On the Generalized Semi-Relativistic

Schrödinger-Poisson System in ℝ^{n}

Walid Abou Salem, Thomas Chen, and Vitali Vougalter

Received: December 21, 2012 Revised: March 23, 2013

Communicated by Heinz Siedentop

Abstract. The Cauchy problem for the semi-relativistic Schrödinger-Poisson system of equations is studied in ℝ^{n}, n ≥ 1, for a wide class of nonlocal interactions. Furthermore, the asymptotic behavior of the solution as the mass tends to infinity is rigorously discussed, and compared with solutions to the non-relativistic Schrödinger-Poisson system.

2010 Mathematics Subject Classification: 82D10, 82C10 Keywords and Phrases: Schrödinger-Poisson system, mean-field dynamics, long-range interaction, functional spaces, density matrices, Cauchy problem, global existence, infinite mass limit.

1 Introduction

1.1 Motivation and heuristic discussion

In this article, we study the global Cauchy problem for the semi-relativistic Schrödinger-Poisson system in ℝ^{n}, n ≥ 1, for a wide class of nonlocal interactions, both in the attractive and repulsive cases. This system is relevant to the description of many-body semi-relativistic quantum particles in the meanfield limit. We consider a system of N semi-relativistic quantum particles in ℝ^{n}, n ≥ 1 with long-range two-body interactions g \frac{1}{N} ∑_{1≤i<j≤N} \frac{1}{|x_{i}−x_{j}|^{γ}}, with 0 < γ ≤ 1 if n ≥ 2, and 0 < γ < 1 if n = 1, and with g ∈ ℝ. In the mean-field limit, one can formally show that the density matrix that describes the mixed state of the system satisfies the Hartree-von Neumann equation

\begin{cases}i∂_{t}ρ(t) = [H_{m} + w_{γ} ⋆ n(t), ρ(t)], x ∈ ℝ^{n}, n ≥ 1, t ≥ 0 \\ H_{m} = \sqrt{m^{2} − Δ} − m, w_{γ} = g \frac{1}{|x|^{γ}} , n(t, x) = ρ(t, x, x), ρ(0) = ρ_{0} \end{cases} ,

(1.1)

where Δ stands for the n-dimensional Laplacian, ⋆ stands for convolution in ℝ^{n}, and m ≥ 0 is the mass.^{1} Since ρ_{0} is a positive, self-adjoint trace-class operator acting on L^{2}(ℝ^{n}), its kernel can be decomposed with respect to an orthonormal basis of L^{2}(ℝ^{n}),

ρ_{0}(x, y) = ∑_{k∈ℕ} λ_{k}ψ_{k}(x)\overline{ψ_{k}(y)}  (1.2)

where {ψ_{k}}_{k∈ℕ} denotes an orthonormal basis of L^{2}(ℝ^{n}). Furthermore,

\underline{λ} := \{λ_{k}\}_{k∈ℕ} ∈ l^{1} , λ_{k} ≥ 0 , ∑_{k} λ_{k} = 1.

We will show that there exists a one-parameter family of complete orthonormal bases of L^{2}(ℝ^{n}), {ψ_{k}(t)}_{k∈ℕ}, for t ∈ ℝ_{+}, such that the kernel of the solution ρ(t) to (1.1) can be represented as

ρ(t, x, y) = ∑_{k∈ℕ} λ_{k}ψ_{k}(t, x)\overline{ψ_{k}(t, y)}.  (1.3)

Substituting (1.3) in (1.1), the one-parameter family of orthonormal vectors {ψ_{k}(t)}_{k∈ℕ} is seen to satisfy the semi-relativistic Schrödinger-Poisson system

i\frac{∂ψ_{k}}{∂t} = H_{m}ψ_{k} + V ψ_{k}, k ∈ ℕ  (1.4)
V [Ψ] = w_{γ} ⋆ n[Ψ], Ψ := {ψ_{k}}_{k=1}^{∞},  (1.5)
n[Ψ(x, t)] = ∑_{k=1}^{∞} λ_{k}|ψ_{k}|^{2} .  (1.6)

The purpose of this note is to show global well-posedness of (1.4) in a suitable Banach space (to be specified below), and to study the asymptotics of the solution as the mass m tends to ∞, which we compare to solutions to the non-relativistic Schrödinger-Poisson system, see [11]. The semi-relativistic Schrödinger-Poisson system of equations in a finite domain of ℝ^{3} and with repulsive Coulomb interactions has been studied recently in [1, 2]. Here, we generalize the result of [1] in several ways. First, the problem is studied in ℝ^{n}, n ≥ 1.

1The rigorous derivation of the semi-relativistic Hartree-von Neumann equation is a topic of future work, see [3, 4] for a derivation of this system of equations in the non-relativistic case.

Documenta Mathematica 18 (2013) 343–357

Semi-relativistic Schrödinger-Poisson system in ℝ^{n} 345

Second, we consider a wide class of nonlocal interactions in both the attractive and repulsive cases, and which includes the repulsive Coulomb case in three spatial dimensions. Third, in the infinite mass limit m → ∞, we prove that solutions to the semi-relativistic, and to the non-relativistic Schrödinger-Poisson systems become indistinguishable; the latter has been studied extensively, see for example [5, 8] and references therein. In the special case when the initial density matrix is a pure state ρ_{0} = |ψ_{0}⟩⟨ψ_{0}|, the Schrödinger-Poisson system becomes a single Hartree equation

i∂_{t}ψ = (\sqrt{m^{2} − Δ} − m)ψ + (w_{γ} ⋆ |ψ|^{2})ψ, ψ(0) = ψ_{0}.

In that sense, our analysis generalizes the results of [10, 7] to the effective dynamics of a mixed state of a semi-relativistic system. The organization of this paper is as follows. In Subsection 1.3 we state our main results. We prove local and global well-posedness in Section 2. Finally, in Section 3, we discuss the asymptotic behavior of the solutions as the mass tends to infinity. For the benefit of a general reader, we recall some useful results about fractional integration and fractional Leibniz rule in Appendix A.

1.2 Notation

• A ≲ B means that there exists a positive constant C independent mass m such that A ≤ C B.

• L^{p} stands for the standard Lebesgue space. Furthermore, L_{I}^{p}B = L^{p}(I; B). ⟨·, ·⟩_{L^{2}} denotes the L^{2}(ℝ^{n}) inner product. We will often use the abbreviated notation L_{T}^{p} for L_{[0,T]}^{p}, in the situation where [0, T] denotes a time interval.

• l^{1} = {{a_{l}}_{l∈ℕ}| ∑_{l≥1} |a_{l}| < ∞}.

• W^{s,p} = (−Δ + 1)^{− \frac{s}{2}} L^{p}, the standard (complex) Sobolev space. When p = 2, W^{s,2} = H^{s}. Ḣ^{s} denotes the homogeneous Sobolev space with norm ∥ψ∥Ḣ^{s} = (⟨ψ, (−Δ)^{s}ψ⟩_{L^{2}})^{\frac{1}{2}} .

• For fixed \underline{λ} ∈ l^{1}, λ_{k} ≥ 0, and for sequences of functions Φ := \{ϕ_{k}\}_{k∈ℕ} and Ψ := {ψ_{k}}_{k∈ℕ}, we define the inner product

⟨Φ, Ψ⟩_{L^{2}} := ∑_{k≥1} λ_{k}⟨ϕ_{k}, ψ_{k}⟩_{L^{2}},

which induces the norm

∥Φ∥_{L^{2}} = (∑_{k≥1} λ_{k}∥ϕ_{k}∥_{L^{2}}^{2})^{\frac{1}{2}} .

The corresponding Hilbert space is L^{2}.

• For fixed \underline{λ} ∈ l^{1}, λ_{k} ≥ 0,

H^{s} = {Ψ = {ψ_{k}}_{k∈ℕ}| ψ_{k} ∈ H^{s}, ∑_{k≥1} λ_{k}∥ψ_{k}∥_{H^{s}}^{2} < ∞}

is a Banach space with norm ∥Ψ∥_{H^{s}} = (∑_{k≥1} λ_{k}∥ψ_{k}∥_{H^{s}}^{2})^{\frac{1}{2}} .

• For fixed \underline{λ} ∈ l^{1}, λ_{k} ≥ 0,

Ḣ^{s} = {Ψ = {ψ_{k}}_{k∈ℕ}| ψ_{k} ∈ Ḣ^{s}, ∑_{k≥1} λ_{k}∥ψ_{k}∥2 ; Ḣ^{s} < ∞}

is a Banach space with norm ∥Ψ∥Ḣ^{s} = (∑_{k≥1} λ_{k}∥ψ_{k}∥2 ; Ḣ^{s})^{\frac{1}{2}} .

1.3 Statement of main results

For s > 0, we define the state space for the Schrödinger-Poisson system by

S^{s} := \{(Ψ, \underline{λ})| Ψ = \{ψ_{k}\}_{k=1} ∈ H^{s} is a complete orthonormal system in L^{2}(ℝ^{n}),

\underline{λ} = \{λ_{k}\}_{k∈ℕ} ∈ l^{1}, λ_{k} ≥ 0\}.

The following is our first main result about the global Cauchy problem.

Theorem 1.1. Consider the system of equations (1.4)-(1.6), with m ≥ 0, with 0 < γ ≤ 1 if n ≥ 2, and 0 < γ < 1 if n = 1. Suppose that (Ψ(0), \underline{λ}) ∈ S^{s}, s ≥ 1/2. If g ≥ 0, or g < 0 with ∥Ψ(0)∥_{L^{2}} small enough, then there is a unique mild solution (Ψ, \underline{λ}) ∈ C([0, ∞], S^{s}).

Remark 1.2. \underline{λ} is time-independent, and hence the evolution can be thought as that of Ψ ∈ H^{s}.

Remark 1.3. Local well-posedness requires less regularity, in particular, s ≥ γ/2, see Proposition 2.2 in Section 2.1. On the other hand, in order to enhance local to global well-posedness, energy conservation is used, and consequently, s ≥ \frac{1}{2} is assumed to ensure finiteness of the energy.

Remark 1.4. It follows from the proof of local well-posedness (Proposition 2.2 in Section 2.1) that there exists a positive time T independent of m ≥ 0 such that ∥Ψ∥_{L_{T}^{∞}H^{s}} ≤ C∥Ψ(0)∥_{H^{s}}, s ≥ γ/2, where C > 0 is independent of m.

Remark 1.5. The solution is continuous in the mass m. In particular, as m ↘ 0, and for T > 0 fixed, Ψ → Ψ^{(0)} strongly in L_{T}^{∞}(H^{s}), s ≥ 1/2, where Ψ^{(0)} satisfies (1.4)-(1.6) with m = 0 and initial condition Ψ(0), see Proposition 2.6 in Sect. 2.

The second result is about the infinite mass limit. Let Γ satisfy the nonrelativistic Schrödinger-Poisson system of equations

i\frac{∂ψ_{k}}{∂t} = −\frac{1}{2m}Δψ_{k} + V ψ_{k}, k ∈ ℕ

V [Ψ] = w_{γ} ⋆ n[Ψ], Ψ := {ψ_{k}}_{k=1}^{∞},

n[Ψ(x, t)] = ∑_{k=1}^{∞} λ_{k}|ψ_{k}|^{2},

with initial condition Ψ(0) = {ψ_{k}(0)}_{k∈ℕ}.

Theorem 1.6. Suppose that the hypotheses of Theorem 1.1 hold. Then there exists τ > 0 such that Ψ − Γ → 0 in L_{τ}^{∞}(H^{s}), s ≥ γ/2, as m → ∞.

In other words, when the mass tends to infinity, the solutions of the semirelativistic, and of the non-relativistic Schrödinger-Poisson systems of equations asymptotically become indistinguishable.

Remark 1.7. The proof of Theorem 1.6 relies on local well-posedness, and this is why the result holds for s ≥ γ/2.

2 Well-posedness

2.1 Local well-posedness

In what follows, we fix \underline{λ} ∈ l^{1}, λ_{l} ≥ 0, l ∈ ℕ. We start by showing that the nonlinearity V [Ψ]Ψ is locally Lipschitz.

Lemma 2.1. For Ψ, Φ ∈ H^{s}, s ≥ γ/2,

∥V [Ψ]Ψ − V [Φ]Φ∥_{H^{s}} ≲ (∥Ψ∥_{H^{s}}^{2} + ∥Φ∥_{H^{s}}^{2})∥Ψ − Φ∥_{H^{s}}.

Proof. The proof relies on the fractional Leibniz rule and fractional integration, see Appendix A. From the Minkowski inequality,

∥V [Ψ]Ψ − V [Φ]Φ∥_{H^{s}} ≲ ∥(V [Ψ] − V [Φ])Ψ∥_{H^{s}} + ∥V [Φ](Ψ − Φ)∥_{H^{s}} (2.1)

We begin by estimating the first term on the right.

∥(V [Ψ] − V [Φ])Ψ∥_{H^{s}} ≲ ∑_{k,l≥1} λ_{k}λ_{l}∥w_{γ} ⋆ (|ψ_{l}|^{2} − |ϕ_{l}|^{2})ψ_{k}∥_{H^{s}}

≲ ∑_{k,l≥1} λ_{k}λ_{l}{∥w_{γ} ⋆ (|ψ_{l}|^{2} − |ϕ_{l}|^{2})∥_{L^{∞}}∥ψ_{k}∥_{H^{s}}

+ ∥w_{γ} ⋆ (|ψ_{l}|^{2} − |ϕ_{l}|^{2})∥_{W^{s, \frac{2n}{γ}}} ∥ψ_{k}∥_{L^{\frac{2n}{n−γ}}} \}

≲ ∑_{k,l≥1} λ_{k}λ_{l}\{∥ψ_{l} − ϕ_{l}∥_{H^{\frac{γ}{2}}} (∥ψ_{l}∥_{H^{\frac{γ}{2}}} + ∥ψ_{l}∥_{H^{\frac{γ}{2}}} )∥ψ_{k}∥_{H^{s}}

+ ∥|ψ_{l}|^{2} − |ϕ_{l}|^{2}∥_{L^{\frac{2n}{2n−γ}}} ∥ψ_{k}∥_{H^{\frac{γ}{2}}} \}
≲ (∥Ψ∥_{H^{s}}^{2} + ∥Φ∥_{H^{s}}^{2})∥Ψ − Φ∥_{H^{s}}.  (2.2)

Here, we used Minkowski inequality in the first line, fractional Leibniz rule (Lemma A.1 in the Appendix) in the second line, Hölder’s inequality, fractional integration (Lemma A.2) and Lemma A.3 in the third line. Similarly,

∥V [Φ](Ψ − Φ)∥_{H^{s}} ≲ ∑_{k,l≥1} λ_{k}λ_{l}∥w_{γ} ⋆ |ϕ_{l}|^{2}(ψ_{k} − ϕ_{k})∥_{H^{s}}

≲ ∑_{k,l≥1} λ_{k}λ_{l}\{∥w_{γ} ⋆ |ϕ_{l}|^{2}∥_{L^{∞}}∥ψ_{k} − ϕ_{k}∥_{H^{s}} + ∥w_{γ} ⋆ |ϕ_{l}|^{2}∥_{W^{s, \frac{2n}{γ}}} ∥ψ_{k} − ϕ_{k}∥_{L^{\frac{2n}{n−γ}}} \}

≲ ∑_{k,l≥1} λ_{k}λ_{l}\{∥ϕ_{l}∥_{H^{\frac{γ}{2}}}^{2} ∥ψ_{k} − ϕ_{k}∥_{H^{s}} + ∥|ϕ_{l}|^{2}∥_{L^{\frac{2n}{2n−γ}}} ∥ψ_{k} − ϕ_{k}∥_{H^{\frac{γ}{2}}} \}

≲ ∥Φ∥_{H^{s}}^{2}∥Ψ − Φ∥_{H^{s}}.  (2.3)

The claim of the lemma follows from inequalities (2.1), (2.2) and (2.3).

Using a standard contraction map argument, the generalized semi-relativistic Schrödinger-Poisson system of equations is locally well-posed.

Proposition 2.2. Consider the system of equations (1.4)-(1.6), with m ≥ 0, 0 < γ ≤ 1 if n ≥ 2, and 0 < γ < 1 if n = 1. Suppose that (Ψ(0), \underline{λ}) ∈ S^{s}, s ≥ γ/2. Then there exists a positive time T such that the unique solution Ψ ∈ C([0, T]; H^{s}). Furthermore, there exists a maximal time τ^{∗} ∈ (0, ∞] such that lim_{t↗τ^{∗}} ∥Ψ(t)∥_{H^{s}} = ∞.

Proof. Given ρ, T > 0, consider the Banach space

B_{T,ρ}^{s} = {Ψ ∈ L_{T}^{∞}(H^{s}) : ∥Ψ∥_{L_{T}^{∞}H^{s}} ≤ ρ}.

Let U^{(m)} = e^{−itH_{m}}, the unitary operator generated by the semi-relativistic Hamiltonian H_{m} = \sqrt{−Δ + m^{2}} − m. We define the mapping N by

N (Ψ)(t) = U^{(m)}(t)Ψ(0) − i ∫_{0}^{t} U^{(m)}(t − t′)V [Ψ(t′)]Ψ(t′)dt′,

which is the solution given by the Duhamel formula. First we show that N is a mapping from B_{T,ρ}^{s} into itself.

∥N (Ψ)∥_{L_{T}^{∞}H^{s}} ≤ ∥Ψ(0)∥_{H^{s}} + T ∥V [Ψ]Ψ∥_{L_{T}^{∞}H^{s}}

≤ ∥Ψ(0)∥_{H^{s}} + T ∑_{k,l≥1} λ_{k}λ_{l}∥w_{γ} ⋆ |ψ_{l}|^{2}ψ_{k}∥_{L_{T}^{∞}H^{s}}

≤ ∥Ψ(0)∥_{H^{s}} + T ∑_{k,l≥1} λ_{k}λ_{l}{∥w_{γ} ⋆ (|ψ_{l}|^{2})∥_{L_{T}^{∞}L^{∞}}∥ψ_{k}∥_{L_{T}^{∞}H^{s}}+

+ ∥w_{γ} ⋆ (|ψ_{l}|^{2})∥_{L_{T}^{∞}W^{s, \frac{2n}{γ}}} ∥ψ_{k}∥_{L_{T}^{∞}L^{\frac{2n}{n−γ}}} \},

where we have used fractional Leibniz rule (Lemma A.1) in the last inequality. It follows from fractional integration (Lemma A.2) and Sobolev embedding H^{\frac{γ}{2}} ↪ L^{\frac{2n}{n−γ}} that

∥N (Ψ)∥_{L_{T}^{∞}H^{s}} ≤ ∥Ψ(0)∥_{H^{s}} + T ∑_{k,l≥1} λ_{k}λ_{l}\{∥ψ_{l}∥_{L_{T}^{∞}H^{\frac{γ}{2}}}^{2} ∥ψ_{k}∥_{L_{T}^{∞}H^{s}}+

+ ∥ψ_{l}∥_{L_{T}^{∞}L^{\frac{2n}{n−γ}}}^{2} ∥ψ_{k}∥_{L_{T}^{∞}H^{s}}\}

≤ ∥Ψ(0)∥_{H^{s}} + T ∑_{k,l≥1} λ_{k}λ_{l}\{∥ψ_{l}∥_{L_{T}^{∞}H^{\frac{γ}{2}}}^{2} ∥ψ_{k}∥_{L_{T}^{∞}H^{s}}\}

≤ ∥Ψ(0)∥_{H^{s}} + T (∑_{l≥1} λ_{l}\{∥ψ_{l}∥_{L_{T}^{∞}H^{\frac{γ}{2}}}^{2} )(∑_{k≥1} λ_{k}∥ψ_{k}∥_{L_{T}^{∞}H^{s}}^{2})^{\frac{1}{2}}

≤ ∥Ψ(0)∥_{H^{s}} + T ∥Ψ∥_{L_{T}^{∞}H^{\frac{γ}{2}}}^{2} ∥Ψ∥_{L_{T}^{∞}H^{s}},

where we have used the fact that λ_{k} ≥ 1 and ∑_{k≥1} λ_{k} = 1 before the last inequality. Since s ≥ \frac{γ}{2} , and since by assumption, Ψ ∈ B_{T,ρ}^{s}, we can choose T and ρ such that

∥Ψ(0)∥_{H^{s}} ≤ \frac{ρ}{2}, T ρ^{2} < \frac{1}{2},

it follows from the last inequality and the Duhamel formula that

∥Ψ∥_{L_{T}^{∞}H^{s}} ≤ 2∥Ψ(0)∥_{H^{s}} ≤ ρ.

Second, since the nonlinearity is locally Lipschitz (Lemma 2.1), N is a contraction map for sufficiently small T.

∥N (Ψ) − N (Φ)∥_{L_{T}^{∞}H^{s}} ≤ T ∥V [Ψ]Ψ − V [Φ]Φ∥_{L_{T}^{∞}H^{s}}
≲ T ρ^{2}∥Ψ − Φ∥_{L_{T}^{∞}H^{s}}.

Local well-posedness follows from a standard contraction mapping argument, see for example, [6].

It follows from local well-posedness that for every k ∈ ℕ, ∥ψ_{k}∥_{L^{2}} is conserved.

Lemma 2.3. Suppose that the hypotheses of Proposition 2.2 hold. Then ∥ψ_{k}(t)∥_{L^{2}} = ∥ψ_{k}(0)∥_{L^{2}}, t ∈ [0, τ^{∗}).

Proof. Multiplying (1.4) by \overline{ψ_{k}} and integrating over space yields

\frac{i}{2}∂_{t}∥ψ_{l}∥^{2} = ⟨ψ_{l}, H_{m}ψ_{l}⟩ + ⟨ψ_{l}, V [Ψ]ψ_{l}⟩.

Taking the imaginary part of both sides of the equation yields ∂_{t}∥ψ_{l}∥^{2} = 0.

The energy functional associated with the semi-relativistic Schrödinger-Poisson system is

E(Ψ) = \frac{1}{2}⟨Ψ, H_{m}Ψ⟩_{L^{2}} + \frac{1}{4}⟨Ψ, V [Ψ]Ψ⟩_{L^{2}}.

Formally, conservation of energy follows from multiplying (1.4) by λ_{l}∂_{t}\overline{ψ_{k}}, integrating over space, and summing over k ≥ 1. To make the argument precise, we need a regularization procedure.

Lemma 2.4. Suppose that the hypotheses of Proposition 2.2 hold. Then E(Ψ(t)) = E(Ψ(0)), t ∈ [0, τ^{∗}), is satisfied for solutions Ψ ∈ C([0, τ^{∗}), H^{s}) with s ≥ \frac{1}{2}.

Proof. Let

J_{ϵ} = (ϵH_{m} + 1)^{−1}, ϵ > 0,

act on the sequence of embedding spaces

⋯ H^{\frac{3}{2}} ↪ H^{\frac{1}{2}} ↪ H^{\frac{−1}{2}} ↪ H^{\frac{−3}{2}} ⋯

It follows from fractional calculus that

(i) J_{ϵ} is a bounded operator from H^{s} to H^{s+1},

(ii) ∥J_{ϵ}Ψ∥_{H^{s}} ≤ ∥Ψ∥_{H^{s}}, and

(iii) J_{ϵ}Ψ → Ψ strongly in H^{s} as ϵ → 0.

Now,

E(J_{ϵ}Ψ(t_{2})) − E(J_{ϵ}Ψ(t_{1})) = ∫_{t_{1}}^{t_{2}} ∂_{t}E(J_{ϵ}Ψ(t)) dt
= Re{ ∫_{t_{1}}^{t_{2}} −i⟨H_{m}J_{ϵ}Ψ(t), H_{m}J_{ϵ}Ψ(t)⟩_{L^{2}}+
+ ⟨H_{m}J_{ϵ}Ψ(t), J_{ϵ}V [Ψ(t)]Ψ(t)⟩_{L^{2}}+
+ ⟨J_{ϵ}V [J_{ϵ}Ψ(t)]J_{ϵ}Ψ(t), H_{m}J_{ϵ}Ψ(t)⟩_{L^{2}}+
+ ⟨J_{ϵ}V [J_{ϵ}Ψ(t)]J_{ϵ}Ψ(t), J_{ϵ}V [Ψ(t)]Ψ(t)⟩_{L^{2}}}.

The first term is trivially zero, since H_{m}J_{ϵ} = J_{ϵ}H_{m}. Let

g_{ϵ}(t) = Re{⟨H_{m}J_{ϵ}Ψ(t), J_{ϵ}V [Ψ(t)]Ψ(t)⟩_{L^{2}}+
+ ⟨J_{ϵ}V [J_{ϵ}Ψ(t)]J_{ϵ}Ψ(t), H_{m}J_{ϵ}Ψ(t)⟩_{L^{2}}+
+ ⟨J_{ϵ}V [J_{ϵ}Ψ(t)]J_{ϵ}Ψ(t), J_{ϵ}V [Ψ(t)]Ψ(t)⟩_{L^{2}}}.

Then

E(J_{ϵ}Ψ(t_{2})) − E(J_{ϵ}Ψ(t_{1})) = ∫_{t_{1}}^{t_{2}} g_{ϵ}(t)dt.

It follows from the above properties (i)-(iii) of J_{ϵ} that lim_{ϵ→0} g_{ϵ}(t) = 0. Furthermore,

g_{ϵ}(t) ≤ ∥V [Ψ(t)]Ψ(t)∥_{L^{2}}∥H_{m}Ψ(t)∥_{L^{2}} + ∥V [Ψ(t)]Ψ(t)∥_{L^{2}}^{2}.  (2.4)

Using Lemma A.3, we have

∥V [Ψ]Ψ∥_{L^{2}} ≲ ∑_{k,l≥1} λ_{k}λ_{l}∥ψ_{l}∥2 ; Ḣ^{\frac{γ}{2}} ∥ψ_{k}∥_{L^{2}}.

The Gagliardo-Nirenberg inequality,

∥ψ_{l}∥Ḣ^{\frac{γ}{2}} ≲ ∥ψ_{l}∥γ ; Ḣ^{\frac{1}{2}} ∥ψ_{l}∥_{L^{2}}^{1−γ},

together with conservation of charge (Lemma 2.3), yields

∥V [Ψ]Ψ∥_{L^{2}} ≲ ∑_{l≥1} λ_{l}∥ψ_{l}∥2γ ; Ḣ^{\frac{1}{2}}

≲ (∑_{l≥1} λ_{l}∥ψ_{l}∥2 ; Ḣ^{\frac{1}{2}} )^{γ}

≲ ∥Ψ∥_{H^{\frac{1}{2}}}^{2γ} ,

where we have used in the second inequality the fact that ∑_{l≥1} λ_{l} = 1, λ_{l} ≥ 0, and f(x) = x^{γ}, 0 < γ < 1, is concave (equality when γ = 1 is trivially satisfied). Substituting back in (2.4) yields

g_{ϵ}(t) ≲ ∥Ψ∥_{H^{\frac{1}{2}}}^{2γ+1} + ∥Ψ∥_{H^{\frac{1}{2}}}^{4γ} ,

which is finite for t < τ^{∗}. By the Dominated Convergence Theorem,

E(Ψ(t_{2})) − E(Ψ(t_{1})) = ∫_{t_{1}}^{t_{2}} lim_{ϵ→0} g_{ϵ}(t)dt = 0 ,

as claimed.

Global well-posedness in H^{s}, for s ≥ \frac{1}{2}, follows from conservation of charge and energy.

Proposition 2.5. Suppose that the hypotheses of Proposition 2.2 hold. Then, if g > 0 or g < 0 with ∥Ψ(0)∥_{L^{2}} small enough, and for s ≥ \frac{1}{2},

∥Ψ(t)∥_{H^{s}} ≤ C∥Ψ(0)∥_{H^{s}}e^{α(E(Ψ(0))+∥Ψ(0)∥_{L^{2}}^{δ})t},

where C, α and δ are positive constants that are independent of m ≥ 0.

Proof. We start by bounding ∥Ψ(t)∥Ḣ^{\frac{γ}{2}} from above, uniformly in time.

⟨Ψ, V [Ψ]Ψ⟩_{L^{2}} = ∑_{l≥1} λ_{l}⟨ψ_{l}, V [Ψ]ψ_{l}⟩

≤ ∥V [Ψ]∥_{L^{∞}}∥Ψ∥_{L^{2}}^{2}

≲ (∑_{k≥1} λ_{k}∥ψ_{k}∥2 ; Ḣ^{\frac{γ}{2}} )∥Ψ∥_{L^{2}}^{2}

≲ (∑_{k≥1} λ_{k}∥ψ_{k}∥2γ ; Ḣ^{\frac{1}{2}} )∥Ψ∥_{L^{2}}^{2}

≲ (∑_{k≥1} λ_{k}∥ψ_{k}∥2 ; Ḣ^{\frac{1}{2}} )^{γ}∥Ψ∥_{L^{2}}^{2}

≲ ∥Ψ∥_{H^{\frac{1}{2}}}^{γ} ∥Ψ∥_{L^{2}}^{2}.

Here, we used Hölder’s inequality in the second line, Lemma A.3 in the third line, the Gagliardo-Nirenberg inequality and conservation of charge in the fourth line, and ∑_{k≥1} λ_{k} = 1, λ_{k} ≥ 0, the fact that x^{γ}, 0 < γ < 1, is concave in the fifth line (equality when γ = 1 is trivially satisfied). Together with conservation of energy (Lemma 2.4), this implies that for g > 0 or g < 0 with ∥Ψ(0)∥_{L^{2}} small enough,

∥Ψ∥Ḣ^{\frac{γ}{2}} ≤ α (E(Ψ(t)) + ∥Ψ(0)∥_{L^{2}}^{δ}) ,  (2.5)

where α and δ are constants independent of the mass m ≥ 0. Now, it follows from the Duhamel formula that

∥Ψ(t)∥_{H^{s}} ≤ ∥Ψ(0)∥_{H^{s}} + ∫_{0}^{t} ∥Ψ(t′)∥2 ; Ḣ^{\frac{γ}{2}} ∥Ψ(t′)∥_{H^{s}} dt′
≤ ∥Ψ(0)∥_{H^{s}} + α (E(Ψ(t)) + ∥Ψ(0)∥_{L^{2}}^{δ}) ∫_{0}^{t} ∥Ψ(t′)∥_{H^{s}} dt′,

where we used Hölder’s and Minkowski inequalities in the first line, and (2.5) in the second line. By Gronwall’s lemma,

∥Ψ(t)∥_{H^{s}} ≤ ∥Ψ(0)∥_{H^{s}}e^{α(E(Ψ(0))+∥Ψ(0)∥_{L^{2}}^{δ})t}

follows.

Proof of Theorem 1.1. It follows from Propositions 2.2 and 2.5 that τ^{∗} = ∞, i.e., the generalized semi-relativistic Schrödinger-Poisson system of equations is globally well-posed.

We now prove the claim of Remark 1.5 about the asymptotic behaviour of the system as the mass tends to zero.

Proposition 2.6. Consider the system of equations (1.4)-(1.6) with initial condition (\underline{λ}, Ψ(0)). Let Ψ^{(0)} denote the solution of the initial value problem with mass m = 0, and fix T > 0. Under the hypotheses of Proposition 2.5, Ψ → Ψ^{(0)} strongly in L_{T}^{∞}(H^{s}), s ≥ 1/2, as m → 0.

Proof. Proposition 2.5 implies that, given T > 0, there exists finite ρ > 0 such that

sup_{m∈[0,1]} ∥Ψ∥_{L_{T}^{∞}H^{s}} < ρ.  (2.6)

We now compare the norm of the difference of Ψ(t) and Ψ^{(0)}(t), t ∈ [0, T]. It follows from the Duhamel formula that

∥Ψ(t) − Ψ^{(0)}(t)∥_{H^{s}} ≲ ∥ (U^{(m)}(t) − U^{(0)}(t)) Ψ(0)∥_{H^{s}}+

+ ∫_{0}^{t}{∥V [Ψ(t′)]Ψ(t′) − V [Ψ^{(0)}(t′)]Ψ^{(0)}(t′)∥_{H^{s}}+

+ ∥ (U^{(m)}(t′) − U^{(0)}(t′)) V [Ψ^{(0)}(t′)]Ψ^{(0)}(t′)∥_{H^{s}}}dt′

≲ mT ∥Ψ(0)∥_{H^{s}} + ∫_{0}^{t} ∥V [Ψ(t′)]Ψ(t′) − V [Ψ^{(0)}(t′)]Ψ^{(0)}(t′)∥_{H^{s}} dt′

+ \frac{mT^{2}}{2} ∥V [Ψ^{(0)}]Ψ^{(0)}∥_{L_{T}^{∞}H^{s}},

where we used Minkowski inequality in the first inequality and Hölder’s inequality in the second. We also used 0 ≤ \sqrt{−Δ + m^{2}} − m ≤ m. It follows from the fact that the nonlinearity is locally Lipschitz (Lemma 2.1) and (2.6) that

∥V [Ψ(t′)]Ψ(t′) − V [Ψ^{(0)}(t′)]Ψ^{(0)}(t′)∥_{H^{s}} ≲ ρ^{2}∥Ψ(t′) − Ψ^{(0)}(t′)∥_{H^{s}},
∥V [Ψ^{(0)}]Ψ^{(0)}∥_{L_{T}^{∞}H^{s}} ≲ ρ^{3}.

Hence

∥Ψ(t) − Ψ^{(0)}(t)∥_{H^{s}} ≲ mρT + mρ^{3}T + ρ^{2} ∫_{0}^{t} ∥Ψ(t′) − Ψ^{(0)}(t′)∥_{H^{s}} dt′.

By Gronwall’s lemma, Ψ → Ψ^{(0)} strongly in L_{T}^{∞}(H^{s}) as m → 0.

3 Asymptotic behaviour of solutions as mass tends to infinity

In this section, we discuss the asymptotics of the solution as the mass m tends to infinity.

Proof of Theorem 1.6. Recall that from the proof of local well-posedness in Section 2.1, there exists T > 0 independent of m such that ∥Ψ∥_{L_{T}^{∞}H^{s}} ≤ C∥Ψ(0)∥_{H^{s}}, s ≥ γ/2, where C is independent of m. Similarly, one can show that there exists T′ > 0 independent of m such that ∥Γ∥_{L_{T′}^{∞} H^{s}} ≤ C∥Ψ(0)∥_{H^{s}}, where C is independent of m. Let τ = min(T, T′). Let Γ̃ = {γ̃_{k}}_{k∈ℕ} satisfy the system of equations

\begin{cases}i∂_{t}Γ̃ = V [Γ̃]Γ̃, \\ V [Γ̃] = w_{γ} ⋆ n[Γ̃], n[Γ̃] = ∑_{k=1}^{∞} λ_{k}|γ̃_{k}|^{2}, \end{cases}

with initial condition Γ̃(0) = Ψ(0). Alternatively, Γ̃ satisfies the integral equation

Γ̃(t) = Ψ(0) − i ∫_{0}^{t} V [Γ̃(t′)]Γ̃(t′)dt′.

354 W. Abou Salem, T. Chen, V. Vougalter Uniqueness of the solution follows from the fact that the nonlinearity is locally Lipschitz (Lemma 2.1). We are going to compare Ψ to Γ̃, and then Γ̃ to Γ.

∥Ψ(t) − Γ̃(t)∥_{H^{s}} ≤ ∥ (U^{(m)}(t) − 1) Ψ(0)∥_{H^{s}}+  (3.1)
+ ∫_{0}^{t} ∥ (U^{(m)}(t − t′) − 1) V [Γ̃(t′)]Γ̃(t′)∥_{H^{s}}dt′+  (3.2)
+ ∫_{0}^{t} ∥V [Ψ(t′)]Ψ(t′) − V [Γ̃(t′)]Γ̃(t′)∥_{H^{s}}dt′.  (3.3)

To estimate the first term on the right-hand-side, we apply the Fourier transform and use Parseval’s Theorem,

∥ (U^{(m)}(t) − 1) Ψ(0)∥_{H^{s}}^{2}

= ∑_{l≥1} λ_{l} ∫_{ℝ^{n}} |e^{−it(\sqrt{m^{2}+|k|^{2}}−m)} − 1|^{2}(1 + |k|^{2})^{s}|ψ̂_{l}(0, k)|^{2}dk

≤ ∑_{l≥1} λ_{l}\{∫_{|k|≤m^{\frac{1}{4}}} |e^{−it(\sqrt{m^{2}+|k|^{2}}−m)} − 1|^{2}(1 + |k|)^{2s}|ψ̂_{l}(0, k)|^{2}dk+

+ ∫_{|k|>m^{\frac{1}{4}}} |e^{−it(\sqrt{m^{2}+|k|^{2}}−m)} − 1|^{2}(1 + |k|)^{2s}|ψ̂_{l}(0, k)|^{2}dk\}

≤ ∑_{l≥1} λ_{l}\{∫_{|k|≤m^{\frac{1}{4}}} \frac{t^{2}|k|^{4}}{(\sqrt{m^{2} + |k|^{2}} + m)^{2}} (1 + |k|)^{2s}|ψ̂_{l}(0, k)|^{2}dk+

+ 4 ∫_{|k|>m^{\frac{1}{4}}} (1 + |k|)^{2s}|ψ̂_{l}(0, k)|^{2}dk\}

≤ \frac{τ^{2}}{4m}∥Ψ(0)∥_{H^{s}}^{2} + 4 ∑_{l≥1} ∫_{|k|>m^{\frac{1}{4}}} (1 + |k|)^{2s}|ψ̂_{l}(0, k)|^{2}dk

→ 0 as m → ∞. Since V [Γ̃]Γ̃ ∈ H^{s}, it follows from the Dominated Convergence Theorem that

lim_{m→∞} ∫_{0}^{t} ∥ (U^{(m)}(t − t′) − 1) V [Γ̃(t′)]Γ̃(t′)∥_{H^{s}}dt′ = 0.

To estimate the third term, let ρ > 0 be a constant such that

sup_{m≥1}(∥Ψ∥_{L_{τ}^{∞}H^{s}} + ∥Γ∥_{L_{τ}^{∞}H^{s}}) + ∥Γ̃∥_{L_{τ}^{∞}H^{s}} ≤ ρ.

It follows from the fact that the nonlinearity is locally Lipschitz that

∥V [Ψ(t′)]Ψ(t′) − V [Γ̃(t′)]Γ̃(t′)∥_{H^{s}} ≤ Cρ^{2}∥Ψ(t′) − Γ̃(t′)∥_{H^{s}},

where C is a positive constant independent of m.

Documenta Mathematica 18 (2013) 343–357

Therefore,

∥Ψ(t) − Γ̃(t)∥_{H^{s}} ≤ f_{m} + Cρ^{2} ∫_{0}^{t} ∥Ψ(t′) − Γ̃(t′)∥_{H^{s}}dt′,

where f_{m} bounds the first two terms on the r.h.s. of (3.1). As shown above, lim_{m→∞} f_{m} = 0 and C is independent of m, so that it application of Gronwall’s lemma yields

lim_{m→∞} ∥Ψ − Γ̃∥_{L_{τ}^{∞}H^{s}} = 0.

Similarly, one can show that

∥Γ(t) − Γ̃(t)∥_{H^{s}} ≤ g_{m} + Cρ^{2} ∫_{0}^{t} ∥Ψ(t′) − Γ̃(t′)∥_{H^{s}}dt′,

where lim_{m→∞} g_{m} = 0 and C is independent of m, and it follows that

lim_{m→∞} ∥Γ − Γ̃∥_{L_{τ}^{∞}H^{s}} = 0.

Since

∥Ψ − Γ∥_{L_{τ}^{∞}H^{s}} ≤ ∥Ψ − Γ̃∥_{L_{τ}^{∞}H^{s}} + ∥Γ − Γ̃∥_{L_{τ}^{∞}H^{s}},

it follows that

lim_{m→∞} ∥Ψ − Γ∥_{L_{τ}^{∞}H^{s}} = 0 ,

as desired.

Acknowledgements

WAS acknowledges the financial support of a Discovery grant from the Natural Sciences and Engineering Research Council of Canada. T.C. was supported by NSF grants DMS-1009448 and DMS-1151414 (CAREER).

A Appendix

The following result about the fractional Leibniz rule can be found in [9].

Lemma A.1.

∥D^{s}(uv)∥_{L^{p}} ≲ ∥D^{s}u∥_{L^{q_{1}}} ∥v∥_{L^{r_{1}}} + ∥u∥_{L^{q_{2}}} ∥D^{s}v∥_{L^{r_{2}}} ,

where \frac{1}{p} = \frac{1}{q_{i}} + \frac{1}{r_{i}} , i = 1, 2.

The second result is about inequality involving fractional integral operators, which can be found, for example, in [12].

Lemma A.2. Let I_{α}, for 0 < α < n, be the fractional integral operator

I_{α}(u) = ∫_{ℝ^{n}} |x − y|^{α−n}u(y) dy.

Then

∥I_{α}(u)∥_{L^{p}} ≲ ∥u∥_{L^{q}}, \frac{1}{p} = \frac{1}{q} − \frac{α}{n}.

We also recall the following useful Hardy-type inequality.

Lemma A.3. Let 0 < γ < n. Then,

sup_{x∈ℝ^{n}} | ∫_{ℝ^{n}} \frac{1}{|x − y|^{γ}} |u(y)|^{2}dy| ≲ ∥u∥2 ; Ḣ^{\frac{γ}{2}} .

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[8] R. Illner, P. F. Zweifel. Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schr ödinger-Poisson systems. Math. Methods Appl. Sci. 17 (1994), 349–376.

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W. Abou Salem Department of Mathematics and Statistics University of Saskatchewan Saskatoon S7N 5E6 Canada walid.abousalem@usask.ca Thomas Chen Department of Mathematics University of Texas at Austin Austin, TX, 78712 USA tc@math.utexas.edu

V. Vougalter University of Cape Town Department of Mathematics and Applied Mathematics Private Bag Rondebosch 7701 South Africa Vitali.Vougalter@uct.ac.za 358

Documenta Mathematica 18 (2013)