb {R}}^3) \rightarrow L^2( {\mathbb {R}}^3)\) satisfying
1.3 Central limit theorem
From a probabilistic point of view, (1.7) implies a law of large numbers, in the sense that for a one-particle self-adjoint operator O on \(L^2( {\mathbb {R}}^3)\) and for every \(\delta >0\)
Here \(O^{(j)}\) denotes the operator on \(L^2( {\mathbb {R}}^{3N})\) acting as O on the jth particle and as identity elsewhere. The proof of (1.37) follows from Markov’s inequality (see [13]). As a next step, we are interested in a central limit theorem. For this, we consider the rescaled random variable
where \(\varphi _{N,t}\) denotes the solution of (1.12) with initial data \(\varphi _{N,0} = \varphi _0\).
We consider initial data \(\psi _{N,0}\) of the form \(\psi _{N,0} = {\mathscr {U}}_{\varphi _0}^* \mathbb {1}^{\le N} T_{N,0}^* \varOmega \) exhibiting Bose–Einstein condensation [11, Theorem 3]. As a consequence, such a initial data satisfy a law of large numbers in the sense of (1.37). Moreover, such initial data obeys a central limit theorem in the sense that
for every \(- \infty< a< b < \infty \). Here, \(G_0\) denotes the centered Gaussian random variable with variance \(\Vert \sigma _0 \Vert _2^2\), where
following from Theorem 1 for time \(t=0\).
Note that initial data of the form \(\psi _{N,0} = {\mathscr {U}}_{\varphi _0}^* \mathbb {1}^{\le N} T_{N,0}^* \varOmega \) describe approximate ground states of trapped systems [9]. In experiments, such initial data are prepared by trapping particles through external fields and by cooling them down to extremely low temperatures so that the system essentially relaxes to its ground state.
The validity of a central limit theorem for the ground state of trapped systems has already been addressed in [32]. To be more precise, [32] considers the ground state of (1.1) for \(\beta =1\), i.e. in the Gross–Pitaevskii regime. The ground state is known to exhibit Bose–Einstein condensation. It is proven that the ground state satisfies a central limit theorem. The arguments of the proof can be adapted to the intermediate regime \(\beta < 1\) using the norm approximation for the ground state obtained in [9].
Now, we consider the time evolution of the initial data \(\psi _{N,0} = {\mathscr {U}}_{\varphi _0}^* \mathbb {1}^{\le N} T_{N,0}^* \varOmega \) with respect to the Schrödinger equation (1.2) and show the validity of a (multi-variate) central limit theorem.
Theorem 1
Let \(\beta \in (0,1)\) and assume V to be radially symmetric, smooth, compactly supported and point-wise nonnegative. Furthermore, fix \(\ell >0\) (independent of N). Let \(\varphi _{t}\) denote the solution of (1.3) and \(\varphi _{N,t}\) the solution of (1.12) both with initial data \(\varphi _{0} \in H^4( {\mathbb {R}}^3)\). Moreover, we denote by \(\psi _{N,t}\) the solution of the Schrödinger equation (1.2) with initial data \(\psi _{N,0} = {\mathscr {U}}_{\varphi _{N,0}}^* \mathbb {1}^{\le N} T_{N,0}^* \varOmega \) (where \({\mathscr {U}}_{\varphi _{N,0}}\) and \(T_{N,0}\) are defined in (1.8) resp. (1.13)). For \(k \in {\mathbb {N}}\), let \(O_1, \dots , O_k\) be bounded operators on \(L^2( {\mathbb {R}}^3)\). We define \(\nu _{j,t} \in L^2( {\mathbb {R}}^3)\) through
where the operators \(U(t;0),V(t;0) \in L^2( {\mathbb {R}}^3 \times {\mathbb {R}}^3)\) are defined in Proposition 1, \(q_t = 1- | \varphi _t \rangle \langle \varphi _t |\) and \(\eta _t\) as defined in (1.20).
Assume \(\Sigma _t \in {\mathbb {C}}^{k \times k}\), given through
is invertible.
Furthermore, let \(g_1, \dots g_k \in L^1( {\mathbb {R}})\) with \({\widehat{g}}_i \in L^1( {\mathbb {R}}, (1+|s|)^4 \text {d}s)\) for all \(i \in \lbrace 1, \dots k \rbrace \) and let \({\mathscr {O}}_{j,N,t}\) denote the random variable (1.38) associated with \(O_j\) for all \(j \in \lbrace 1, \dots , k \rbrace \). For every \(\alpha < \min \lbrace \beta /2, (1-\beta ) /2 \rbrace \), there exists \(C>0\) such that
where \(\gamma = \min \lbrace \beta , (1-\beta ) \rbrace \).
A similar result has been established in [6, 13] for the mean-field regime characterized through weak interaction of the particles. It is shown that fluctuations around the nonlinear Hartree equation of bounded self-adjoint one-particle operators satisfy a (multi-variate) central limit theorem. We show that this result is true in the intermediate regime, where the interaction is singular, too. In particular, the correlation structure which becomes of importance in the intermediate regime does not affect the validity of a central limit theorem. However, it affects the covariance matrix (1.41) through the Bogolioubiv transform \(T_{t}\).
Similarly as in [13, Corollary 1.3 ], Theorem 1 implies a Berry–Esséen-type central limit theorem. To be more precise, we consider a bounded self-adjoint operator O on \(L^2( {\mathbb {R}}^3)\) and the random variable
For every \(\alpha < \min \lbrace \beta /2, (1-\beta ) /2 \rbrace \) and \(- \infty< a< b < \infty \), there exists a constant \(C>0\) such that
where \(G_t\) is the centered Gaussian random variable with variance \(\Vert \sigma _t \Vert _2^2\) and \(\sigma _t \in L^2( {\mathbb {R}}^3)\) is defined through
Note that Theorem 1 resp. (1.42) implies that fluctuations around the nonlinear Hartree equation with singular interaction satisfy a (multi-variate) central limit theorem. Comparing with \(\sigma _0\) from (1.40), the fluctuations enter in the variance \(\sigma _t\) through the operators U(t; 0), V(t; 0) as defined in Proposition (1) and the Bogoliubov transformation (1.21).
Moreover, note that the covariance matrix (1.41) resp. the variance (1.43) are completely determined by the Bogoliubov transform \(T_{t}\) defined in (1.21) and the quadratic fluctuation dynamics \({\mathscr {U}}_2 (t;0)\) defined in (1.27). Theorem 1 resp. the properties (1.36) of the operators U(t; 0), V(t; 0) show that the solution of the Schrödinger equation (1.2) modulo the extraction of the condensate is approximately a quasi-free state for quasi-free initial data. This observation coincides with results in [24, 28, 29].
2 Proof of results
2.1 Preliminaries
The proof of Theorem 1 is based on the norm approximation (1.31) from [11]. In the following, we collect useful properties of the unitaries used therein.
To this end, we define the more general quadratic dynamics \({\mathscr {U}}_{\mathrm {gen}}(t;s)\).
Definition 3
Let \({\mathscr {U}}_{\mathrm {gen}}(t;s)\) be the dynamics satisfying
where the generator \({\mathscr {G}}_{\mathrm {gen},t}\) is of the form
with
for constants \(c,C>0\).
In the following, we prove the results for the dynamics \({\mathscr {U}}_{\mathrm {gen}}(t;s)\). As the next Lemma shows, the results then apply to \({\mathscr {U}}_2 (t;s)\), too.
Lemma 1
The dynamics \({\mathscr {U}}_2(t;s)\) defined in Definition 1 is of the form of \({\mathscr {U}}_{\mathrm {gen}}(t;s)\) defined in Definition 3.
Proof
By the definition (1.27) of \({\mathscr {G}}_{2,t}\), we split
and consider each of the summands separately. First, we consider \({\mathscr {G}}_{2,t}^{\mathscr {V}}\) defined in (1.28), which is again split into four terms. The first one, \({\mathscr {G}}_{2,t}^{{\mathscr {V}},1}\) of the r.h.s. of (1.28), satisfies assumption (2.3) since on the one hand
and
following from (1.19) and (1.23). For the same reasons, the second term \({\mathscr {G}}_{2,t}^{{\mathscr {V}},2}\) of the r.h.s. of (1.28) satisfies assumption (2.3), too. For the third term \({\mathscr {G}}_{2,t}^{{\mathscr {V}},3}\), the definition of \(K_{2,t}\) implies
again from (1.19) and (1.23). The fourth term \({\mathscr {G}}_{2,t}^{{\mathscr {V}},4}\) satisfies the assumption (2.2) due to (1.19).
Furthermore, a transformation of variables shows
Therefore, \({\mathscr {G}}^{\lambda }_{2,t}\) is of form (2.2).
Moreover, for the term \({\mathscr {G}}_{2,t}^{{\mathscr {K}}} - {\mathscr {K}}\), we observe with (1.23) and (1.25)
The remaining bounds follow in the same way. Note that (1.24) implies the bound
Moreover, by definition (1.17) of the limiting kernel, \(\omega _\infty \in L^p ( {\mathbb {R}}^3)\) for all \(p<3\). Hence, the remaining terms of \({\mathscr {G}}_{2,t}^{{\mathscr {K}}}\) satisfy the assumptions, too.
We are left with the first term of the r.h.s. of (2.4). We write \(T_{t} = e^{ -B( \eta _{t} )}\). The properties (1.14) of the Bogoliubov transformation lead to
Since \(\Vert {\dot{\eta }}_t \Vert _2 \le C e^{C|t|}\) from (1.24), these terms satisfy assumption (2.3), too. \(\square \)
As proven in [11, Proposition 8], any moments of the number of particles operator are approximately preserved with respect to conjugation with the Bogoliubov transformation \(T_{N,t}\). To be more precise for every fixed \(k \in {\mathbb {N}}\) and \(\delta >0\), there exists \(C>0\) such that
As the following Lemma shows, the moments of number of particles operator are propagated in time with respect to the quadratic \({\mathscr {U}}_{\mathrm {gen}}( t;0)\).
Lemma 2
Let \({\mathscr {U}}_{\mathrm {gen}}(t;s)\) be as defined in Definition 3 and \(\psi \in {\mathscr {F}}\). For every \(k \in {\mathbb {N}}\), there exists a constant \(C>0\) such that for all \(t\in {\mathbb {R}}\)
Proof
We compute the derivative
Using the commutation relations and the definition (2.2), we find
For the first term of the right-hand side, the commutation relations yield
where C depends on \(k \in {\mathbb {N}}\). The second of the r.h.s. of (2.6) follows in the same way. Hence, there exists \(C>0\) such that
Hence, the Gronwall inequality implies
\(\square \)
For \(f \in L^2( {\mathbb {R}}^3)\), let \(\phi _a(f) = a^*(f) + a(f)\). In [13, Proposition 3.4], it is shown that for every \(k \in {\mathbb {N}}\) and \(\delta \in {\mathbb {R}}\), there exists a constant \(C>0\) such that
for all \(\psi \in {\mathscr {F}}\) and \(\alpha \ge 1\). Hereafter, we denote \(\text {d}\varGamma (H) = \sum _{j=1}^N H^{(j)}\) for a bounded operator H on \(L^2( {\mathbb {R}}^3)\). A similar estimate holds true for when replacing the creation and annihilation operators \(a(f), a^*(f)\) with the modified ones \(b^*(f), b(f)\) defined in (1.9). Let \(\phi _b (f) = b^*(f) + b(f)\). In fact, as proven in [32, Lemma 3.2], for every \(k \in {\mathbb {N}}\), there exists a constant \(C>0\) such that
for all \(\xi \in {\mathscr {F}}^{\le N}_+(t)\) and \(\alpha \ge 1\).
2.2 Proof of Proposition 1
It follows from Lemma 1 that it is enough to prove Proposition 1 with respect the dynamics \({\mathscr {U}}_{\mathrm {gen}} (t;s)\).
First, we prove that for \(f \in L^2( {\mathbb {R}}^3)\) the Fock space vectors \({\mathscr {U}}^*_{\mathrm {gen}}(t;s) a^*(f) {\mathscr {U}}_{\mathrm {gen}}(t;s) \varOmega \) and \({\mathscr {U}}_{\mathrm {gen}}^*(t;s) a(f) {\mathscr {U}}_{\mathrm {gen}}(t;s) \varOmega \) are elements of the one-particle sector. The following Lemma is a generalization of [14, Lemma 8.1].
Lemma 3
Let \({\mathscr {U}}_{\mathrm {gen}} (t;s)\) be the dynamics defined Definition 3. Then for all \(f \in L^2( {\mathbb {R}} )\),
where either \(a^\sharp (f) = a(f)\) or \(a^\sharp (f) = a^*(f)\) and where \({\mathscr {P}}_1\) denotes the projection onto the one-particle sector of the Fock space \({\mathscr {F}}\).
Proof
The proof follows the arguments of the proof of [14, Lemma 8.1]. For \(m \in {\mathbb {N}}\), \(m \not = 1\), we define for arbitrary m-particle wave function \(\psi \in {\mathscr {F}}\) with \(\Vert \psi \Vert =1\) the function
Since \(m \not = 1\), we observe that \(F(s)=0\) and furthermore
using the notation \(f_t = e^{-it \varDelta } f\). Since \(e^{-i \varDelta t}\) is a unitary operator, we find
Then,
and the definition of \({\mathscr {G}}_{\mathrm {gen},t}\) in (2.2) leads to
The assumption (2.3) implies on the one hand
and on the other hand
Hence,
and analogously,
Note that these bounds are independent of \(f \in L^2( {\mathbb {R}}^3)\). Thus,
Using the bounds \(\Vert a^\sharp (f) \psi \Vert \le \Vert f \Vert _2 \Vert ( {\mathscr {N}}+1)^{1/2} \psi \Vert \), we obtain
Here, we used Lemma 2 for the last estimate. Since \(F(s)=0\), the Gronwall inequality implies \(F(t) =0\) for all \(t \in {\mathbb {R}}\). \(\square \)
Proof of Proposition 1
We prove the Proposition with respect to the dynamics \({\mathscr {U}}_{\mathrm {gen}}(t;s)\) defined in Definition 3. Then, Proposition 1 follows from Lemma 1.
The proof follows the arguments of the proof of [6, Theorem 2.2]. Let \({\mathscr {P}}_k\) denote the projection onto the k-particle sector \({\mathscr {F}}_k\) of the Fock space. It follows from Lemma that 3
for all \(f \in L^2( {\mathbb {R}}^3)\) and \(k \not = 1\). Thus, there exist linear operators \(U(t;s), V(t;s): L^2( {\mathbb {R}}^3) \rightarrow L^2( {\mathbb {R}}^3)\) such that
where \(J:L^2( {\mathbb {R}}^3) \rightarrow L^2( {\mathbb {R}}^3) \) denotes the anti-linear operator defined by \(Jf= {\overline{f}}\) for all \(f \in L^2( {\mathbb {R}}^3)\). The operators U(t; s) and V(t; s) are bounded in \(L^2( {\mathbb {R}}^3)\). This follows from Lemma 2, since
and
We define the bounded operator \(\varTheta \) on \( L^2( {\mathbb {R}}^3) \oplus L^2( {\mathbb {R}}^3)\) through
Then
for all \(f,g \in L^2( {\mathbb {R}}^3)\). For fixed \(\psi \in {\mathscr {D}}( {\mathscr {K}} + {\mathscr {N}} )\), \(g \in L^2( {\mathbb {R}}^3 )\), \(s \in {\mathbb {R}}\) and any bounded operator \({\mathscr {M}}\) on \({\mathscr {F}}\) with \({\mathscr {M}}{\mathscr {D}} ( {\mathscr {K}} + {\mathscr {N}} ) \subset {\mathscr {D}} ( {\mathscr {K}} + {\mathscr {N}} )\), we define furthermore
Here, \(a^\sharp , a^\flat \) are either creation or annihilation operators. Since \(e^{-i {\mathscr {K}} t} a^\sharp (f) e^{i {\mathscr {K}} t} = a^\sharp ( e^{it \varDelta } f)\) and \(\Vert e^{it \varDelta } f \Vert _2 = \Vert f\Vert _2\) for all\( f \in L^2( {\mathbb {R}}^3)\), we can write
The commutation relations imply that \(F(s)=0\). Furthermore,
using the notation \(f_t = e^{-i \varDelta t} f\). Analogous calculations as in the proof of Lemma 3 show that
The assumption (2.3) implies \(\Vert h_{i,t} \Vert _2 \le C e^{C |t|} \Vert f \Vert _2 \) for \(i =1,2\). Thus,
for all \(f \in L^2( {\mathbb {R}}^3)\) and therefore
Since \(F(s)=0\), the Gronwall inequality implies \(F(t) = 0 \) for all \(t \in {\mathbb {R}}\). Hence,
for every \(f_1,f_2,h_1,h_2 \in L^2( {\mathbb {R}}^3)\) and every bounded operator \({\mathscr {M}}\) on the Fock space \({\mathscr {F}}\) such that \({\mathscr {M}} {\mathscr {D}} \left( {\mathscr {K}} + {\mathscr {N}}\right) \subset {\mathscr {D}} \left( {\mathscr {K}} + {\mathscr {N}} \right) \). We claim that
for all \(\psi \in {\mathscr {D}}\left( {\mathscr {K}} + {\mathscr {N}} \right) \) with \(\Vert \psi \Vert =1\). Combining (2.9) with (2.11), we find
where S is defined in 1.34. It follows that
for all \(f_1,h_1,f_2,h_2 \in L^2( {\mathbb {R}}^3)\). Consider now
On the one hand, (2.9) shows that \(R \varOmega =0\), and on the other hand, it follows from (2.12), that R commutes with any creation and annihilation operator. Since states of the form \(a^*(f_1) \dots a^*(f_n) \varOmega \) build a basis of the Fock space \({\mathscr {F}}\), we conclude
for all \(f,g \in L^2( {\mathbb {R}}^3)\).
Now, we are left with proving (2.11). For this, note that (2.10) implies
where \(P_\psi \) resp. \(P_\varOmega \) denote the projection on the subspace of \({\mathscr {F}}\) spanned by \(\psi \) resp. \(\varOmega \). Therefore, on the one hand
and on the other hand,
Assuming that \(\langle \psi , \varOmega \rangle \not =0\), claim (2.11) follows. If \(\langle \psi , \varOmega \rangle =0\), we repeat the same arguments with \({\widetilde{\psi }} = \frac{1}{\sqrt{2}} ( \psi + \varOmega )\). This leads to (2.11).
It remains to prove the properties (1.35). Since for all \(f,g \in L^2( {\mathbb {R}}^3)\)
the first property follows. Furthermore, from
we deduce the second property. \(\square \)
2.3 Proof of Theorem 1
The proof uses ideas introduced in [32]. We consider the expectation value
The norm approximation (1.31) from [11] implies that for every \(\alpha < \min \lbrace \beta /2, (1-\beta )/2 \rbrace \) there exists \(C>0\) such that
We are hence left with computing the expectation value
We split this computation in several Lemmata.
Lemma 4
(Action of the unitary \({\mathscr {U}}_{\varphi _{N,t}}\)) Let \(T_{N,t}\) and \({\mathscr {U}}_2(t;0)\) be as defined in (1.13) resp. (1.26). Moreover, let \(\xi _{N,t} = T_{N,t}^* {\mathscr {U}}_{2}(t;0) \varOmega \). Then, using the same notations as in Theorem 1, there exists \(C>0\) such that
Proof
We recall that for \(f \in L^2( {\mathbb {R}}^3)\) we denote \(\phi _b (f) = b^*(f) + b(f)\) with the modified creation and annihilation operators \(b^*(f), b(f)\) defined in (1.9).
In order to show Lemma 4, we define for \(j \in \lbrace 1, \ldots , k \rbrace \)
We observe that
where \(p_{N,t} = | \varphi _{N,t} \rangle \langle \varphi _{N,t}\vert \) and \(q_{N,t} = 1- p_{N,t}\). The properties (1.8) of the unitary \({\mathscr {U}}_{ \varphi _{N,t}}\) imply
Hence,
We compute
Using the fundamental theorem of calculus, we can write the difference as an integral
The estimate \(\Vert \text {d}\varGamma ( A) \psi \Vert \le \Vert A \Vert \; \Vert {\mathscr {N}} \psi \Vert \) leads to
Since \(\Vert q_{N,t}{\widetilde{O}}_{m,N,t} q_{N,t}\Vert \le \Vert O_m \Vert \) and \(\Vert q_{N,t}O_j \varphi _{N,t}\Vert \le \Vert O_j \Vert \), we find with (2.8)
for \(\alpha \ge 1\). Recall that \(\xi _{N,t} = T_{N,t}^* {\mathscr {U}}_2 (t;0) \varOmega \). It follows from (2.5) and Lemma 2 that
for a constant \(C>0\) uniform in N. Hence,
\(\square \)
Lemma 5
( Replace modified creation and annihilation operators with standard ones) Let \(T_{N,t}\) and \({\mathscr {U}}_2(t;0)\) be as defined in (1.13) resp. (1.26). Moreover, let \(\xi _{N,t} = T_{N,t}^* {\mathscr {U}}_{2}(t;0) \varOmega \). Then, with the same notations as in Theorem 1, there exists \(C>0\) such that
Proof
Recall that
with the standard creation and annihilation operators \(a^*(f), a(f)\), while
with the modified creation and annihilation operators defined in (1.9). To this end, we compute
By definition of the modified creation and annihilation operators (1.9), we obtain
Since \(\Vert a^*(f) \xi \Vert \le \Vert f \Vert _2 \; \Vert ( {\mathscr {N}} +1)^{1/2} \xi \Vert \) resp. \(\Vert a(f) \xi \Vert \le \Vert f \Vert _2 \; \Vert {\mathscr {N}}^{1/2} \xi \Vert \) and \(\Vert q_{N,t}O_m \varphi _{N,t}\Vert _2 \le \Vert O_m \Vert \), we find
Now, Lemma 2 together with (2.7) and (2.5) implies
\(\square \)
Lemma 6
(Replace modified Hartree equation with nonlinear Schrödinger equation) Let \(T_{N,t}\) and \({\mathscr {U}}_2(t;0)\) be as defined in (1.13) resp. (1.26). Moreover, let \(\xi _{N,t} = T_{N,t}^* {\mathscr {U}}_{2}(t;0) \varOmega \). Then, with the same notations as in Theorem 1, there exists \(C>0\) such that
Proof
By linearity of the operator \(\phi _a (f)\), we compute
As
the estimate (1.18) implies
with \(\gamma =\min \lbrace \beta , 1-\beta \rbrace \). Hence, the bound
leads to
We conclude again with Lemma (2.7), Lemma (2.5), and Lemma 2
\(\square \)
Lemma 7
(Action of \(T_{N,t}\)) Let \(T_{N,t}\) and \({\mathscr {U}}_2(t;0)\) be as defined in (1.13) resp. (1.26). Moreover, let \(\xi _{N,t} = T_{N,t}^* {\mathscr {U}}_{2}(t;0) \varOmega \) and \(\xi _{t} =T_{N,t} \xi _{N,t}= {\mathscr {U}}_{2,t} \varOmega \). Then, using the same notations as in Theorem 1, there exists \(C>0\) such that
with \(h_{j,t} = \cosh ( \eta _t ) q_t O \varphi _t + \sinh ( \eta _t ) \overline{q_t O_j \varphi _t}\) and \(\eta _t\) as defined in (1.20).
Proof
We compute using the properties (1.14) of the Bogoliubov transformation
with \(\eta _{N,t}\) as defined in (1.20). In the following, we denote \(h_{j,N,t} =\cosh ( \eta _{N,t} ) q_t O \varphi _t + \sinh ( \eta _{N,t} ) \overline{q_t O_j \varphi _t} \). Since
we need to consider
We observe using (1.24)
Thus, the estimate (1.22) leads to
Using \(\Vert h_{j,t} \Vert _2 \le C \Vert O_j \Vert \), the same arguments as in step 3 lead to
\(\square \)
Lemma 8
(Computing the expectation value) Let \({\mathscr {U}}_2(t;0)\) be as defined in (1.26). Let \(\xi _t = {\mathscr {U}}_{2,t} \varOmega \). Then, using the same notations as in Theorem 1,
Proof
Since \(\xi _t = {\mathscr {U}}_{2,t} \varOmega \), we are left with computing
As proven in Proposition 1, the unitary \({\mathscr {U}}_{2,t}\) gives rise to a Bogoliubov transformation. Hence, there exists bounded operators U(t; 0), V(t; 0) on \(L^2( {\mathbb {R}}^3)\) such that
In the following, we denote
Note that the Baker–Campbell–Hausdorff formula implies on the one hand
for \(f,g \in L^2( {\mathbb {R}}^3 )\), i.e.,
with \(\nu _t= \sum _{j=1}^k \nu _{j,t}\). On the other hand, the Baker–Campbell–Hausdorff formula applied to the creation and annihilation operator implies
Hence, we write the expectation value (2.14) as
Let \(\Sigma _t \in {\mathbb {C}}^{k \times k}\) be given through
then
By assumption, the matrix \(\Sigma _t \) is invertible. Hence,
\(\square \)
Summarizing the results from Lemmas 4–8, we finally obtain
with \(\gamma = \min \lbrace \beta , 1- \beta \rbrace \). This proves Theorem 1.
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Open access funding provided by Institute of Science and Technology (IST Austria). S.R. would like to thank Benjamin Schlein for many fruitful discussions.
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Simone Rademacher acknowledges partial support from the NCCR SwissMAP. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 754411.
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Rademacher, S. Central limit theorem for Bose gases interacting through singular potentials. Lett Math Phys 110, 2143–2174 (2020). https://doi.org/10.1007/s11005-020-01286-w
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DOI: https://doi.org/10.1007/s11005-020-01286-w