Spectral Analysis of Mean-Field Hamiltonian ofNonrelativistic Many-Body Systems
Hironobu Sasaki, Shoji Shimizu, Akito Suzuki
Abstract
We conduct the spectral analysis of a time-dependent Schrödinger
operator that can be considered as the quantized Hamiltonian of a nonrelativistic many-body system in the mean-field regime. The basic properties of these Hamiltonians in general two-particle interactions satisfying suitable decay conditions are shown. Specifically, we determine thelocation of the essential spectrum and the finiteness or infiniteness ofthe discrete spectrum. The Hartree equation analysis plays an essential
role in our study. 1Key words: Mean-field Hamiltonian, Hartree equation.

1Introduction

In this paper we conduct a spectral analysis of a time-dependent Schrödingeroperator Ht on L2 (Rdx ) defined by
Ht = −∆ + V ∗ |φt |2 .
Here, ∆ is the generalized Laplacian on L2 (Rd ), V is a real-valued function on
Rd , and φt : Rd → C is a time-global solution of the Hartree equationi∂t φt = −∆φt + (V ∗ |φt |2 )φt(1.1)

where ∗ denotes the convolution in Rd . Therefore, the term V ∗ |φt |2 can beformally expressed as∫2V ∗ |φt | (x) =V (x − y)|φt |2 (y)dy, x ∈ Rd .
RdThe operator Ht is the quantum counterpart of the classical mean-field Hamiltonian, which describes a particle influenced by the collective force of manyother particles with pairwise potential V and macroscopic density ρt (x) =|φt (x)|2 (see e.q., [G]). The state of this system, described by the density operator D(t) = ⟨φt , ·⟩φt , is governed by the von Neumann equationi∂t D(t) = [Ht , D(t)],12010 Mathematics Subject Classification: 35J10, 35Q40, 35Q41

1which is equivalent to the Hartree equation (1.1). The operator Ht has recently
been applied for the mean-field analysis of many-boson systems (see e.q., [AB]and references therein). We are interested in the relationship between themean-field Hamiltonian Ht and the Hartree solution φt . The Hartree equation
has been thoroughly investigated in partial diﬀerential equation theory, but
the spectral property of Ht is less well-known. In this paper, we investigate
the basic properties of Ht using several results from the Hartree theory.
The present paper is organized as follows: In Section 2, we first investigate
the properties of the potential V ∗ |φt |2 from a suitable two-particle interaction
V : Rd → R and any initial value satisfying φ ∈ H 2 (Rd ) of equation (1.1).
Next, we prove some spectral properties of Ht in a generalized spatial dimension
d. In particular, we prove the stability theorem for the essential spectrum(Theorem 2.5) and the existence of a ground state for Ht (Corollary 2.6). Using
the general perturbation theory, recently developed by Arai [Ar], we prove that
discrete eigenvalues of Ht are stable (Theorem 2.9, Corollary 2.10). Finally, inSection 3, we show the finiteness or infiniteness of discrete eigenvalues of Htfor d = 3 (Theorems 3.1–3.2).

2

Mean-Field Hamiltonian in General Dimensions

In the following, we denote Lp (Rd ) (resp. H s (Rd )) as Lp (resp. H s ) for 0 < p ≤∞ and s ∈ R. Unless otherwise stated, we assume that V : Rd → R satisfies
the following two conditions:(A) V is even; i.e., V (x) = V (−x) for any x ∈ Rd .
(B) There exist some V1 ∈ Lp and V2 ∈ L∞ such that V = V1 + V2 , wherep = 2 if d = 1, 2, 3 and p > d/2 if d ≥ 4.
The following proposition is well-known:
Proposition 2.1. For any φ0 ∈ H 2 , there exists a time-global solution φt of(1.1) uniquely defined on a suitable open interval Iφ0 of R containing the origin
such that φt ∈ C(Iφ0 , H 2 ) ∩ C 1 (Iφ0 , L2 ). Furthermore, mass and energy areconserved:∥φt ∥L2 = ∥φ∥L2 ,t ∈ Iφ0 ,∫12V ∗ |φt |2 (x)|φt |2 (x)dx
E(φt ) : = ∥∇φt ∥L2 +2 Rd= E(φ0 ), t ∈ Iφ0 .
2(2.1)

The proof is shown in Chapter 4 of [Caz].
In the following, we set
Vt = V ∗ |φt |2 ,t∈Rand⟨x⟩θ = (1 + |x|2 )θ/2 ,x ∈ Rd , θ ∈ R.

Before proving the self-adjointness of Ht , we need the following two lemmas.
Since the first lemma is well-known, its proof is omitted (see e.q., [Is]).
Lemma 2.2. Let V : Rd → R satisfy (B). Then we have
∥V f ∥L2 ≤ t− 2p ∥⟨x⟩−2 ∥Lp ∥V ∥Lp ∥1 − t∆f ∥L2d(2.2)

for any t > 0 and f ∈ D(−∆).
Lemma 2.3. Let q > 1 and V ∈ Lq . Then Vt ∈ Lq and∥Vt ∥Lq ≤ ∥V ∥Lq ∥φ0 ∥2L2(2.3)

holds for any t ∈ Iφ0 . Moreover, we have
∥Vt − Vs ∥Lq ≤ 2∥V ∥Lq ∥φ0 ∥L2 ∥φt − φs ∥L2 , t, s ∈ Iφ0 .

(2.4)

Proof. Since |φt |2 ∈ L1 and ∥|φt |2 ∥L1 = ∥φ0 ∥2L2 , the Hölder-Young inequality
gives∥Vt ∥Lq ≤ ∥V ∥Lq ∥|φt |2 ∥L1= ∥V ∥Lq ∥φ0 ∥2L2 ,which implies (2.3). To prove (2.4), we first calculate (2.5) by the Minkowski
inequality as follows:∥Vt − Vs ∥Lq = ∥V ∗ (|φt |2 − |φs |2 )∥Lq= ∥V ∗ {(|φt | − |φs |)(|φt | + |φs |)}∥Lq(2.5)≤ ∥V ∗ {(|φt | − |φs |)|φt |}∥Lq + ∥V ∗ {(|φt | − |φs |)|φs |}∥Lq≤ ∥V ∥Lq ∥(|φt | − |φs |)|φt |∥L1 + ∥V ∥Lq ∥(|φt | − |φs |)|φs |∥L1 .
The final inequality of (2.5) again uses the Hölder-Young inequality. Applying
the Schwarz inequality and the conservation of mass, the final right-hand sideof the above inequality becomes∥V ∥Lq ∥|φt | − |φs |∥L2 ∥φt |∥L2 + ∥V ∥Lq ∥|φt | − |φs |∥L2 ∥φs |∥L2
≤ ∥V ∥Lq ∥φt − φs ∥L2 ∥φt ∥L2 + ∥V ∥Lq ∥φt − φs ∥L2 ∥φs |∥L2= 2∥V ∥Lp ∥φt − φs ∥L2 ∥φ0 ∥L2 .
Combining (2.5) and (2.6), we obtain (2.4).
3(2.6)

Proposition 2.4. For any V : Rd → R satisfying assumptions (A) and (B),the following properties hold:(i) Ht is a self-adjoint operator on L2 with D(Ht ) = D(−∆) for any t ∈ Iφ0 .
(ii) For any fixed t0 ∈ Iφ0 , Ht converges to Ht0 in the norm resolvent sense ast → t0 .
Proof. Property (i) follows directly from Lemmas 2.2 and 2.3 and the KatoRellich theorem. To prove (ii), we calculate(Ht − z)−1 − (Hs − z)−1 = (Ht − z)−1 (Vs − Vt )(Hs − z)−1for z ∈ C \ R. Hence by Lemma 2.7 below, we have∥(Ht − z)−1 − (Hs − z)−1 ∥ ≤ ∥(Ht − z)−1 (Vs − Vt )(Hs − z)−1 ∥αC(s, z)≤∥φt − φs ∥L2 .
|Imz|
Since φt ∈ C 1 (Iφ0 , L2 ) by Proposition 2.1, the proof is complete.
Imposing stronger assumptions on V , we can identify the essential spectrum
σess (Ht ). For this purpose, we specify a class of potentials V .
We say that V ∈ Lp + L∞,ϵ if for any ϵ > 0 there exist V1 ∈ Lp and V2 ∈ L∞
such that
V = V1 + V2 , ∥V2 ∥L∞ < ϵ.
Theorem 2.5. Assume that V : Rd → R satisfies (A) and belongs to the classof Lp (Rd ) + L∞,ϵ (Rd ) for p = 2 (d = 3), p > d/2 (d ≥ 4). Then we haveσess (Ht ) = [0, ∞).

(2.7)

Proof. As is well-known (see e.q., [ReSi IV]), it is suﬃcient to show that Vt ∈Lp (Rd ) + L∞,ϵ (Rd ). By the Hölder inequality and the conservation of mass, wehave∥V2 ∗ |φt |2 ∥L∞ ≤ ∥V2 ∥L∞ ∥φt ∥2L2≤ ϵ∥φ0 ∥2L2 .

(2.8)

Combining this inequality with Lemma 2.3 completes the proof.
As a corollary, we note the existence of a ground state of Ht .
Corollary 2.6. Let V be given as in Theorem 2.5. Also, assume that E(φ0 ) <0. Then, Ht has a ground state for any t ∈ Iφ0 .

4Proof. By the min-max principle (see e.q., [ReSi, IV]), finding a unit vectorψ ∈ D(Ht ) ⊂ H 2 such that ⟨ψ, Ht ψ⟩ < 0 is suﬃcient, since we know thatinf σess (Ht )=0 by Theorem 2.5. However, since energy is conserved, we have
⟨φt , Ht φt ⟩ = ⟨φt , −∆φt ⟩ + ⟨φt , Vt φt ⟩≤ 2⟨φt , −∆φt ⟩ + ⟨φt , Vt φt ⟩= 2E(φt ) = 2E(φ0 ) < 0.
Taking ψ = φt /∥φt ∥, the proof is completed.
Next, we develop a stability theorem for a discrete eigenvalue of Ht . For
this purpose, we adopt the general perturbation theory established in [Ar].
Let us denote the resolvent set of a self-adjoint operator T by ρ(T ). Forz ∈ ρ(T ), we set
RT (z) = (T − z)−1 .
We first prove the following lemma:Lemma 2.7. There exists a positive constant α dependent only on V and φ0such that∥(Vt − Vs )RHs (z)∥ ≤ αC(s, z)∥φt − φs ∥L2holds for all s, t ∈ Iφ0 and z ∈ ρ(H). Here we set
C(s, z) = ∥RHs (z)∥ + ∥Hs RHs (z)∥.
Remark 2.8. Since supz∈K C(s, z) < ∞ holds for all compact subsets K ⊂ρ(H), it follows thatlim ∥(Vt − Vs )RHs (z)∥ = 0(2.9)
t→suniformly in z on each compact subset K ⊂ ρ(H).
Proof. Since∥(Vt − Vs )RHs (z)∥ ≤ ∥(Vt − Vs )(−∆ + 1)−1 ∥∥(−∆ + 1)RHs (z)∥=: B1 B2 ,(2.10)

we only need to estimate B1 and B2 . We first estimate B1 . Setting t = 1 in(2.2) together with (2.4) we see that
B1 ≤ ∥⟨x⟩−2 ∥Lp ∥Vt − Vs ∥Lp≤ 2∥⟨x⟩−2 ∥Lp ∥V ∥Lp ∥φ0 ∥L2 ∥φt − φs ∥L2 .

5(2.11)

Next we estimate B2 . First we note that, by (2.2) and (2.3),
d∥Vs f ∥L2 ≤ ∥⟨x⟩−2 ∥Lp ∥Vs ∥Lp {t1− 2p ∥ − ∆f ∥ + t− 2 ∥f ∥}dd≤ ∥⟨x⟩−2 ∥Lp ∥V ∥Lp ∥φ0 ∥2L2 {t1− 2p ∥ − ∆f ∥L2 + t− 2 ∥f ∥L2 }dholds for any f ∈ D(−∆). Since 1 − d/(2p) > 0 by assumption (B), we obtain
1∥Vs f ∥L2 ≤ ∥ − ∆f ∥L2 + C∥f ∥L22for any f ∈ D(−∆), where C > 0 depends only on V . Hence we see that∥ − ∆RHs (z)∥ = ∥(Hs − Vs )RHs (z)∥≤ ∥Hs RHs (z)∥ + ∥Vs RHs (z)∥1≤ ∥Hs RHs (z)∥ + ∥ − ∆RHs (z)∥ + C∥RHs (z)∥,2which implies the inequality
B2 ≤ 2∥Hs RHs (z)∥ + (2C + 1)∥RHs (z)∥.

(2.12)

Combining (2.10), (2.11), and (2.12), we complete the proof.
In stating the following result, we require some notations defined in [Ar].
Suppose that t0 ∈ Iφ0 is arbitrary and fixed. Suppose also that Ht0 has an
isolated eigenvalue E0 ∈ R. Let r0 be a constant satisfying
0 < r0 <minE∈σ(Ht0 )\{E0 }|E − E0 |.

Then, we define
Cr0 (E0 ) = {z ∈ C||z − E0 | < r0 }.
Note that Cr0 (E0 ) ⊂ R \ σ(Ht0 ). By virtue of Remark 2.8, there exists δ > 0such that()−1supz∈Cr0 (E0 )∥(Vt − Vt0 )RHt0 (z)∥ <1+rsupz∈Cr0 (E0 )∥RHt0 (z)∥for any t ∈ (t0 − δ, t0 + δ).
We now state the following:
Theorem 2.9. Suppose that the multiplicity m(E0 ) of E0 is finite. Then,for all t ∈ (t0 − δ, t0 + δ), Ht has exactly m(E0 ) eigenvalues in the interval(E0 − r0 , E0 + r0 ), including multiplicities, and σ(Ht ) ∩ (E0 − r0 , E0 + r0 )consists of these eigenvalues alone.
6

Corollary 2.10. Suppose in addition that m(E0 ) = 1 and let Ω0 be a normalized eigenvector of Ht0 with an eigenvalue E0 , i.e., Ht0 Ω0 = E0 Ω0 and∥Ω0 ∥ = 1. Then, for all t ∈ (t0 − δ, t0 + δ), Ht has a simple eigenvalue Et inthe interval (E0 − r0 , E0 + r0 ) given by∑⟨Ω0 , Vt − Vt0 Ω0 ⟩ + ∞n=1 cn (t)∑∞Et = E0 +,1 + n=1 dn (t)where(−1)n+1cn (t) : =2πi∫dz⟨Ω0 , [(Vt − Vt0 )RHt (z)]n+1 Ω0 ⟩,Cr0 (E0 )n+1 ∫dn (t) : =(−1)
2πidz
Cr0 (E0 )⟨Ω0 , [(Vt − Vt0 )RHt (z)]n Ω0 ⟩E0 − zand σ(Ht ) ∩ (E0 − r0 , E0 + r0 ) = {Et }. Moreover, a normalized eigenvector of
Ht is given as∑Ω0 + ∞
n=1 Ωt,nΩt = √,∑∞1 + n=1 dn (t)where(−1)n+1Ωt,n :=2πi∫dz(Ht0 − z)−1 [(Vt − Vt0 )RHt (z)]n Ω0 .

Cr0 (E0 )Proof. Both of the above statements are direct consequences of Theorem A.3
and Corollary A.4 in [Ar]. In that reference, equation (2.9) is given as Hypothesis (A).

3

Mean-Field Hamiltonian in Three Spatial Dimensions

In this section we consider the case d = 3, and show the finiteness or infinitenessof the discrete spectrum σdisc (Ht ) of Ht .
First we treat the finiteness. We assume that(C) V ≤ 0, V ∈ L1 (R3 ) ∩ L2 (R3 ).
Clearly, condition (C) implies condition (B) in Section 2.
Theorem 3.1. Assume that V satisfies (A) and (C). Then σdisc (Ht ) ⊂(−∞, 0). Moreover, the number of discrete eigenvalues N (Ht ) can be estimated by a(∥V ∥L1 ∥φt ∥L3 )2/3 , where a > 0 is some universal constant.
7Proof. The first statement immediately follows from Theorem 2.5. The second
follows from the Lieb-Cwickel-Rosenbljum bound (see e.g., [Si2])
2N (Ht ) ≤ a∥Vt ∥ 3 3
L2

and the estimate∥Vt ∥L 32 ≤ ∥V ∥L1 ∥φt ∥L3given by the Hölder-Young inequality.
Finally, we consider the infiniteness of σdisc (Ht ) withV (x) := −a, a > 0, γ ∈ (0, 2)|x|γ(3.1)

and φ0 in S(R3 ), denoting the space of rapidly decreasing C ∞ functions on R3 .
For such V and φ0 , the following proposition holds:
Proposition 3.2. There exists a unique solution φt ∈ C 1 (R, S(R3 )) of (1.1).
The proof is provided in Proposition 3.1 of [HaOz].
We also need the following lemma:
Lemma 3.3. For any pair (α, β) satisfying 0 < α < d < β, we have∫|x − y|−α ⟨y⟩−β ≤ C⟨x⟩−α .
RdFor the proof, the reader is referred to Appendix 2, Lemma 1(c) in [AS] andLemma 17.2 in [Is].
We can now state the following:
Theorem 3.2. Let V : R3 → R be of the form (3.1). Then, Ht is a selfadjoint operator on L2 and σess (Ht ) = [0, ∞). Moreover, σdisc (Ht ) ⊂ (−∞, 0)
is infinite.
Proof. Let us consider the first part of the statement. By the proof of Lemma
2.5 in [HaOz], we have
∥Vt ∥L∞ ≤ C∥∇φt ∥γ ∥φt ∥2−γ .
Moreover, by the virtue of Lemma 3.3, we see that∫∫42−γ
2|x − y|−γ ⟨y⟩−4 dy|x − y| |φt | (y)dy ≤ ∥⟨y⟩ |φt | ∥L∞
R3R3−γ≤ C⟨y⟩ ,8since φt ∈ S(R3 ) is true by Proposition 3.2. Therefore, we know that Vt ∈ L∞and Vt tends to 0 as |x| → ∞, implying that Vt ∈ L2 + L∞,ϵ . Hence, as whenproving Theorem 2.5, we have completed the proof of the first part.
It remains to prove the infiniteness of σdisc (Ht ). By a well-known theorem(see e.q., [Ar, Si2]), it is suﬃcient to show that there exists C, R > 0 such thatVt (x) ≤ −C, |x| ≥ R.
|x|γ

Following the proof of Proposition 5 in [Si1], we need only to show that∫|x − y|γ |φt (x − y)|2 |y|−γ dy → 0 as |x| → ∞.
(3.2)
|y|≥r0Since|y|γ |φt (y)|2 ≤ ∥⟨y⟩4+γ |φt (y)|2 ∥L∞ ⟨y⟩−4 ,the above integral can be estimated by∫∫γ2−γ|x − y| |φt (x − y)| |y| dy =R3R3≤C∫|y|γ |φt (y)|2 |x − y|−γ dy⟨y⟩−4 |x − y|−γ

R3−γ

≤ C⟨x⟩ ,which demonstrates the truth of (3.2).

References[AB]: Z. Ammari, S. Breteaux, Propagation of chaos for many-boson systems
in one dimension with a point pair-interaction, Asmpt. Anal., 76, 123–170.
[AS]: P. Asholm, G. Schmidt, Spectral and scattering theory for Schrödinger
operators, Arch. Rat. Mech. Anal. 40, 281–311 (1971).
[Ar]: A. Arai, Spectral analysis of an eﬀective Hamiltonian in nonrelativistic
quantum electrodynamics, Ann.Henri Poincaré., 12, 119–152 (2011).
[G]: F. Golse, The Mean-Field Limit for the Dynamics of Large Particle Systems. Journées Équations aux dérivées partielles, GDR 2434 (2003).
[HaOz]: N. Hayashi, T. Ozawa, Smoothing eﬀect of some Schrödinger equations, J. Funct. Anal., 85, 307–348 (1989).
[Is]: H. Isozaki, Many-body Schrödinger equations, Maruzen, 2004 (in Japanese).

9[ReSi IV]: M. Reed, B. Simon, Methods of modern mathematical physics IV,
Analysis of operators., New York, Academic Press, 1978.
[Si1]: B. Simon, On the infiniteness or finiteness of the number bound statesof an N -body quantum systems, I., Helv. Phys. Acta, 43, 607–630 (1970).
[Si2]: B. Simon, Functional integration in quantum physics, New York, Academic Press, 1979.

H.SasakiDepartments of Mathematics and Informatics
Chiba University
Chiba 263-8522
Japane-mail: sasaki@math.s.chiba-u.ac.jp
S.Shimizu
Osaka City University Advance Mathematical Institute3-3-138 Sugimoto, Sumiyohi-ku
Osaka 558-8585
Japane-mail: shimizu.ims@gmail.com
A.Suzuki
Department of Mathematics, Faculty of engineering
Shinshu University
Nagano 380-8553
Japane-mail: akito@shinshu-u.ac.jp

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