Convergence Rates from Yukawa to Coulomb Interaction in the Thomas-Fermi-von Weizs\"acker Model

0.43 MB·

by  Faizan Q. Nazar

CONVERGENCE RATES FROM YUKAWA TO COULOMB
INTERACTION IN THE THOMAS–FERMI–VON WEIZSA¨CKER
MODEL
F. Q. NAZAR
Abstract. We establish uniform convergence, with explicit rate, of the solution to
the Thomas–Fermi–von Weizs¨acker (TFW) Yukawa model to the solution of the TFW
Coulomb model, for general condensed nuclear configurations. As a consequence, we
6 show the convergence of forces from the Yukawa to the Coulomb model. These results
1 relyonanextensionofNazar&Ortner(2015)totheYukawasetting. Auxiliaryresultsof
0
independent interest shown also include new existence, uniqueness and stability results
2
for the Yukawa ground state.

n
a
J

6
1. Introduction

]
h One of the challenges in molecular simulation is treating the interaction of charged
p
particles using the Coulomb potential. Due to the long-range of the Coulomb potential
-
h 1 , the Yukawa potential Y (x) = e−a|x|, for a > 0, is often used as a short-ranged
t |x| a |x|
a approximation [6, 5, 15, 4, 17]. The Yukawa potential also appears in the Thomas–Fermi
m
theory of impurity screening, where the parameter a > 0 represents the inverse screening
[
length of a metal [13, 14, 1].

1
The aim of this paper is to establish the uniform convergence of the Yukawa ground
v
7 state to the Coulomb ground state, in the Thomas–Fermi–von Weizsa¨cker (TFW) model.
8 The main technical result estimates the rate of convergence. A rigorous statement is given
1
in Theorem 3.5.
1
0
Theorem. Let m L∞(R3) represent a nuclear charge distribution satisfying
.
1 ∈
0 1
m 0 and lim inf m(z) dz = + .
6
1 ≥ R→∞ R x∈R3ZBR(x) ∞
:
v Let the corresponding Coulomb ground state electron density and electrostatic potential,
i denoted by u,φ : R3 R, satisfy the TFW equations,
X
→
r 5
a ∆u+ u7/3 φu = 0,
− 3 −
∆φ = 4π(m u2),
− −
and for a > 0, let the corresponding Yukawa ground state, denoted by (u ,φ ), satisfy the
a a
TFW Yukawa equations
5
∆u + u7/3 φ u = 0,
− a 3 a − a a
∆φ +a2φ = 4π(m u2).
− a a − a
Then there exists C > 0 such that
u u + φ φ Ca2. (1.1)
a W2,∞(R3) a W2,∞(R3)
k − k k − k ≤
Date: January 7, 2016.
FQN is funded by the MASDOC doctoral training centre, EPSRC grant EP/H023364/1.
1
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 2
To the best of the author’s knowledge, this is the first result that provides a rate of
convergence for ground states from Yukawa to Coulomb interaction, for any electronic
structure model.
An important consequence of (1.1) is an estimate for the rate of convergence of forces
in the TFW model, when passing from the Yukawa to Coulomb interaction. Given a
countable collection of nuclei Y = (Yj)j∈N R3 and a > 0, the TFW Yukawa and
⊂
Coulomb energy densities, (Y;x) and (Y;x) respectively, can be defined. It follows
a
E E
from (1.1) that
∂ ∂
Ea E (Y;x) dx Ca2. (1.2)
∂Y − ∂Y ≤
(cid:12)ZR3 (cid:18) k k(cid:19) (cid:12)
(cid:12) (cid:12)
A rigorous statement of this result is given in Theorem 4.1.
(cid:12) (cid:12)
(cid:12) (cid:12)
Inaforthcomingarticle[7], theaimwill betogeneralise theanalysisofvariationalprob-
lems for the mechanical response to defects in an infinite crystal [8] to electronic structure
models, using the TFW model with Coulomb interaction. The uniform convergence of
forces from Yukawa to Coulomb suggests that one could construct an approximate me-
chanical response problem using the Yukawa interaction. This could be more efficient for
the purposes of numerical simulations as it replaces the long-range Coulomb interaction
with the short-ranged Yukawa interaction. The result (1.2) suggests that the error in the
electron density may propagate into an O(a2) error in the equilibrium configuration. This
will be explained in future work.
The remainder of this article is organised as follows: In Section 2 the definition of the
TFW model is recalled and the relevant existing results are summarised. In Section 3
the main technical results are stated, including the rigorous statement of the convergence
result (1.1). Applications are presented in Section 4, followed by the detailed proofs of
the results in Section 5. An additional technical argument is given in the Appendix, that
extends uniqueness of the Yukawa ground state to all a > 0.
Remark 1. The analytical approach presented closely follows and adapts the study of
the TFW equations in [6, 11]. An overview of the TFW equations can be found in [11]
and [17] provides a background on the Yukawa potential and its various applications.
Tothebestoftheauthor’sknowledge, theclosestexisting resultto(1.1)intheliterature
is [6, Proposition 2.30], which shows u u strongly in H1 (R3) as a 0, for periodic
a → loc → (cid:3)
and neutral TFW systems, but does not estimate the rate.
Acknowledgements. TheauthorthanksVirginieEhrlacherandXavierBlancforhelpful
discussions about the TFW model in the Yukawa setting.
2. The TFW Yukawa Model
For p [1, ] define the function spaces
∈ ∞
Lp (R3) := f : R3 R K R3 compact,f Lp(K) and
loc { → |∀ ⊂ ∈ }
Lp (R3) := f Lp (R3) sup f < .
unif { ∈ loc | k kLp(B1(x)) ∞}
x∈R3
For k N, Hk (R3),Hk (R3) are defined analogously. For a multi-index α = (α ,α ,α ),
∈ loc unif 1 2 3
define the partial derivative ∂α = ∂α1∂α2∂α3. Throughout this paper, α,β denote three-
1 2 3
dimensional multi-indices.
The Coulomb interaction, for f,g L6/5(R3), is given by
∈
f(x)g(y)
D (f,g) = dx dy = f 1 (y)g(y) dy.
0 x y ∗ |·|
R3 R3 R3
Z Z | − | Z (cid:16) (cid:17)
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 3
and is finite due to the Hardy–Littlewood–Sobolev estimate [2]. The Yukawa interaction
is a short-range approximation to the Coulomb interaction, with the Yukawa potential
Y (x) = e−a|x|, for a > 0, replacing the Coulomb potential 1 . The parameter a > 0
a |x| |x|
controls the range of the interaction, in particular one formally recovers the long-ranged
Coulomb interaction as a 0. The Yukawa interaction, for a > 0 and f,g L2(R3), is
→ ∈
given by
f(x)e−a|x−y|g(y)
D (f,g) = dx dy = (f Y )(y)g(y) dy,
a a
x y ∗
R3 R3 R3
Z Z | − | Z
which is finite as Cauchy-Schwarz’ and Young’s inequality for convolutions imply
D (f,g) Y f g Ca−2 f g .
a a L1(R3) L2(R3) L2(R3) L2(R3) L2(R3)
| | ≤ k k k k k k ≤ k k k k
Let a > 0 and m L2(R3),m 0, denote the charge density of a finite nuclear cluster,
then the correspon∈ding TFW Y≥ukawa energy functional is defined, for v H1(R3), by
∈
1
ETFW(v,m) = C v 2+C v10/3 + D (m v2,m v2). (2.1)
a W |∇ | TF 2 a − −
R3 R3
Z Z
The function v corresponds to the positive square root of the electron density. The first
two terms of (2.1) model the kinetic energy of the electrons while the third term models
the Coulomb energy. This definition of the Coulomb energy is only valid for smeared
nuclei. The energy (2.1) can be rescaled to ensure that C = C = 1.
W TF
To construct the electronic ground state for aninfinite arrangement of nuclei (e.g., crys-
tals),itisnecessarytorestrictadmissiblenuclearchargedensitiestom L1 (R3),m 0,
∈ unif ≥
satisfying
(H1) sup m(z) dz < ,
∞
x∈R3ZB1(x)
1
(H2) lim inf m(z) dz = .
R→∞x∈R3 R ZBR(x) ∞
The property (H1) guarantees that no clustering of infinitely many nuclei occurs at any
point in space whereas (H2) ensures that there are no large regions that are devoid of
nuclei.
For each m satisfying (H1)–(H2), [11, Theorem 6.10] guarantees the existence and
uniqueness of a ground state (u,φ) satisfying
5
∆u+ u7/3 φu = 0, (2.2a)
− 3 −
∆φ = 4π(m u2), (2.2b)
− −
Similarly, as remarked in [6, Chapter 6], it also follows that for sufficiently small a > 0,
the existence and uniqueness of the Yukawa ground state (u ,φ ), solving
a a
5
∆u + u7/3 φ u = 0, (2.3a)
− a 3 a − a a
∆φ +a2φ = 4π(m u2), (2.3b)
− a a − a
The equation (2.2b) arises from the Coulomb interaction, as 1 is the Green’s function
4π|·|
for the Laplacian on R3, while (2.3b) is obtained for the Yukawa problem, as 1 Y is the
4π a
Green’s function for ∆+a2 on R3, a > 0.
−
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 4
Definition 1. In this article, for any nuclear configuration m satisfying (H1)–(H2), the
ground state corresponding to m refers to the unique solution (u,φ) to (2.2). For a > 0,
the Yukawa ground state corresponding to m refers to the unique solution (u ,φ ) to
a a
(2.3). (cid:3)
3. Main Results
3.1. Regularity estimates. This section generalises the TFW pointwise stability esti-
mate and its consequences [11] from the Coulomb to the Yukawa setting.
The proofs of the main results in the next section require uniform regularity estimates
for Yukawa systems refining those shown in [6], provided that a (0,a ] for some a > 0.
0 0
∈
The main regularity estimate (3.1) relies on uniform variants of (H1)–(H2), so the
class of nuclear configurations , defined in [11], is used. Given M,ω ,ω > 0, let
L2 0 1
M
ω = (ω ,ω ) and define
0 1
(M,ω) = m L2 (R3) m M,
ML2 ∈ unif k kL2unif(R3) ≤
(cid:26) (cid:12)
(cid:12)
(cid:12) R > 0 inf m(z) dz ω R3 ω . (3.1)
(cid:12) 0 1
∀ x∈R3ZBR(x) ≥ − (cid:27)
As each nuclear distribution m (M,ω) satisfies (H1)–(H2), [6, Chapter 6] guar-
L2
∈ M
antees the existence of corresponding ground states (u ,φ ) for sufficiently small a. The
a a
proof of [6, Proposition 2.2, Chapter 6] is adapted to extend existence and uniqueness of
Yukawa ground states to all a > 0. In addition, the uniformity in upper and lower bounds
on m (M,ω) yields regularity estimates and lower bounds on these ground states
L2
∈ M
which are also uniform.
Proposition 3.1. Let a > 0 and m (M,ω), then for any 0 < a a there exists
0 L2 0
∈ M ≤
(u ,φ ) solving (2.3), satisfying u 0 and
a a a
≥
u + φ C(a ,M), (3.2)
k akHu4nif(R3) k akHu2nif(R3) ≤ 0
where the constant C(a ,M) is increasing in both a and M.
0 0
Proposition 3.1 can be generalised to obtain existence of Yukawa ground states cor-
responding to finite nuclear configurations, for sufficiently small a > 0. The following
result will be used in Proposition 4.2 to compare the Yukawa ground state with its finite
approximation.
Proposition 3.2. For any nuclear distribution m : R3 R , satisfying
≥0
→
m M,
k kL2unif(R3) ≤
there exists a = a (m) > 0 such that for all 0 < a a , there exists (u ,φ ) solving
0 0 0 a a
≤
(2.3), satisfying u 0 and
a
≥
u + φ C(M). (3.3)
k akHu4nif(R3) k akHu2nif(R3) ≤
If m c > 0 for some x R3 and R ,c , then a = a (R ,c ) > 0.
BR0(x) ≥ 0 ∈ 0 0 0 0 0 0
PrRoposition 3.3. Let a > 0 and m (M,ω), then for all 0 < a a the corre-
0 L2 0
∈ M ≤
sponding Yukawa ground state (u ,φ ) is unique and there exists c > 0 such that the
a a a0,M,ω
electron density u satisfies
a
inf u (x) c > 0. (3.4)
x∈R3 a ≥ a0,M,ω
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 5
Assuming higher regularity of the nuclear distributions implies higher regularity of the
ground state. Therefore define for k N
0
∈
(M,ω) = m Hk (R3) m M,
MHk ∈ unif k kHuknif(R3) ≤
(cid:26) (cid:12)
(cid:12)
(cid:12) R > 0 inf m(z) dz ω R3 ω .
(cid:12) 0 1
∀ x∈R3ZBR(x) ≥ − (cid:27)
Arguing by induction and applying the uniform lower bound (3.4) yields the following
result.
Corollary 3.4. Let a > 0, k N and m (M,ω), then for all 0 < a a the
0 0 Hk 0
∈ ∈ M ≤
corresponding Yukawa ground state (u ,φ ) satisfies
a a
u + φ C(a ,k,M,ω). (3.5)
k akHukn+if4(R3) k akHukn+if2(R3) ≤ 0
3.2. Uniform Yukawa estimates. The main result of this article is a uniform estimate
comparing the Yukawa and Coulomb ground states corresponding to the same nuclear
configuration. This result is essentially a consequence of [11, Theorems 3.4 and 3.5].
In the following, (u,φ) = (u ,φ ) denotes the corresponding Coulomb ground state
0 0
solving (2.2), i.e the ground state with Yukawa parameter a = 0.
Theorem 3.5. Suppose a > 0, k N , m (M,ω) and let (u,φ) denote the
0 0 Hk
∈ ∈ M
correspondingCoulomb ground state. For0 < a a , let (u ,φ ) denote the corresponding
0 a a
≤
Yukawa ground state, then there exists C = C(a ,k,M,ω) > 0 such that
0
u u + φ φ Ca2. (3.6)
a Wk+2,∞(R3) a Wk+2,∞(R3)
k − k k − k ≤
Remark2. Theerrortermin(3.6)arisesfromtheadditionaltermintheYukawaequation
(2.3b), as opposed to due to a difference in nuclear distributions in [11, Theorems 3.4 and
3.5]. For this reason, the author believes that an analogous result to Theorem 3.5 also
(cid:3)
holds for point charge nuclei.
Theorem 3.5 can be generalised to compare two Yukawa ground states (u ,φ ),
a1 a1
(u ,φ ) corresponding to the same nuclear configuration, where the parameters a ,a
a2 a2 1 2
differ.
Corollary 3.6. Let a > 0, k N , m (M,ω) and suppose 0 < a a a ,
0 0 Hk 1 2 0
∈ ∈ M ≤ ≤
then let (u ,φ ),(u ,φ ) denote the corresponding Yukawa ground states. There exists
a1 a1 a2 a2
C = C(a ,k,M,ω) > 0 such that
0
u u + φ φ C a2 a2 . (3.7)
k a1 − a2kWk+2,∞(R3) k a1 − a2kWk+2,∞(R3) ≤ 2 − 1
3.3. Pointwise Yukawa estimates. Theorems 3.7 and 3.8 ex(cid:0)tend [11(cid:1), Theorems 3.4
and 3.5] to the Yukawa model and require the class of test functions
H = ξ H1(R3) ξ(x) γ ξ(x) x R3 (3.8)
γ
∈ |∇ | ≤ | |∀ ∈
(cid:26) (cid:12) (cid:27)
(cid:12)
for some γ > 0. Observe that e−γe|·| H(cid:12) for any 0 < γ γ.
(cid:12)γ
∈ ≤
Theorem 3.7. Let m (M,ω), and let m : R3 R satisfy
1 L2 2 ≥0
∈ M e→
m M′,
k 2kL2unif(R3) ≤
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 6
then there exists a = a (ω,m ) > 0 such that for all 0 < a a there exist solutions
1 1 2 1
≤
(u ,φ ) and (u ,φ ) to (2.3) corresponding to m ,m , where (u ,φ ) satisfies
1,a 1,a 2,a 2,a 1 2 2,a 2,a
u 0 and
2,a
≥
u + φ C(M′), (3.9)
2,a H4 (R3) 2,a H2 (R3)
k k unif k k unif ≤
independently of a. Define
w = u u , ψ = φ φ , R = 4π(m m ),
1,a 2,a 1,a 2,a m 1 2
− − −
then there exist C = C(M,M′,ω),γ = γ(M,M′,ω) > 0, such that for any ξ H
γ
∈
∂α1w 2+ ∂α2ψ 2 ξ2 C R ξ2. (3.10)
m
| | | | ≤
R3 R3
Z (cid:18)|αX1|≤4 |αX2|≤2 (cid:19) Z
In particular, for any y R3,
∈
∂αw(y) 2+ ψ(y) 2 C R (x) 2e−2γ|x−y| dx. (3.11)
m
| | | | ≤ | |
R3
|α|≤2 Z
X
Theorem 3.7 can be generalised to obtain higher-order pointwise estimates, but this
requires that m ,m (M,ω) for some k N to ensure that both infu ,infu > 0.
1 2 Hk 0 1 2
∈ M ∈
Theorem 3.8. Let a > 0, k N , m ,m (M,ω) and for 0 < a a , let
0 0 1 2 Hk 0
∈ ∈ M ≤
(u ,φ ),(u ,φ ) denote the corresponding Yukawa ground states. Define
1,a 1,a 2,a 2,a
w = u u , ψ = φ φ , R = 4π(m m ),
1,a 2,a 1,a 2,a m 1 2
− − −
then there exist C = C(a ,k,M,ω),γ = γ(a ,M,ω) > 0, independent of a, such that for
0 0
any ξ H
γ
∈
∂α1w 2+ ∂α2ψ 2 ξ2 C ∂βR 2ξ2. (3.12)
m
| | | | ≤ | |
R3 R3
Z (cid:18)|α1X|≤k+4 |α2X|≤k+2 (cid:19) Z |Xβ|≤k
In particular, for any y R3,
∈
∂α1w(y) 2+ ∂α2ψ(y) 2 C ∂βR (x) 2e−2γ|x−y| dx. (3.13)
m
| | | | ≤ | |
R3
|α1X|≤k+2 |αX2|≤k Z |Xβ|≤k
4. Applications
4.1. Yukawa and Coulomb forces. Let η C∞(B (0)) be radially symmetric and
∈ c R0
satisfy η 0 and η = 1 describe the charge density of a single (smeared) nucleus, for
≥ R3
some fixed R0 > 0. For any countable collection of nuclear coordinates Y = (Yj)j∈N
(R3)N, let the corrResponding nuclear configuration be defined by ∈
m (x) = η(x Y ). (4.1)
Y j
−
j∈N
X
A natural space of nuclear coordinates, related to the spaces is
Hk
M
(M,ω) := Y (R3)N m (M,ω) . (4.2)
L2 Y L2
Y { ∈ | ∈ M }
For any Y (M,ω) and a > 0, there exists a unique Yukawa ground state (u ,φ )
L2 a a
∈ Y
corresponding to m = m . Two definitions for the energy density for an infinite system
Y
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 7
are provided, for bounded Ω R3:
⊂
1
(Y;x) dx := u 2 + u10/3 + φ (m u2), (4.3)
E1,a |∇ a| a 2 a − a
ZΩ ZΩ ZΩ ZΩ
1
(Y;x) dx := u 2 + u10/3 + φ 2 +a2 φ2 , (4.4)
E2,a |∇ a| a 8π |∇ a| a
ZΩ ZΩ ZΩ (cid:18)ZΩ ZΩ (cid:19)
which satisfy (Y; ), (Y; ) L1 (R3).
Suppose noEw1,athat·ΩE2,aR3 ·is∈a cuhnaifrge-neutral volume [20], that is, if n is the unit
⊂
normal to ∂Ω, then φ n = 0 on ∂Ω. Recall (2.3b),
a
∇ ·
∆φ +a2φ = 4π(m u2),
− a a − a
it then follows that
1 1 1
φ 2 +a2 φ2 = ( ∆φ +a2φ )φ + φ φ n = φ (m u2),
8π |∇ a| a 8π − a a a a∇ a · 2 a − a
(cid:18)ZΩ ZΩ (cid:19) ZΩ Z∂Ω ZΩ
hence
(Y;x) dx = (Y;x) dx.
1,a 2,a
E E
ZΩ ZΩ
Similarly, for finite systems and Ω = R3, the two energies (4.3)–(4.4) agree. Thus ,
1,a 2,a
E E
are two energy densities which are well-defined for infinite configurations.
Given Y (M,ω), similarly define the Coulomb energy densities (Y; ), (Y; )
L2 1 2
L1 (R3) [1∈1]Y E · E · ∈
unif
1
(Y; ) := u 2 +u10/3 + φ(m u2), (4.5)
1
E · |∇ | 2 −
1
(Y; ) := u 2 +u10/3 + φ 2. (4.6)
2
E · |∇ | 8π|∇ |
By comparing the Yukawa and Coulomb energy densities, (4.3)–(4.4) with (4.5)–(4.6)
respectively, then applying Theorem 3.5 and Proposition 3.2 yields the convergence of the
energy densities: for all 0 < a a
0
≤
+ C(a ,M)a2. (4.7)
kE1,a −E1kL2unif(R3) kE2,a −E2kHu1nif(R3) ≤ 0
In (4.7), the regularity of the difference is limited by the nuclear distribution
1,a 1
m L2 (R3), whereas this term does notEapp−peEar in , hence the latter possesses
∈ unif E2,a−E2
additional regularity.
The next result shows that the force generated by a nucleus converges when passing
from the Yukawa to the Coulomb model.
Theorem 4.1. Let a > 0, Y (M,ω) and i 1,2 , then for all 0 < a a and
0 L2 0
k N, the Yukawa force density∈∂Y (Y, ) L1(R∈3){exis}ts and satisfies ≤
∈ YkEi,a · ∈
∂ ∂ ∂m (x)
1,a 2,a Y
E (Y;x) dx = E (Y;x) dx = φ (x) dx. (4.8)
a
∂Y ∂Y ∂Y
R3 k R3 k R3 k
Z Z Z
In addition, the Coulomb force density ∂ (Y, ) L1(R3) also exists and there exists
YkEi · ∈
C = C(a ,M,ω) > 0 such that for all 0 < a a
0 0
≤
∂ ∂
Ei,a Ei (Y;x) dx Ca2. (4.9)
∂Y − ∂Y ≤
(cid:12)ZR3(cid:18) k k(cid:19) (cid:12)
(cid:12) (cid:12)
The expression (4.8) sh(cid:12)ows that the forces generated(cid:12) by the energy densities and
1,a
(cid:12) (cid:12) E
are identical. Also, (4.9) establishes an O(a2) convergence of forces when passing
2,a
E
from the Yukawa to the Coulomb model.
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 8
4.2. Thermodynamic limit estimates. The following result extends [11, Proposition
4.1] to the Yukawa setting, providing an estimate for comparing the infinite Yukawa
ground state with its finite approximation, over compact sets, thus providing explicit
rates of convergence for the thermodynamic limit. This is discussed in Remark 3.
Interpreted differently, the result yields estimates onthe decay of the perturbation from
the bulk electronic structure at a domain boundary.
Proposition 4.2. Let m (M,ω), Ω R3 be open and suppose there exists m :
L2 Ω
R3 R such that m =∈ Mm on Ω and m⊂ M (e.g., m = mχ ). Then
→ ≥0 Ω k ΩkL2unif(R3) ≤ Ω Ω
there exists a = a (ω,m ) > 0 such that for all 0 < a a there exists a ground state
0 0 Ω 0
≤
(u ,φ ) corresponding to m and (u ,φ ) solving (2.3) with m = m , u 0 and
a a Ω,a Ω,a Ω Ω,a
≥
C = C(a ,M,ω),γ = γ(a ,M,ω) > 0, independent of a and Ω, such that for all y Ω
0 0
∈
∂α(u u )(y) + (φ φ )(y) Ce−γdist(y,∂Ω). (4.10)
a Ω,a a Ω,a
| − | | − | ≤
|α|≤2
X
Remark 3. Let R > 0 and R , then applying Proposition 4.2, with Ω = B (0) and
n ↑ ∞ Rn
m = m and0 < a a = a (ω)givesarateofconvergenceforthefiniteapproximation
Ω Rn ≤ 0 0
(u ,φ ), solving (2.3), to the ground state (u ,φ )
a,Rn a,Rn a a
u u + φ φ Ce−γ(Rn−R). (4.11)
k a − a,RnkW2,∞(BR(0)) k a − a,RnkL∞(BR(0)) ≤
This strengthens the result that (u ,φ ) converges to (u ,φ ) pointwise almost ev-
a,Rn a,Rn a a
(cid:3)
erywhere along a subsequence [6].
4.3. Pointwise stability and neutrality estimates. The following results extend [11,
Corollary 4.2, Theorem 4.3] to the Yukawa model. Corollary 4.3 shows that the decay
properties of the nuclear perturbation are inherited by the response of the Yukawa ground
state, and Corollary 4.4 shows the neutrality of nuclear perturbations for the TFW equa-
tions in the Yukawa setting.
Corollary 4.3. Let a > 0, k N , m ,m (M,ω) and 0 < a a , then let
0 0 1 2 Hk 0
∈ ∈ M ≤
(u ,φ ),(u ,φ ) denote the corresponding Yukawa ground states and define
1,a 1,a 2,a 2,a
w = u u , ψ = φ φ , R = 4π(m m ).
1,a 2,a 1,a 2,a m 1 2
− − −
(1) (Exponential Decay) If R Hk(R3) and spt(R ) B (0), or there exists γ′ > 0
m m R
∈ ⊂
such that ∂βR (x) Ce−γ′|x|, then there exist C = C(a ,k,M,ω),γ =
|β|≤k| m | ≤ 0
γ(a ,M,ω) > 0 depending also on R or γ′ such that
0
P
∂α1w(x) + ∂α2ψ(x) Ce−γ|x|. (4.12)
| | | | ≤
|α1X|≤k+2 |αX2|≤k
(2) (AlgebraicDecay) If there exist C,r > 0 such that ∂βR (x) C(1+ x )−r
|β|≤k| m | ≤ | |
then there exists C = C(a ,r,k,M,ω) > 0 such that
0
P
∂α1w(x) + ∂α2ψ(x) C(1+ x )−r. (4.13)
| | | | ≤ | |
|α1X|≤k+2 |αX2|≤k
(3) (Global Estimates) If R Hk(R3) then there exists C = C(a ,k,M,ω) > 0 such
m 0
∈
that
w + ψ C R . (4.14)
Hk+4(R3) Hk+2(R3) m Hk(R3)
k k k k ≤ k k
Corollary 4.4. Let a > 0, m ,m (M,ω) and 0 < a a , then define ρ :=
0 1 2 L2 0 12,a
∈ M ≤
m u2 m +u2 .
1 − 1,a − 2 2,a
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 9
(1) If spt(m m ) B (0), or there exist C,γ > 0 such that (m m )(x)
1 2 R′ 1 2
Ce−γe|x|, th−en ρ ⊂ L1(R3) and there exist C,γ > 0, independe|nt of−a, such th|a≤t,
12,a
∈
for all R > 0,
e
ρ Ce−γR. (4.15)
12,a
≤
(cid:12)ZBR(0) (cid:12)
(cid:12) (cid:12)
(2) If there exists C,r > 0 su(cid:12)ch that (m(cid:12) m )(x) C(1+ x )−r then there exists
1 2
(cid:12) | (cid:12)− | ≤ | |
C > 0, independent of a, such that, for all R > 0,
ρ C(1+R)2−r. (4.16)
12,a
≤
(cid:12)ZBR(0) (cid:12)
(cid:12) (cid:12)
(3) If m m L2(R3) (e(cid:12).g., r > 3/2(cid:12) in (2)) then ρ L2(R3) and
1 2 12,a
− ∈ (cid:12) (cid:12) ∈
1
lim ρ (k) dk = 0, (4.17)
12,a
ε→0 |Bε(0)| ZBε(0)
where ρ denotes the Fourier transform of ρ .
12,a b 12,a
5. Proofs
b
The following technical lemma is used in Proposition 5.3 to show u > 0 but will
a,Rn
also be useful to show a uniform lower bound for the ground state electron density u in
a
Lemma 6.1 in the Appendix.
Lemma 5.1. Let 0 < a a and m (M,ω), then for R > 0 define ψ
1 ≤ 2 ∈ ML2 n Rn ∈
C∞(B (0)) satisfying ψ 0 and ψ = 1 on B (0) and m = m χ .
c 4Rn Rn ≥ Rn 2Rn Rn · BRn(0)
Then there exists C = C (a ,a ,ω) > 0 and R = R (a ,a ,ω) > 0 such that for all
0 0 1 2 0 0 1 2
a a a and R R
1 2 n 0
≤ ≤ ≥
ψ 2 D (m ,ψ2 ) C R3. (5.1)
|∇ Rn| − a Rn Rn ≤ − 0 n
R3
Z
Proof of Lemma 5.1. Let a a a . By the construction of ψ
1 ≤ ≤ 2 Rn
ψ 2 = ψ 2 C R−2 C R . (5.2)
|∇ Rn| |∇ Rn| ≤ n ≤ 1 n
Z ZB4Rn(0)rB2Rn(0) ZB4Rn(0)rB2Rn(0)
Additionally, it follows that
D (m ,ψ2 ) = (m Y )ψ2 (m Y )(x) dx
a Rn Rn Rn ∗ a Rn ≥ Rn ∗ a
ZR3 ZB2Rn(0)
e−a|y|
= m (x y) dx dy
Rn − y
ZR3 ZB2Rn(0)∩BRn(y) ! | |
e−a|y|
= m (x) dx dy. (5.3)
Rn y
ZR3 ZB2Rn(−y)∩BRn(0) ! | |
First consider for R′ > 0
e−a|y| R′ 4π
dy = 4π re−ar dr = 1 e−aR′(1+aR′) ,
y a2 −
ZBR′(0) | | Z0 (cid:16) (cid:17)
hence choosing R′ = (4a)−1 ensures that
e−a|y| 4π
dy = 1 5e−1/4 =: C a−2, (5.4)
y a2 − 4 2
ZB1/4a(0) | |
(cid:0) (cid:1)
CONVERGENCE FROM YUKAWA TO COULOMB IN THE TFW MODEL 10
where C > 0. Now choose R (4a)−1, then the triangle inequality implies for y
2 n
≥ | | ≤
(4a)−1, B ( y) B (0), hence as m (M,ω)
2Rn − ⊃ Rn ∈ ML2
m (x) dx m(x) dx ω R3 ω . (5.5)
Rn ≥ ≥ 0 n − 1
ZB2Rn(−y)∩BRn(0) ZBRn(0)
Combining the inequalities (5.3)–(5.5) gives
e−a|y|
D (m ,ψ2 ) = m (x) dx dy
a Rn Rn Rn y
ZR3 ZB2Rn(−y)∩BRn(0) ! | |
e−a|y|
m (x) dx dy
≥ Rn y
ZB1/4a(0) ZB2Rn(−y)∩BRn(0) ! | |
e−a|y|
m (x) dx dy C a−2(ω R3 ω ). (5.6)
≥ Rn y ≥ 2 0 n − 1
ZB1/4a(0) ZBRn(0) ! | |
Now define C = C2ω0 > 0 and R R := max 1,(4a )−1,(C1+C2ω1a−12)1/2 , then com-
0 2a22 n ≥ 0 { 1 C0 }
bining (5.2) and (5.6) yields the desired estimate (5.1) for any a a a and R R
1 2 n 0
≤ ≤ ≥
ψ 2 D (m ,ψ2 ) C R +C ω a−2 2C R3
|∇ Rn| − a Rn Rn ≤ 1 n 2 1 − 0 n
Z
C(cid:0) R3 2C R3 =(cid:1) C R3. (cid:3)
≤ 0 n − 0 n − 0 n
5.1. Proof of regularity estimates.
Proposition 5.2. Let m : R R satisfy
≥0
→
m M,
k kL2unif(R3) ≤
and R , then define the truncated nuclear distribution m = m χ . There
n ↑ ∞ Rn · BRn(0)
exists R = R (m),a = a (m) > 0 such that for all R R and 0 < a a , the unique
0 0 0 0 n 0 0
≥ ≤
solution to the minimisation problem
ITFW(m ) = inf ETFW(v,m ) v L2(R3),v L10/3(R3),v 0 (5.7)
a Rn a Rn ∇ ∈ ∈ ≥
(cid:26) (cid:12) (cid:27)
(cid:12)
yields a unique solution (u ,φ ) to (cid:12)
a,Rn a,Rn (cid:12)
5
7/3
∆u + u φ u = 0, (5.8a)
− a,Rn 3 a,Rn − a,Rn a,Rn
∆φ +a2φ = 4π m u2 . (5.8b)
− a,Rn a,Rn Rn − a,Rn
which satisfy the following estimates, with constan(cid:0)ts independen(cid:1)t of R :
n
u C(M), (5.9)
k a,RnkHu4nif(R3) ≤
φ C(M), (5.10)
k a,RnkHu2nif(R3) ≤
and u > 0 on R3 whenever m 0. In particular, if m c > 0 for some
a,Rn Rn 6≡ BR0(x) ≥ 0
x R3 and R ,c > 0, then a = a (R ,c ) > 0.
0 0 0 0 0 0 R
∈
In the case m (M,ω), Proposition 5.2 can be extended to all a > 0. The
L2
∈ M
following result will be used to prove Proposition 3.1.