Radiative corrections to the binding energy for a spin 1 / 2 charged - - PowerPoint PPT Presentation

Radiative corrections to the binding energy for a spin 1/2 charged particle

(Toulon 2014)

Semjon Wugalter Joint works with Jean-Marie Barbaroux (University of Toulon )

Semjon Wugalter NRQED binding energy

• “Quantitative estimates on the binding energy for Hydrogen in non-relativistic QED. II. The

spin case.”, arXiv :1306 :4464 (2013) J.-M. Barbaroux, S. W.

• “Contribution of the Spin-Zeeman term to the binding energy for hydrogen in non-relativistic

QED”, Annals of the University of Bucharest (2013) J.-M. Barbaroux, S.W.

• “On the ground state energy of the translation invariant Pauli-Fierz model. II.”, Documenta

Mathematica (2012). J.-M. Barbaroux., S.W.

• “Non-analyticity of the ground state energy of the Hamiltonian for Hydrogen atom in

non-relativistic QED”, Journal of Physics A : Mathematical and Theoretical (2010) J.-M. Barbaroux., S. W.

• “Quantitative estimates on the binding energy for Hydrogen in non-relativistic QED”,

Annales Henri Poincaré (2010). J.-M. Barbaroux., Thomas Chen, Vitali Vougalter, S.W.

• “On the ground state energy of the translation invariant Pauli-Fierz model”, Proc. Amer.
• Math. Soc., 136 (3), 1057-1064 (2008).

J.-M. Barbaroux., Thomas Chen, Vitali Vougalter, S.W.

• “Quantitative estimates on the enhanced binding for the Pauli-Fierz operator”, J. Math. Phys.,
• vol. 46, no12 (2005).

J.-M. Barbaroux., Helmut Linde, S. W.

• “Binding conditions for atomic N-electron systems in non-relativistic QED”, Ann. Henri

Poincaré. 4 (6), 1101 - 1136 (2003). J.-M. Barbaroux., Thomas Chen, S. W.

Semjon Wugalter NRQED binding energy

1

Pauli-Fierz Operator

2

Ground state - Binding energy

3

Unitary transform - Self-Energy - Change of units

4

Preliminary results

5

Increase of the binding energy - spin case

Semjon Wugalter NRQED binding energy

Description

We study the Hamiltonian in NRQED for an atom.

◮ System with 1 electron, described as quantum, non relativistic,

pointwise particle with charge −e and spin 1

2 ◮ The electron interacts with the quantized magnetic field ◮ One static pointwise nucleus, with positive charge - The electron

interacts with the field of the nucleus via the Coulomb potential.

◮ One also study the free case (“self-energy” operator).

Semjon Wugalter NRQED binding energy

Introduction - Schrödinger Hamiltonian

Hamiltonian One electron interacting with a pointwise nucleus of charge e, (Z = 1). Hp = −∆ + V Coulomb Potential : V(x) = − e2

|x|

Electron mass m = 1/2 ; Planck constant = 1 ; velocity of light c = 1. Fine structure constant : α = e2 ≈ 1/137 Binding energy Energy necessary to remove the electron to spatial infinity : Σ(0) − Σ(V) = inf σ(−∆) − inf σ(−∆ − α/|x|).

Semjon Wugalter NRQED binding energy

Introduction - Schrödinger Hamiltonian

⊲ Coulomb uncertainty principle :

• R3 1

|x||ψ(x)|2dx ≤ ∇ψ ψ

⊲ ψ, (−∆ + V)ψ ≥ ∇ψ2 − α∇ψ ψ =

• ∇ψ2 − e2

2 ψ 2

• ≥0

−α2 4 ψ2 ⊲ Σ(V) = inf σ(−∆ + V) = − α2

4 .

Binding energy Σ(0) − Σ(V) = inf σ(−∆) − inf σ(−∆ + V) = α2 4 .

Semjon Wugalter NRQED binding energy

Coupling to the quantized radiation field : Pauli-Fierz Hamiltonian Hamiltonian (Coulomb gauge) N = 1 electron

Coulomb potential case : Pauli-Fierz operator

HPF =

• −i∇x ⊗ If +√αA(x)

2

• kinetic energy

−

αZ |x| ⊗ If

• Coulomb electrostatic potential

+ (q−1) √ασ · B(x)

• Zeeman term

+

Iel ⊗ Hf

radiation field energy operator

Free case : “self-energy” operator

Tself.en. = (−i∇x + √αA(x))2

• kinetic energy

+(q−1) √ασ · B(x)

• Zeeman term

+

Hf

• Field energy

System of units : Electron mass m = 1/2 ; Planck const. = 1 ; speed of light c = 1 ; fine structure constant : α = e2 ≈ 1/137

Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian

Hilbert space H = Hpart ⊗ Fs

◮ Hpart = L2(R3, Cq) : Hilbert space for N = 1 electron. R3 is

configuration space, Cq for spin q = 1 : “spinless” particule ; q = 2 : electron (with spin)

◮ Fs : Bosonic Fock space

Fs = Ωf C ⊕

∞

• n=1

n

s

  L2(R3, C2)

• ne photon space (momentum variable×2 polarizations transv.)

 

• F(n)

s n-photon space

◮ Vacuum : Ωf .

Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian

◮ creation/annihilation operators : a∗ λ(k), aλ(k). Fulfils C.C.R :

[aλ(k), a∗

λ′(k)] = δλ,λ′δ(k − k′), [a♯ λ(k), a♯ λ′(k)] = 0,

aλ(k)Ωf = 0

◮ Field energy :

Hf =

• λ=1,2
• ω(k)a∗

λ(k)aλ(k)dk,

ω(k) = |k| Hf =

n Hf (n), Hf Ωf = 0,

(H(n)

f

Φ(n))(k1, k2, · · · , kn) = n

j=1 |kj|Φ(n)(k1, k2, · · · , kn) ◮ Photon number operator :

Nf =

• λ=1,2
• a∗

λ(k)aλ(k)dk

i.e., (Nf Φ)(n)(k1, · · · , kn) = nΦ(n)(k1, · · · , kn).

Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian

◮ Magnetic vector potential : (Coulomb gauge)

A(x) = A−(x) + A+(x) =

• λ=1,2

χΛ(|k|) 2π|k|

1 2

ǫλ(k)eik.xaλ(k)dk + h.c.

◮ Polarization vectors : ǫλ(k), ǫ1(k) · ǫ2(k) = 0, k · ǫλ(k) = 0. ◮ UV (Ultraviolet) cutoff : χΛ(|k|) ◮ Coupling between electron and quantized magnetic field

√ασ · B, with B = Curl A, B(x) =

• λ=1,2
• R3

χΛ(|k|) 2π|k|1/2 k × iελ(k)eikxaλ(k)dk + h.c. , and σ = (σ1, σ2, σ3), σi are 2 × 2 Pauli matrices.

Semjon Wugalter NRQED binding energy

Pauli-Fierz Hamiltonian

HPF =(P−√αA(x))2+V+(q−1)√ασ · B(x)+Hf , V = − α |x|, P = i∇x And also HPF = Hp + Hf + HI(α) where Hp= (−∆ + V) ⊗ If Hf = field energy operator HI(α)= interaction = −2Re √αP.A(x)+αA(x)2+√ασ · B(x)

Semjon Wugalter NRQED binding energy

Self-adjointness

Hamiltonian

• The Hamiltonian HPF is self-adjoint, with domain D(Hpart. + Hf )
• Stability of the first kind :

inf σ(HPF) > −∞

• Stability of the second kind : N electrons and M nuclei with charge Zk

(k = 1, ..., M) inf σ(HPF) ≥ −C(Λ, max{Zk}) (M + N)

Ground state

inf σ(HPF) is an eigenvalue of multiplicity q .

Semjon Wugalter NRQED binding energy

Binding energy

Hamiltonian HPF = T + V ⊗ If sur H = L2(R3) ⊗ F T = (−i∇x ⊗ If + √αA(x))2 + Iel ⊗ Hf − cn.o.α Binding energy : Σα(0) − Σα(V) = inf σ(T) − inf σ(T + V) Remark : HPF = T + V = −∆x + V + Hf + (−2Re √αA(x) · i∇x + αA(x)2 − cn.o.α)

• :=HI(α)

Semjon Wugalter NRQED binding energy

What is expected ?

HPF = −∆x + V + Hf + HI(α)

1

The free particle binds a larger quantity of (low-energetic) photons than the confined particle.

2

The binding energy should increase : Σα(0) − Σα(V) > Σ(0) − Σ(V) = α2 4

Semjon Wugalter NRQED binding energy

Unitary transform

U = eiPf ·x

The e− momentum variable is shifted by Pf =

λ

• k a∗

λ(k)aλ(k)dk.

The photon “position” is shifted by x.

• U(i∇x)U∗ = i∇x − Pf

(i∇x acquires the meaning of the total momentum, i.e., momentum of particle + field).

• UA(x)U∗ = A(0)

and UB(x)U∗ = B(0).

• UTU∗ = (P − Pf − √αA(0))2 + (q−1)σ.B(0) + Hf − cn.o.α,

P := i∇x.

• U(T + V)U∗ = U HPF U∗ =
• P − Pf − √αA(0)

2 + (q−1)σ.B(0) + Hf

• T

− α

|x| − cn.o.α

Semjon Wugalter NRQED binding energy

Unitary transform

Equivalent Hamiltonian H := U(T + V)U∗ =

• P − Pf − √αA(0)

2 + (q−1)σ.B(0) + Hf

• T

− α |x| − cn.oα = (P2 − α |x|)

• Schrödinger operator

+ (Pf + √αA(0))2 + (q−1)σ.B(0) + Hf − cn.oα

• Self-Energy with total momentum 0, T(0)

−2Re P.(Pf + √αA(0))

Semjon Wugalter NRQED binding energy

Operator Self-Energy

Self-energy at fixed total momentum : The operator T = −∆ + Hf + HI(α) commutes with the total momentum Ptot, Ptot = (i∇x ⊗ If ) + (Iel ⊗ Pf ), Pf =

• λ
• k a∗

λ(k)aλ(k)dk

T ≃ ⊕

R3 T(p)dp

T(p) acting on H0 ≃ Cq ⊗ F Theorem ( [F’74], [CF’07], [C’08] ) inf σ(T(0)) = inf σ(T) = Σα(0) Σα(0) is an eigenvalue of T(0) : T(0)ΨGS

0 = Σα(0)ΨGS 0 .

Semjon Wugalter NRQED binding energy

Change of units

HPF = (i∇x − √αA(x))2 + (q−1)σ.B(x) + Hf − α |x| − cn.oα Change of variables (change of units) X = αx K =

1 α2 k

For W the associated unitary transform, W HPF W∗ = α2HBFS HBFS = (i∇X − α

3 2 A(αX))2 + (q−1)α 3 2 σ.B(αX) + Hf − 1

|X| − cnoα. UV cutoff ∼ 1 in "BFS" units implies UV cutoff ∼ α2 in atomic units.

Semjon Wugalter NRQED binding energy

Increase of the binding energy - some rigorous results

◮ [HiSp’01] (large α, dipol approximation) ◮ [HVV’02] ( small α ) ◮ [HS’03] (q = 2, αZ fixed) ◮ [CVV’03] : Σα(0) − Σα(V) > α2/4 (α small, q = 1 or 2). ◮ [HHSp’05] : Σα(0) − Σα(V) up to the order α3, with error O(α

7 2 log α) (scalar boson,

q = 1) ◮ [BFP’06] : Expansion in α for inf σ(Hα) and its associated ground state, for arbitrary N inf σ(HBFS) = ǫ0 +

2N

• k=1

ǫk(α)αk/2 + o(αN) "The quantity inf σ(HBFS) is not analytic nor even smooth at α = 0, but its derivatives of sufficiently high order diverge as α → 0" ◮ [GH’09, HH’11] : Analyticity in α of the ground state energy inf σ(HBFS) (q = 1).

Semjon Wugalter NRQED binding energy

Spinless case

Theorem [BCVV’10] Σα(0) − Σα(V) = α2 4

• Σ(0)−Σ(V)

+ e(3)α3

increase of bind. energy

+ e(4)α4 + e(5) α5 log α

infrared divergence

+o(α5 log α)

e(3)= 2 3π ∞ χ2

Λ(t)

1 + t dt > 0 e(4)= 1 6 A−(0)(Hf + P2

f )(−1)A+(0) · A−(0)Ωf , (Hf + P2 f )−1Ωf

+ 1 12

3

• i=1

(P2

f + Hf )− 1 2 Pi f (P2 f + Hf )−1A+(0) · A+(0)Ωf 2 −

1 2 A−(0).(Hf + P2

f )−1A+(0)Ωf 2

+4a2

0(−∆ −

1 |x| + 1 4 )− 1

2 Q⊥∆f12 ,

a0 =

• k2

1 + k2 2

4π2|k|2 1 |k|2 + |k| χΛ(|k|)dk1dk2dk3 e(5)= 4 π (−∆ − 1 |x| + 1 4 )

1 2 ∇f12 = 0

Semjon Wugalter NRQED binding energy

Spinless case

Self-energy (Translationnaly invariant case) [BCVV’08] Φ2 := −(Hf + P2

f )−1A+(0) · A+(0)Ωf ,

Φ3 := −(Hf + P2

f )−1Pf · A+(0)Φ2 ,

Φ1 := −(Hf + P2

f )−1Pf · A−(0)Φ2 .

Θtrial := Ωf + αΦ2 + 2α

3 2 Φ1 + 2α 3 2 Φ3

Then Σα(0) = inf σ(T) = inf σ(T(0)) = Θtrial , T(0)Θtrial + O(α4) = −α2Φ22

∗ + α3(2A−(0)Φ22−4Φ32 ∗−4Φ12 ∗) + O(α4) ,

where φ, ψ∗ = φ, (Hf + P2

f )ψ.

For the true GS ΨGS

0 of T(0), ΨGS 0 = Θtrial

+ R0, with R0 = O(α), R0∗ = O(α2).

Semjon Wugalter NRQED binding energy

Estimate of the self-energy up to α4

Ground state energy of T(0) - improved estimate [BV’11] We have Σα(0) = d(2)α2 + d(3)α3 + d(4)α4 + O(α5) , with

d(2) := −Φ22

∗,

d(3) := 2A−Φ22 − 4Φ32

∗ − 4Φ12 ∗

d(4) := −

• 2A−Φ22 − 4Φ12

∗ − 4Φ32 ∗

Φ2∗ 2 +8ℜΦ1, A− · A−Φ3+8A−Φ12 +8A−Φ32 −16˜ Φ22

∗ −16Φ42 ∗ +Φ22Φ22 ∗ ,

Φ2 :=−(Hf + P2

f )−1A+ · A+Ωf ,

Φ3 := −(Hf + P2

f )−1Pf · A+Φ2 ,

Φ1 := −(Hf + P2

f )−1Pf · A−Φ2 ,

˜ Φ2 :=−PΦ2

⊥(Hf + P2 f )−1

• Pf · A+Φ1 + Pf · A−Φ3 +

1 2 A+ · A−Φ2

• Φ4 :=−(Hf + P2

f )−1

• Pf · A+Φ3 +

1 4 A+ · A+Φ2

• ,

where PΦ2

⊥ is the orthogonal projection onto {ϕ ∈ F | ϕ, Φ2∗ = 0}.

Semjon Wugalter NRQED binding energy

Corollary The ground state energy Σα(V) of H fulfils Σα(V) = ˜ d(2)α2 + ˜ d(3)α3 + ˜ d(4)α4 + ˜ d(5)α5 log α + o(α5 log α) .

Semjon Wugalter NRQED binding energy

Increase of the binding energy - spin case

◮ Remaining difficulties from spinless case

• Standard Kato’s perturbation theory is useless
• α-dependence in V (Coulomb potential) and in the interaction

term HI(α)

• The magnetic potential A(x) contains a frequency space

singularity 1/|k|

1 2 : Infrared divergence problem

◮ Additional difficulties

• Additional term √ασ.B(x) with a priori “high order”
• Twice degenerate ground state for T and H
• Weaker (but optimal) photon number estimate on ground states

for T and H. ΨGS, Nf ΨGS ≤ cα .

Semjon Wugalter NRQED binding energy

Self-energy - spin case

Theorem [BV’12] - Self-energy T = (i∇x − Pf − √αA(0))2 + √ασ · B(0) + Hf − cn.o.α Σα(0)= inf σ(T) = inf σ(T(0)) = −αΓ12

∗ + −α2(2A−Γ12−Γ22 ∗+Γ12 ∗Γ12) + O(α3)

Ground state of T(0) : ΨGS

0 = Ωf + √αΓ(a,b) 1

+ αΓ(a,b)

2

+ R

Γ(a,b)

1

= −(Hf + P2

f )−1σ.B+(0)Ωf

• a

b

• Γ(a,b)

2

= −(Hf + P2

f )−1[σ.B+(0)Γ(a,b) 1

+ 2A+(0).Pf Γ(a,b)

1

+ A(0)+.A(0)+Ωf

• a

b

• ]

Remark : Photon number estimate ΨGS

0 , Nf ΨGS 0 = O(α) (instead of O(α2)

in the spinless case).

Semjon Wugalter NRQED binding energy

Increase of the binding energy - Spin case

Theorem [BV’13 (ArXiv)] Σα(0) − Σα(V) = inf σ(T(0)) − inf σ(H) = 1 4α2 + (e(3) + e(3),Zeeman)α3 + O(α3+ 1

3 )

with

e(3) = 2 3π ∞ χ2

Λ(t)

1 + t dt > 0 e(3),Zeeman = 2 3π ∞ t2χ2

Λ(t)

(1 + t)3 dt > 0 ◮ No α term in the binding energy (as expected) ◮ An additional α3 term due to the Zeeman term in H compared to the

spinless particle case

Semjon Wugalter NRQED binding energy

Strategy of the proof - increase of the binding energy

The general strategy follows the one of [BCVV’10] for the spinless case : Iterative procedure Its implementation is however very different.

Semjon Wugalter NRQED binding energy

Strategy of the proof - increase of the binding energy

Step 0 : The trial state Ψtrial = ΨGS

0 fα yields :

Σα(0) − Σα(V) ≥ α2/4 ◮ Let fα be the GS of (−∆ − α/|x|), with e.v. −e0 = −α2/4. ◮ Let ΨGS be the G.S. of the operator T(0) =

• −Pf − √αA(0)

2 + √αB(0) + Hf ◮ Normalized trial state : Θtrial = fα(x)ΨGS ∈ L2(R3) ⊗ F Σα(V) ≤ Θtrial, H

• U(T+V)U∗

Θtrial = Θtrial ,

• : (P−Pf − √αA(0))2 : +√αB(0) + Hf − α

|x|

• Θtrial

= (P2 − α |x| )fαΨGS

0 , fαΨGS

• −α2/4

+ fαΨGS

0 , T(0)fαΨGS

• Σα(0)

− 2Re P · (Pf + √αA(0))fαΨGS

0 , fαΨGS

• 0 by sym. ∂/∂xifα, fα = 0

Semjon Wugalter NRQED binding energy

Strategy of the proof To derive sharp upper and lower bound, we “perturb” around ΨGS

0 fα.

Step 1 : Lower bound on Σα(0) − Σα(V) by choosing a “good” trial function : Ψtrial =ΨGS

0 fα + α

1 2 2P · Pf (Hf + P2

f )−1Γ1fα

+ α

1 2 2(Hf + P2

f )−1P · A+Ωf

1

• ΨGS

= Ωf + √αΓ(1,0)

1

+ αΓ(1,0)

2

+ R Γ(1,0)

1

= −(Hf + P2

f )−1σ.B+(0)Ωf

• 1
• Γ(1,0)

2

= −(Hf + P2

f )−1[σ.B+(0)Γ(1,0) 1

+ 2A+(0).Pf Γ(1,0)

1

+ A(0)+.A(0)+Ωf

• 1
• ]

Step 2 : Photon number bound estimates : ΨGS, Nf ΨGS = O(α2) O(α)

Semjon Wugalter NRQED binding energy

Strategy of the proof

Proposition

Let K = (i∇ − √αA(x))2 + √ασ · B(x) + Hf − α |x| , be the Pauli-Fierz

• perator defined without normal ordering. Let ΨGS ∈ T(0) be a ground state
• f K,

K ΨGS = E ΨGS , normalized by ΨGS = 1 . Let Nf :=

• λ=1,2
• a∗

λ(k) aλ(k) dk

denote the photon number operator. Then, there exists a constant c independent of α, such that for any sufficiently small α > 0, the estimate ΨGS , Nf ΨGS ≤ c α (1) is satisfied.

Semjon Wugalter NRQED binding energy

Strategy of the proof

• Proof. The strategy of the proof of Proposition 1 is similar to the proof of the

photon number bound in [BCVV, 2010]. However, due to the occurrence of the spin-Zeeman term √ασ.B, the photon number is one order larger in powers of the fine structure constant. If we denote by ˜ H := : K : the operator K with normal ordering, we have (i∇ − √αA(x))ΨGS2 = ΨGS, KΨGS + ΨGS, α |x|ΨGS − ΨGS, Hf ΨGS − 2ℜ√αΨGS, σ · B−(x)ΨGS ≤ ΨGS, ˜ HΨGS + cn.o.α + ΨGS, α |x|ΨGS.

Semjon Wugalter NRQED binding energy

Strategy of the proof

Since ℜφ, (i∇−√αA(x))2−α/|x|+Hf +2√ασ·B−(x)φ = φ, ˜ Hφ+cn.o.αφ2 , and since from [GLL] we have, for α small enough −ΨGS, Hf ΨGS − 2ℜ√αΨGS, σ · B−(x)ΨGS ≤ 0 .

Semjon Wugalter NRQED binding energy

Strategy of the proof

Now we have, with P := i∇, and the inequality −2ℜP.A−(x) ≥ −P2 − (A−(x))2, 0 ≥ ΨGS, ˜ H ΨGS = ℜ

• ΨGS,
• − ∆ − 4√αP · A−(x) + α : A(x) :2

+ 2√ασ · B−(x) + Hf − α |x|

• ΨGS

≥ −2ΨGS, α |x|ΨGS + ΨGS, (1 − 8√α)(−∆)ΨGS + 8√αΨGS, −∆ΨGS − 2√αΨGS, P2ΨGS − 2√αΨGS, (A−(x))2ΨGS + 1 4ΨGS, Hf ΨGS + 1 2ΨGS, Hf ΨGS + 2ΨGS, α : (A−(x))2 : ΨGS + 1 4ΨGS, Hf ΨGS + 2√αℜΨGS, σ · B−(x)ΨGS + ΨGS, α |x|ΨGS .

Semjon Wugalter NRQED binding energy

Strategy of the proof

In the right hand side, the sum of the first and the second term is bounded below by −cα2 for some constant c > 0, the sum of the third and fourth term is positive, and the sum of the fifth and sixth term is positive using [GLL]. Again using [GLL], we obtain that the sum of the seventh and eight term is larger than −cα2 and the sum of the ninth and tenth term is positive. Therefore, we obtain ΨGS, α |x|ΨGS ≤ cα2 . This implies (i∇ − √αA(x))ΨGS2 ≤ cα . We set v := i∇ − √αA(x) . Using [aλ(k), Hf ] = |k|, [aλ(k), v] = ǫλ(k) 2π|k|

1 2 ζ(|k|)eik.x

Semjon Wugalter NRQED binding energy

Strategy of the proof

Applying the pull-through formula yields aλ(k)EΨGS = aλ(k)KΨGS =

• (Hf + |k|)aλ(k) − 1

|x|aλ(k) + [aλ(k), v]v + v[aλ(k), v] + v2aλ(k) + √ασ · B(x)aλ(k) − i√αζ(|k|) σ · (ǫλ(k) ∧ k) |k|

1 2 2π

eik.x ΨGS . Thus aλ(k) ΨGS = − √αζ(|k|) 2π|k|

1 2

2 K + |k| − E

• i∇−√αA(x)
• · ǫλ(k)eik.x

+ iσ · ǫλ(k) ∧ k 2π eik.x

• ΨGS .

Semjon Wugalter NRQED binding energy

Strategy of the proof

We obtain

• aλ(k)ΨGS

≤ c √α ζ(|k|) |k|

3 2

• i∇ − √αA(x)
• ΨGS

+ c √α ζ(|k|) |k|

1 2

• ΨGS
• ≤ c
• α

|k|

3 2 +

√α |k|

1 2

• ζ(|k|)ΨGS ,

This a priori bound exhibits the L2-critical singularity in frequency space. It does not take into consideration the exponential localization of the ground state due to the confining Coulomb potential.

Semjon Wugalter NRQED binding energy

Strategy of the proof

To account for the latter, following the proof of [BCVV, 2010], we use two results from the work of Griesemer, Lieb, and Loss, [GLL], which provides the bound

• aλ(k)ΨGS
• < c √α ζ(|k|)

|k|

1 2

• |x|ΨGS
• .

Moreover,

• exp[β|x|]ΨGS
• 2

≤ c

• 1 +

1 Σ0 − E − β2

• ΨGS 2 ,

for any β2 < Σ0 − E = O(α2) . For the 1-electron case, Σ0 is the infimum of the self-energy operator, and E is the ground state energy of ˜ H.

Semjon Wugalter NRQED binding energy

Strategy of the proof

Choosing β = O(α) yields, |x|ΨGS ≤ |x|4ΨGS

1 4 ΨGS 3 4 ≤ (4!) 1 4

β

• exp[β|x|]ΨGS
• 1

4 ΨGS 3 4

≤ c β

• 1 +

1 Σ0 − E − β2 1

8 ΨGS

≤ c1α− 5

4 .

Thus,

• aλ(k)ΨGS
• < cα− 3

4 ζ(|k|)

|k|

1 2

. We see that binding to the Coulomb potential weakens the infrared singularity by a factor |k|, but at the expense of a large constant factor α−2.

Semjon Wugalter NRQED binding energy

Strategy of the proof

We arrive at ΨGS , Nf ΨGS =

• aλ(k)ΨGS
• 2

dk ≤

• |k|<δ

c α− 3

2

|k| dk +

• δ≤|k|≤Λ

c α2 |k|3 + c α |k|

• dk

≤ cα− 3

2 δ2 + cα2 log δ + cα

≤ cα , for δ = α

5 4 . This proves the result.

Semjon Wugalter NRQED binding energy

A priori estimates on states “orthogonal” to fα

Step 3 : A priori estimates on states “orthogonal” to fα : For ϕ s.t. ϕ(n)(k1, λ1, ...kn, λn; . ), fα(.)

• a

b

• = 0, (a.e. ki, all λi, all a and b)

ϕ, Hϕ ≥ (Σα(0) − α2 4 )ϕ2 + 3 32α2ϕ2

• 1

2 dist. between first 2 levels of Hpart

+ νH

1 2

f ϕ2

Semjon Wugalter NRQED binding energy

Estimates up to the order α

Step 4 : We prove Σα(0) − Σα(V) ≤ cα2. More precisely. Given a ground state ΨGS of H such that Π0ΨGS = 1, we decompose it into a part parallel to fα and a part orthogonal to fα, where fα is the ground state of the Schrödinger operator −∆ − α/|x| . Namely, we define φ ∈ C2 ⊗ F and G ∈ H by ΨGS = fαφ + G , with G ∈ H orthogonal to the ground state fα in the sense of Definition given in this talk. Next we define a splitting for the state φ. Given ˜ a ˜ b

• ∈ C2, let

Γ(˜

a,˜ b) 1

:= −(Hf + P2

f )−1σ.B+Ωf

˜ a ˜ b

• .

With this definition, Γ(1,0)

1

equals the one photon component Γ1 of an approximate ground state of T(0) as defined by [BV, 2012].

Semjon Wugalter NRQED binding energy

Estimates up to the order α

Then we consider the following decomposition for φ : Let a, b and γ0 be defined by Π0φ = γ0

• a

b

• Ωf ,

with |a|2 + |b|2 = 1 , and let γ1 and R1 de defined by Π1φ = (√αγ1Γ(a,b)

1

+ R1), R1, Γ(a,b)

1

∗ = 0 . Here the bilinear form ., .∗ acts on (C2 ⊗ F)2. For G we define the following decomposition : Π0G =: g , and for Γ1(g) := −(Hf + P2

f )−1σ.B+g ,

similarly we split Π1G as Π1G =: √αβ1Γ1(g) + L1 where β1 and L1 are uniquely defined by the condition Γ1(g), L1∗ = 0 .

Semjon Wugalter NRQED binding energy

Estimates up to the order α

For Λ := sup

ζ(r)=0

|r| where ζ(.) is the ultraviolet cutoff, we define M(Π1G) = PL12 if |β1| < 8(1 + Λ) (P − Pf )Π1G2 if |β1| ≥ 8(1 + Λ) ,

Semjon Wugalter NRQED binding energy

Estimates up to the order α

Proposition i) For some c > 0 we have Σ0 − Σ ≤ cα2 . ii) For ΨGS a ground state of H such that Π0ΨGS = 1 we have H

1 2

f Π≥2ΨGS = O(α) ,

∇g = O(α) , R1∗ = O(α), L1∗ = O(α) , (γ1 − γ0) = O(α

1 2 ) ,

(β1 − 1)g = O(α

1 2 ) ,

(P − Pf )Π≥2ΨGS = O(α) . and M(Π1G) = O(α2) .

Semjon Wugalter NRQED binding energy

Estimates up to the order α

Step 5 : Upper bound up to α2 with error O(α

8 3 ). This yields refined norm

estimates and photon number estimates on remainder. Proposition [Refined photon number bound] We have Π≥2ΨGS, R1, L1 = O(α

5 6 ) .

Semjon Wugalter NRQED binding energy

Estimates up to the order α

The assumption Π0ΨGS = 1 imply that |γ0| is bounded by 1. Therefore, from Proposition above we obtain that |γ1| is bounded. Similarly, we have β1g bounded. For R := ΨGS − √αγ1Γ(a,b)

1

fα − √αβ1Γ1(g), we get

• |k|≤α

1 3

aλ(k)R2dk ≤ 3

• |k|≤α

1 3

aλ(k)ΨGS2dk + 3α|γ1|2

• |k|≤α

1 3

aλ(k)Γ(a,b)

1

2dk + 3α|β1|2

• |k|≤α

1 3

aλ(k)Γ1(g)2dkdx Moreover explicit computations shows that

• λ=1,2
• |k|≤α

1 3

aλ(k)Γ(a,b)

1

2dk ≤ cα

2 3 ,

and similarly

• λ=1,2
• |k|≤α

1 3

aλ(k)Γ1(g)2dk dx ≤ cg2α

2 3 .

Semjon Wugalter NRQED binding energy

Estimates up to the order α

Therefore boundedness of |γ1| and |β1| g yields

• |k|≤α

1 3

aλ(k)R2dk ≤ 3

• |k|≤α

1 3

aλ(k)ΨGS2dk

• + cα

5 3 .

Thus, using the bounds given in the proof of the photon number estimate, we

• btain
• |k|≤α

1 3

aλ(k)R2dk ≤ 3

• |k|≤α

7 4

cα− 3

2

|k| dk +

• α

7 4 ≤|k|≤α 1 3

α2 |k|3 + α |k| dk

• + cα

5 3 ≤ cα 5 3 .

Semjon Wugalter NRQED binding energy

Estimates up to the order α

This implies R, Nf R =

• λ=1,2
• |k|≤α

1 3

aλ(k)R2dk +

• λ=1,2
• |k|>α

1 3

aλ(k)R2dk ≤ cα

5 3 +

• λ=1,2
• |k|>α

1 3

|k| α− 1

3 aλ(k)R2dk

≤ cα

5 3 + α− 1 3 H 1 2

f R2 ≤ cα

5 3 ,

where in the last inequality we used H

1 2

f R2 = H

1 2

f Π≥2ΨGS2 + H

1 2

f fαR12 + H

1 2

f L12

≤ H

1 2

f Π≥2ΨGS2 + R12 ∗ + L12 ∗ ,

and the estimates of Proposition. The identity R, Nf R = Π≥2ΨGS, Nf Π≥2ΨGS + fαR1, Nf fαR1 + L1, Nf L1 , conclude the proof.

Semjon Wugalter NRQED binding energy

Estimates up to the order α

Step 6 : Upper bound up to α2 with error O(α

8 3 ). This yields again refined

norm estimates and photon number estimates on remainder. Step 7 : Upper bound up to α3 with error O(α3+ 1

3 ).

Semjon Wugalter NRQED binding energy

THANK YOU !

Semjon Wugalter NRQED binding energy