Related Works

1. In the Thomas-Fermi-von Weizs¨acker Theory of Atoms and Molecules [17], they introduce the energy functional

E_p^{T F W}(ρ) = Z

|∇ρ^1/2(x)|²dx + 1 p

Z

ρ^p(x)dx − Z

V (x)ρ(x)dx + D(ρ, ρ) for 1 < p < ∞. They are concerned with the following problem

Min{E_p^{T F W}(ρ)|ρ ∈ L¹∩ L^p, ρ(x) ≥ 0, ∇ρ^1/2 ∈ L², Z

ρdx = λ}. (1.4) The main result in [17] is the following:

Theorem 1.1 There is a critical value 0 < λ_c < ∞ depending only on p and V such that

(a) If λ ≤ λ_c, problem (1.4) has a unique solution, (b) If λ > λ_c, problem (1.4) has no solution, (c) When p ≥ ⁴₃, then λc ≥ Z ≡Pk

i=1zi,

(d) When p ≥ ⁵₃ and k=1(atomic case), then λ_c > Z.

2. In the Thomas-Fermi-Dirac-von Weizs¨acker Theory [2], they introduce the en-ergy functional

E_p^{T F DW}(ρ) = A Z

|∇ρ^1/2(x)|²dx + γ p

Z

ρ^p(x)dx −3C_e 4

Z

ρ^4/3(x)dx

− Z

V (x)ρ(x)dx + D(ρ, ρ) + U

for p > ⁴₃. U is the repulsive electrostatic energy of the nuclei,

U = X

1≤i<j≤k

z_iz_j|R_i− R_j|⁻¹.

A > 0 is an adjustable constant. Originally, A was taken to be unity. γ = (6π²)^2/3h²(2mq^2/3)⁻¹ where h = ~/2π, ~=Planck’s constant, and m is the elec-tron mass. q is the number of spin states (=2 for elecelec-trons). C_e is a positive constant. In the original theory (see Dirac, [18]), the value Ce = (6/πq)^1/3 was used for reasons which will be explained in Lieb, E.H.[2].

The function space is

G_p = {ρ|ρ ∈ L³ ∩ L^p, ∇ρ^1/2 ∈ L², D(ρ, ρ) < ∞}.

The Thomas-Fermi-Dirac-von Weizs¨acker Theory[2] has not been as extensively studied as the above theory. They basically proved the existence of ground state solutions and discuss some properties of the solution by α method. The same technique will be used in our work.

2 Main Results

In this paper, we obtain the following lemmas and theorems. The proof of the lem-mas and theorems will be given in the next four sections. First, we prove two lemlem-mas which are useful in the proof of min{E_α(ρ)|ρ ∈ G}, the existence of ground state solution of (1.3).

Lemma 3.1 For every ε > 0, there is a constant Cε, depending on V and ε such that

Z

V (x)ρ(x)dx ≤ εkρk3+ CεD(ρ, ρ)^1/2 for every ρ ≥ 0.

Lemma 3.2 There exist positive constants a₀ and C₀ such that E_α(ρ) ≥ a₀(kρk₃+ k∇ρ^1/2 k²₂ +D(ρ, ρ)) +

Z

J_α(ρ)dx − C₀.

Based on above two lemmas, we prove the existence of min{E_α(ρ)|ρ ∈ G}.

Theorem 4.1 There exists a minimizer ρ₀ for E_α(ρ) on G. Every such ρ₀ ∈ L¹, and R ρ₀ ≤ some constant which is independent of ρ₀.

First, the proof of Theorem 4.1 is concerned with a minimizing sequence ρ_n ∈ G.

Such minimizing sequence make sense since E : G → ¯R is well defined where ¯R is the range of the mapping E_α.

Next, we derive the Euler equation satisfied by ρ₀. Set ψ = ρ^1/2₀ .

Theorem 4.2 ρ^1/2₀ satisfies the Euler equation:

[−4 + W (x)]ψ = 0 where

W = −4

3ψ^2/3−1

3 · ψ^2/3

1 + ψ^−2/3 +5

3ψ^4/3ln(1 + ψ^−2/3) + α − V + B(ψ²).

Now we prove the existence of ground state solution of (1.2). First, we prove one theorem which is useful in the proof of the existence of ground state solution of (1.2).

Theorem 5.1 If ψ ∈ ˜G satisfies the Euler equation and ψ(x) ≥ 0 for all x, then ψ is continuous. Moreover, ψ ∈ C^0,µ for all µ < 1.

Based on the above theorem, we prove the existence of ground state solution of (1.2).

Theorem 5.2 The two functions E(λ), E_α(λ) are finite and satisfy E(λ) = E_α(λ) − αλ.

The solutions of (1.2) and (1.3) are the same. The following theorems discuss the properties of ψ in (1.3). In fact, the minimizer of (1.2) also have the same properties.

Theorem 6.1 If ψ ∈ ˜G satisfies the Euler equation and ψ(x) ≥ 0 for all x, ei-ther ψ ≡ 0 or ψ > 0 on R³\Sk

i=1{R_i}.

Define φ(g) to be φ(g) =

Z

|∇g|²dx − Z

g^8/3dx + Z

g^10/3ln(1 + g⁻²³)dx + D(g², g²).

Theorem 6.2 If V = 0 in φ, there are C^∞ functions of compact support such that φ(g) < 0.

Theorem 6.3 ψ is bounded on R³ and ψ(x) → 0 as |x| → ∞. Also, ψ ∈ H² (i.e., ψ, ∇ψ and 4ψ ∈ L²).

Theorem 6.4 Let ψ be the positive solution to Euler equation. Then for every 0 < t < α, there exists a constant M such that

ψ(x) ≤ M exp[−t^1/2|x|].

3 Some Basic Properties of E

_α

This section are concerned with some properties of E_α which are useful in the proof of min{Eα(ρ)|ρ ∈ G}.

Lemma 3.1 For every ε > 0, there is a constant C_ε, depending on V and ε such that Z

V (x)ρ(x)dx ≤ εkρk₃+ C_εD(ρ, ρ)^1/2 (3.1) for every ρ ≥ 0.

Proof: Let δ > 0 be a small constant and let ζ(x) be a smooth function such that 0 ≤ ζ ≤ 1 and On the other hand define the operator B to be

B(ρ)(x) =

|x−y|dy is the Newtonian potential in R³(see [19],p18),

−4B(ρ) = 4πρ. (3.4)

We deduce from (3.5) and Sobolev’s inequality(see [19],p148) that

k B(ρ)k₆ ≤ CD(ρ, ρ)^1/2. (3.6)

for some constant C. Consequently, by (3.4) and (3.6), Z

V₂(x)ρ(x)dx = Z

V₂(x) 1

4π(−4B(ρ)(x))dx

= 1

4π Z

∇B(ρ)(x)∇V₂(x)dx

= − 1 4π

Z

4V₂(x)B(ρ)(x)dx

≤ 1

4π||4V₂||_6/5||B(ρ)||₆

≤ C₁ k 4V₂ k_6/5 D(ρ, ρ)^1/2 (3.7) where C₁ = _4π¹ C is a constant. Note that V (x) is the Green’s function for the Laplacian, we have

4V₂(x) ∈ C_o^∞. Combining (3.3) and (3.7), we obtain

Z

V (x)ρ(x)dx = Z

V₁(x)ρ(x)dx + Z

V₂(x)ρ(x)dx ≤ εkρk₃+ C_εD(ρ, ρ)^1/2.



Lemma 3.2 There exist positive constants a₀ and C₀ such that E_α(ρ) ≥ a₀(kρk₃+ k∇ρ^1/2 k²₂ +D(ρ, ρ)) +

Z

J_α(ρ)dx − C₀. (3.8)

Proof: Use (3.1), Young’s inequality and Sobolev’s inequality, Since ε can be arbitrary small and C is a fixed number, ^1−εC₂ and ^1−εC_2C are positive.

Choose

4 Minimization of E

_α

(ρ) with ρ ∈ G —The Euler Equation

Theorem 4.1 There exists a minimizer ρ₀ for E_α(ρ) on G. Every such ρ₀ ∈ L¹, and R ρ0 ≤ some constant (independent of ρ₀).

Proof: Let ρ_n ∈ G be a minimizing sequence. By Lemma 2.2, we have kρnk3 ≤ C, k∇ρ^1/2_n k2 ≤ C, D(ρn, ρn) ≤ C.

Therefore, we may extract a subsequence, still denoted by ρ_n, such that as n → ∞,

ρ_n → ρ₀ weakly in L³, (4.1)

ρ_n → ρ₀ a.e., (4.2)

∇ρ^1/2_n → ∇ρ^1/2₀ weakly in L². (4.3) ((4.2)relies on the fact that if Ω is a bounded smooth domain then H¹(Ω) is relatively compact in L²(Ω). (4.1) and (4.3) imply that {ρ^1/2n } is bounded in H¹(Ω). Hence {ρ^1/2n } has a subsequence converging in L²(Ω) and a.e.). Next, we prove that E_α is weakly lower semicontinuous at ρ₀. Note that ∇ρ^1/2n → ∇ρ^1/2₀ weakly in L², we have

lim inf

n→∞

Z

|∇ρ^1/2_n |²dx ≥ Z

|∇ρ^1/2₀ |²dx.

Note that J_α(ρ_n) ≥ 0 and D(ρ_n, ρ_n) ≥ 0, by Fatou’s Lemma, we have lim inf

n→∞

Z

J_α(ρ_n)dx ≥ Z

J_α(ρ₀)dx and

lim inf

n→∞ D(ρn, ρn) ≥ D(ρ0, ρ0).

We now prove that

Z

V (x)ρ_n(x)dx → Z

V (x)ρ₀(x)dx.

We write V = V1+ V2 as in Lemma 3.1. By (3.2) and (4.1), as n → ∞, we have Z

V₁(x)ρ_n(x)dx → Z

V₁(x)ρ₀(x)dx.

On the other hand

Therefore, we prove the existence of min{E_α(ρ)|ρ ∈ G}. It is easy to see from (3.8) that the bound onR ρ₀ is independent of ρ₀.



Next, we derive the Euler equation satisfied by ρ^1/2₀ . Set ψ = ρ^1/2₀ .

Theorem 4.2 ρ^1/2₀ satisfies the Euler equation:

[−4 + W (x)]ψ = 0 (4.4)

Therefore, we find for every ζ ∈ ˜G

So

By (4.5)(4.6)(4.7)(4.8)(4.9)(4.10), we get ψ satisfies the Euler equation



− 4 − 4

3ψ^2/3− 1

3· ψ^2/3

1 + ψ^−2/3 +5

3ψ^4/3ln(1 + ψ^−2/3) + α − V + B(ψ²)

 ψ = 0.



5 Existence of Ground State Solution

By the definition, we also know that B(ψ²) ≥ 0. Therefore, we have

−4ψ ≤ f

where f = V ψ ∈ L^2−ε_loc for any ε > 0 and in particular f ∈ L^s_loc for some s > ³₂. By Lemma 3.2, ψ ∈ H¹. We may, therefore, apply a result of Stampacchia (see [20], Theorem5.2) to conclude that ψ ∈ L^∞_loc. Going back to (4.4) and using the fact that ψ ∈ L^∞_loc, we now see that 4ψ ∈ L^3−ε_loc (since ||V ψ||_3−ε ≤ ||V ||_3−ε||ψ||∞). By Sobolev imbedding theorem (see Adams, [21], p.98), we have ψ ∈ C^0,µ for all µ < 1.



Theorem 5.2 The two functions E(λ), E_α(λ) are finite and satisfy E(λ) = E_α(λ) − αλ.

It is clear thatR |∇ρ^1/2|²dx, R ρ^4/3dx, and D(ρ, ρ) are finite (since ρ ∈ G). And Z

ρ^5/3ln(1 + ρ⁻¹³)dx <

Z

ρ^5/3dx + Z

ρ^4/3dx < ∞.

Let B_δ(R_i) is the ball of radius δ and centered at R_i for i = 1, 2, ..., k. δ is chosen such that all the ball B_δ(R_i) are disjoint. It is clear thatR

Bδ(Ri)V (x)ρ(x)dx is finite(since ρ is continuous). On the other hand, R

R³\B_δ(Ri)V (x)ρ(x)dx is also finite. Therefore, E(ρ) and E_α(ρ) are finite.

Finally, choose any ρ(x) ∈ G such that R ρ(x)dx = λ. We have E(ρ) ≥ E(λ).

So

E_α(ρ) = E (ρ) + αλ ≥ E(λ) + αλ.

Hence,

E_α(λ) ≥ E(λ) + αλ. (5.1)

On the other hand, choose any ρ(x) ∈ G such that R ρ(x)dx = λ. We have Eα(λ) ≤ Eα(ρ) = E (ρ) + αλ.

So

E_α(λ) ≤ E(λ) + αλ. (5.2)

By (5.1) and (5.2), we have

E(λ) = E_α(λ) − αλ.



6 Properties of the minimizer ψ

i=1{R_i}. The conclusion follows from Harnack’s inequality (see Gilbarg, [19]).



Theorem 6.2 If V = 0, there are C^∞ functions of compact support such that φ(g) <

0.

where

ln(1 + 1

b^4/3f^2/3(y)) = ln(1 + 1

f^2/3(y)) − 2

3· f^−5/3

1 + f^−2/3(b^4/3f^2/3− f^2/3) + ...

We find that the power of b in b^11/3

Z

f^10/3(y) ln



1 + 1

b^4/3f^2/3(y)

 dy

is lager than b^11/3. Therefore, the power of b in R g^8/3dx is the smallest one. Hence, for some sufficiently small, but positive b, φ(g) < 0.



The next two proofs will use the idea that:

1. If G is the Green’s function of an operator L, then the solution for f of the equation Lf = h is given by f (x) =R h(s)G(x, s)ds.

2. (see [22] p.46)

Let k be a complex number such that Imk≥ 0. The function φ₀(x, y) := 1

4π

e^ik|x−y|

|x − y| x, y ∈ R³ x 6= y is a solution to the Helmholtz equation

4φ + k²φ = 0

with respect to x for any fixed y. Because of polelike singularity at x 6= y, the function φ₀ is called a fundamental solution to the Helmholtz equation.

Theorem 6.3 ψ is bounded on R³ and ψ(x) → 0 as |x| → ∞. Also, ψ ∈ H² (i.e., ψ, ∇ψ and 4ψ ∈ L²).

Proof: It follows from Appendix that for ψ ≥ 0,

−4

3ψ^2/3− 1

3 · ψ^2/3

1 + ψ^−2/3 + 5

3ψ^4/3ln(1 + ψ^−2/3) + α ≥ 0.

By the definition, we also know that B(ψ²) ≥ 0. Therefore, we have

−4ψ ≤ V ψ.

So

(−4 + 1)ψ ≤ (V + 1)ψ.

Since ψ is continuous (Theorem 5.1), it follows that (V+1)ψ ∈ L² and thus ψ ≤ (−4 + 1)⁻¹[(V + 1)ψ](By maximal principle, see [19] Chapter 3). Set F = (−4 + 1)⁻¹[(V + 1)ψ]. Then

(−4 + 1)F = (V + 1)ψ ∈ L². So

F = Γ₀∗ (V + 1)ψ

where Γ₀ is the Green’s function of −4 + 1. As it is well known, F is bounded and goes to zero as |x| → ∞. Therefore, ψ is bounded and goes to zero as |x| → ∞.

Finally, note that

ψ^5/3

1 + ψ^−2/3 < ψ^5/3 f or ψ ≥ 0 and

ψ^7/3ln(1 + ψ^−2/3) < ψ^7/3+ ψ^5/3.

We know that ψ^5/3 and ψ^7/3≤ dψ for some constant d (since ψ(x) → 0 as |x| → ∞), and B(ψ²)ψ ∈ L² (since ψ ∈ L³ and B(ψ²) ∈ L⁶). So

−4

3ψ^5/3− 1 3

ψ^5/3

1 + ψ^−2/3 + 5

3ψ^7/3ln(1 + ψ^−2/3) + αψ − V ψ + B(ψ²)ψ ∈ L². Therefore, we conclude that 4ψ ∈ L² and so ψ ∈ H².



Theorem 6.4 Let ψ be the positive solution to Euler equation(4.4). Then for every 0 < t < α, there exists a constant M such that

ψ(x) ≤ M exp[−t^1/2|x|].

Proof: By (4.4) we have,

(−4 + W )ψ = 0.

So

ψ = −(−4 + t)⁻¹(W − t)ψ.

As |x| → ∞, we know that V → 0 and B(ψ²) → 0 (since ψ(x) → 0 as |x| → ∞).

From Appendix equation(A.1), as |x| → ∞, we also know that

−4

3ψ^2/3−1 3

ψ^2/3

1 + ψ^−2/3 +5

3ψ^4/3ln(1 + ψ^−2/3) + α → α.

Hence, as |x| → ∞, we have W → α. For |x| > some R, W − t > 0 (since 0 < t < α and W → α > 0 as |x| → ∞). We know that

Y (x − y) := e⁻

√t|x−y|

4π|x − y|

is the Green’s function of −4 + t (see [22] p.46). Therefore, since ψ > 0, ψ(x) = −

Z

Y (x − y)(W (y) − t)ψ(y)

= −

 Z

|y|≤R

Y (x − y)[W (y) − t]ψ(y)dy + Z

|y|>R

Y (x − y)[W (y) − t]ψ(y)dy



≤ − Z

|y|≤R

Y (x − y)[W (y) − t]ψ(y)dy

= − Z

|y|≤R

e⁻

√ t|x−y|

4π|x − y|[W (y) − t]ψ(y)dy.

Hence,

ψ(x) ≤ M exp[−t^1/2|x|].



A Appendix

On the other hand,

J⁰(ρ) = −4

A.2 Claim that J

α

(ρ) and J

_α⁰

(ρ) ≥ 0 for α ≥

¹₂

is a constant

On the other hand,

ρ→∞lim J_α⁰(ρ) = lim

在文檔中 新exchange correlation energy的基態能量 (頁 13-33)