On the strong Scott conjecture for Chandrasekhar atoms Konstantin - - PowerPoint PPT Presentation

On the strong Scott conjecture for Chandrasekhar atoms

Konstantin Merz 1 Joint work with Rupert L. Frank 2, 3, Heinz Siedentop 2, and Barry Simon 3

1 Technische Universit¨

at Braunschweig, Germany

2 Ludwig–Maximilians–Universit¨

at M¨ unchen, Germany

2California Institute of Technology Pasadena, CA, U.S.A.

Luminy, October 22, 2019

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Chandrasekhar operator ( = m = e = 1)

CZ :=

Z

• ν=1
• −c2∆ν + c4 − c2 − Z

|xν|

• +
• 1≤ν<µ≤Z

1 |xν − xµ| in

Z

• ν=1

L2(R3) where c is the velocity of light. Under x → x/c, CZ is unitarily equivalent to c2  

Z

• ν=1

√ −∆ν + 1 − 1 − γ |xν|

• +
• 1≤ν<µ≤Z

c−1 |xν − xµ|   with γ := Z/c. ◮ CZ is bounded from below, if and only if γ ≤ 2/π (Kato’s inequality) ◮ Study of properties of ground states as Z, c → ∞ keeping γ ≤ 2/π fixed.

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Ground state energy as Z, c → ∞ with γ fixed

inf spec(CZ) = E TF(Z) + 1 4−sC(γ)

• Z 2 + O(Z 47/24).

Here, E TF(Z) := inf{ETF[ρ] : 0 ≤ ρ ∈ L5/3(R3), ρ(x)ρ(y)

|x−y| dx dy < ∞} = eTFZ 7/3

is the non-relativistic Thomas–Fermi energy, ρTF

Z (x) = Z 2ρTF 1

(Z 1/3x), and sC(γ) := γ−2 Tr

• p2 + 1 − 1 − γ

|x|

• −

− p2 2 − γ |x|

• −
• > 0

The leading order was proven by Sørensen (’05), the Scott correction (Z 2) by Frank–Siedentop–Warzel (’08) and Solovej–Sørensen–Spitzer (’10). What can be said about the density on these length scales? ρ(x) := N

• R3(N−1)
• ψ(x, x′)
• 2 dx′

for a ground state ψ of CZ

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One-particle density as Z, c → ∞, Z/c = γ

(1) Z −2ρ(Z − 1

3 x) → ρTF

1

(x) weakly and in Coulomb norm where ρTF

Z (x) = Z 2ρTF 1

(Z

1 3 x) (with H. Siedentop)

(2) Strong Scott conjecture: ground state density on the length scale Z −1 (with R. L. Frank, H. Siedentop, and B. Simon)

r ρ(r)

Z3 Z2 Z−1 Z−1/3 1 (Z/r) 3 2

• n scale Z−1 ≪ r ≪ Z− 1

3 Hydrogen Thomas–Fermi

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Ground state density on the length scale Z −1

Let ψ be a ground state of CZ, ρ(x) := N

• R3(N−1) |ψ(x, x′)|2 dx′, and

ρℓ(r) := Nr 2(2ℓ + 1)−1 ℓ

m=−ℓ

• R3(N−1)
• S2 Yℓ,m(ω)ψ(rω, x′)dω
• 2

dx′. The normalized eigenfunctions of the hydrogenic operator C H

ℓ,γ =

• − d2

dr 2 + ℓ(ℓ + 1) r 2 + 1 − 1 − γ r in L2(R+, dr). are denoted by ψH

n,ℓ. Then we define ρH ℓ (r) := ∞ n=0 |ψH n,ℓ(r)|2 and

4πr 2ρH(r) := ∞

ℓ=0(2ℓ + 1)ρH ℓ (r).

Theorem

Let γ ∈ (0, 2/π), ℓ ∈ N0, and U = U1 + U2 with U1 ∈ r −1L∞

comp and U2 ∈ Dγ.

Then, for Z, c → ∞ with Z/c = γ fixed, lim

Z→∞

∞ c−3ρℓ(r/c)U(r) dr = ∞ ρH

ℓ (r)U(r) dr

lim

Z→∞

• R3 c−3ρ(x/c)U(|x|) dx =
• R3 ρH(|x|)U(|x|) dx.

For instance, U(r) = r −11{r<1} + r −3/2−ε1{r≥1}.

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Hydrogenic density

Does the limit even exist? (And in which sense?) To formulate the result precisely, we introduce [0, 1] → [0, 2/π] σ → Φ(σ) := (1 − σ) tan πσ 2 . Note that Φ is strictly monotone on [0, 1] with Φ(0) = 0 and limσ→1 Φ(σ) = 2/π. Thus, there is a unique σγ ∈ [0, 1] such that Φ(σγ) = γ.

Theorem

Let γ ∈ (0, 2/π) and 1/2 < s < min{3/2 − σγ, 3/4}. Then ρH(r) s,γ r 2s−31{r≤1} + r −3/21{r≥1} . Recalling ρH

Z,c(x) = c3ρH(cx) and ρTF Z (x) = Z 2ρTF 1

(Z 1/3x) ∼ (Z/|x|)3/2, this shows that there is a transition between the length scales Z −1 and Z −1/3.

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Remarks

◮ The result is well-known in the non-relativistic case, see Iantchenko–Lieb–Siedentop (’96). They even proved pointwise convergence which is not expected in the relativistic case since the hydrogenic eigenfunctions are not necessarily bounded. ◮ Heilmann–Lieb (’95) studied ρH in the non-relativistic case very well. They showed that ρH decreases monotonically and proved an asymptotic expansion as r → ∞. ◮ Although we show ρH(r) r −3/2, the constant is implicit (and presumably far from optimal) and we are lacking a corresponding lower

• bound. Moreover, the singularities and monotonicity of ρH are still

unknown.

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Strategy of the proof

We follow the lines of Iantchenko–Lieb–Siedentop (’96) by employing a linear-response argument. Let CZ,λ := CZ − λ Z

ν=1 c2U(c|xν|)Πℓ,ν, w.l.o.g.

U ≥ 0, and λ > 0. By the Scott correction, lim

Z→∞

∞ c−3ρℓ(r/c)U(r) dr = 1 2ℓ + 1 lim

λց0

lim

Z→∞ Tr

|ψψ|(CZ − CZ,λ) λc2

• ≤ lim

λց0

lim

Z→∞

Tr(C H

ℓ,Z − λUc(r))− − Tr(C H ℓ,Z)− + aZ

47 24

λc2 with C H

ℓ,Z :=

• c2
• − d2

dr 2 + ℓ(ℓ + 1) r 2

• + c4 − c2 − Z

r in L2(R+, dr). Goal: Interchange lim infλց0 and Tr to apply standard perturbation theory, i.e., the Feynman–Hellmann theorem or compute the derivative! To prove the convergence of c−3ρ(x/c), we also need a bound like Tr(Cℓ − V − λU(r))− − Tr(Cℓ − V )− λ(ℓ + 1/2)−2−ε where 0 ≤ V ≤ γ/r.

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Strategy of the proof

We follow the lines of Iantchenko–Lieb–Siedentop (’96) by employing a linear-response argument. Let CZ,λ := CZ − λ Z

ν=1 c2U(c|xν|)Πℓ,ν, w.l.o.g.

U ≥ 0, and λ > 0. By the Scott correction, lim

Z→∞

∞ c−3ρℓ(r/c)U(r) dr = 1 2ℓ + 1 lim

λց0

lim

Z→∞ Tr

|ψψ|(CZ − CZ,λ) λc2

• ≤ lim

λց0

Tr(C H

ℓ,γ − λU(r))− − Tr(C H ℓ,γ)−

λ with C H

ℓ,γ :=

• − d2

dr 2 + ℓ(ℓ + 1) r 2 + 1 − 1 − γ r in L2(R+, dr). Goal: Interchange lim infλց0 and Tr to apply standard perturbation theory, i.e., the Feynman–Hellmann theorem or compute the derivative! To prove the convergence of c−3ρ(x/c), we also need a bound like Tr(Cℓ − V − λU(r))− − Tr(Cℓ − V )− λ(ℓ + 1/2)−2−ε where 0 ≤ V ≤ γ/r.

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Generalized Feynman–Hellmann theorems

Assume that A is self-adjoint with A− trace class. Assume that B is non-negative, relatively form bounded with respect to A and that that there is 1/2 ≤ s ≤ 1 such that for some M > | inf spec(A)|, (A + M)−sB(A + M)−s is trace class. Assume that at least one of the following conditions hold. ◮ lim supλ→0

• (A + M)s(A − λB + M)−s

< ∞. ◮ There is a s′ such that max{s′, 1/2} < s < 1 and an a > 0 such that B2s ≤ a(A + M)2s′ . Then the one-sided derivatives of λ → S(λ) := Tr(A − λB)− satisfy Tr Bχ(−∞,0)(A) = D−S(0) ≤ D+S(0) = Tr Bχ(−∞,0](A) . In particular, S(λ) is differentiable at λ = 0, if and only if B|kerA = 0. In our case, A = C H

ℓ,γ (which has no zero eigenvalue) and B = U in L2(R+).

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Remarks

(1) If inf σess(A) > 0, the result follows from the classic Feynman–Hellmann theorem. (2) S(λ) is convex by the variational principle, i.e., D±S exist. Moreover, Tr Bχ(−∞,0](A) = Tr((A + M)2sχ(−∞,0](A))((A + M)−sB(A + M)−s) < ∞ by the form trace class condition. (3) The assumption B2s ≤ a(A + M)2s′ for max{1/2, s′} < s < 1 implies lim supλ→0

• (A + M)s(A − λB + M)−s

< ∞ (see also Neidhardt–Zagrebnov (’99)). (4) We will work with s > 1/2 since U1/2(C H

ℓ,γ + M)−1/22 2 ∼

∞ dr U(r) ∞ dk krJℓ+1/2(kr)2 √ k2 + 1 − 1 + M diverges logarithmically. (5) Decisive inequality (Frank, M., Siedentop): for −γ/|x| ≤ V ≤ −˜ γ/|x| with 2/π ≥ γ ≥ ˜ γ and s < 3/2 − σγ, we have |p|2s (|p| + V )2s .

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THANK YOU FOR LISTENING!

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