SLIDE 1
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
Serena Cenatiempo - Gran Sasso Science Institute, L’Aquila joint work with Chiara Boccato, Christian Brennecke and Benjamin Schlein The Analysis of Complex Quantum Systems: Large Coulomb Systems and Related Matters CIRM - October 21, 2019
SLIDE 2 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
System of interest: N interacting bosons
N trapped bosons, described by HN =
N
N
V
L2
sym(R3N)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 2/22
SLIDE 3 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
System of interest: N interacting bosons
N trapped bosons, described by HN =
N
N
V
L2
sym(R3N)
The interaction is characterized by the scattering length a, defined through the zero energy scattering function f
f (x) − − − − →
|x|→∞ 1
For short range potentials f (x) = 1 − a |x| for |x| > R
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 2/22
SLIDE 4 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
System of interest: N interacting bosons
N trapped bosons, described by HN =
N
N
V
L2
sym(R3N)
The interaction is characterized by the scattering length a, defined through the zero energy scattering function f
f (x) − − − − →
|x|→∞ 1
For short range potentials f (x) = 1 − a |x| for |x| > R 8πa =
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 2/22
SLIDE 5 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
System of interest: N interacting bosons
N trapped bosons, described by HN =
N
N
V
L2
sym(R3N)
The interaction is characterized by the scattering length a, defined through the zero energy scattering function f
f (x) − − − − →
|x|→∞ 1
For short range potentials f (x) = 1 − a |x| for |x| > R 8πa =
Experimentally a can be measured via the zero energy cross section: σ0 = 4πa2
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 2/22
SLIDE 6 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Physical phenomenon: Bose-Einstein condensation
N-particle wave function ψN(x1, . . . , xN) ∈ L2(R3N) − − − − →
large N
Condensate wave function ϕ ∈ L2(R3) Reduced one-particle density matrix γ(1)
ψN (x; y) =
- dx2 . . . dxN ψN(x, x2, . . . , xN) ψN(y, x2, . . . , xN)
For every compact operator A on L2(R3) ψN, (A ⊗ 1)ψN = TrAγ(1)
ψN
Complete condensation in the state ψN: γ(1)
ψN −
− − − →
N→∞ |ϕϕ|
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 3/22
SLIDE 7 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Physical phenomenon: Bose-Einstein condensation
N-particle wave function ψN(x1, . . . , xN) ∈ L2(R3N) − − − − →
large N
Condensate wave function ϕ ∈ L2(R3) Reduced one-particle density matrix γ(1)
ψN (x; y) =
- dx2 . . . dxN ψN(x, x2, . . . , xN) ψN(y, x2, . . . , xN)
For every compact operator A on L2(R3) ψN, (A ⊗ 1)ψN = TrAγ(1)
ψN
Complete condensation in the state ψN: γ(1)
ψN −
− − − →
N→∞ |ϕϕ|
For bosons this also implies: γ(k)
ψN −
− − − →
N→∞ |ϕϕ|⊗k i.e. the expectation of any
k-particle observable in the state ψN can be computed using ϕ⊗k.
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 3/22
SLIDE 8 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Mathematical problems
Statics: prove the appearance of condensation in gas of interacting bosons at low temperature & inve- stigate low energy states Expectation: Bogoliubov theory
Anderson et al., BEC in a vapor of Rb-87 (1995) cond-mat/0503044 (2005)
a Dynamics: after cooling the gas to very low temperatures the traps are switched off and the evolution of the condensate is observed. Effective non-linear one-body equation vs many-body Schr¨
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 4/22
SLIDE 9 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Mathematical problems
Statics: prove the appearance of condensation in gas of interacting bosons at low temperature & inve- stigate low energy states Expectation: Bogoliubov theory
Anderson et al., BEC in a vapor of Rb-87 (1995) cond-mat/0503044 (2005)
a Dynamics: after cooling the gas to very low temperatures the traps are switched off and the evolution of the condensate is observed. Effective non-linear one-body equation vs many-body Schr¨
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 4/22
SLIDE 10 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Homogeneous dilute Bose gases
N bosons enclosed in a cubic box of side length L, periodic b.c. HN = −
N
∆xj +
V
ρa3 ≪ 1 Results in the thermodynamic limit i.e. N, L → ∞ and ρ = N/L3 fixed ◮ occurrence of condensation
- hard-core bosons at half filling [Dyson-Lieb-Simon,‘78]
- renormalization group ongoing program:
[Benfatto ‘94], [Balaban-Feldman-Kn¨
◮ thermodynamic functions
- ground state energy: [Dyson‘57], [Lieb-Yngvason ‘98],
[Erd¨
- s-Schlein-Yau ‘08], [Giuliani-Seiringer ‘09], [Yau-Yin ‘13],
[Brietzke-Solovej, Brietzke-Fournais- Solovej, Fournais- Solovej ‘19] ◮ low lying excitation spectrum superfluidity
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 5/22
SLIDE 11 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
The Gross-Pitaevskii regime
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- ◮ If V (x) has scattering length a, then N2V (Nx) has
scattering length aGP = a/N − → dilute regime ρa3
GP = O(N−2)
◮ States with small energy are characterized by a correlation structure
- n length scales of aGP ∼ N−1 −
→ understand role of correlations
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 6/22
SLIDE 12 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
The Gross-Pitaevskii regime
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- ◮ If V (x) has scattering length a, then N2V (Nx) has
scattering length aGP = a/N − → dilute regime ρa3
GP = O(N−2)
◮ States with small energy are characterized by a correlation structure
- n length scales of aGP ∼ N−1 −
→ understand role of correlations Relevance: ◮ effective description of the strong and short range interactions among atoms in typical Bose-Einstein condensation experiments
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 6/22
SLIDE 13 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
The Gross-Pitaevskii regime
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- ◮ If V (x) has scattering length a, then N2V (Nx) has
scattering length aGP = a/N − → dilute regime ρa3
GP = O(N−2)
◮ States with small energy are characterized by a correlation structure
- n length scales of aGP ∼ N−1 −
→ understand role of correlations Relevance: ◮ effective description of the strong and short range interactions among atoms in typical Bose-Einstein condensation experiments ◮ scaling regime leading to a rigorouns derivation of the Gross-Pitaevskii equation i∂tϕ(t) = −∆ϕ(t) + 8πa |ϕ(t)|2ϕ(t)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 6/22
SLIDE 14 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
The Gross-Pitaevskii regime
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- ◮ If V (x) has scattering length a, then N2V (Nx) has
scattering length aGP = a/N − → dilute regime ρa3
GP = O(N−2)
◮ States with small energy are characterized by a correlation structure
- n length scales of aGP ∼ N−1 −
→ understand role of correlations Relevance: ◮ effective description of the strong and short range interactions among atoms in typical Bose-Einstein condensation experiments ◮ scaling regime leading to a rigorouns derivation of the Gross-Pitaevskii equation i∂tϕ(t) = −∆ϕ(t) + 8πa |ϕ(t)|2ϕ(t) ◮ HN equivalent to the Hamiltonian for N bosons in a box with side length N interacting through a fixed potential V , i.e. ρ = N/L3 = N−2
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 6/22
SLIDE 15 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
The Gross-Pitaevskii regime
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- [Lieb-Seiringer-Yngvason,‘00] The ground state energy of HN is
EN = 4πa N + o(N) [Lieb-Seiringer,‘02] The one particle reduced density γ(1)
ψN associated to the
ground state vector of HN is such that in trace norm γ(1)
ψN −
− − − →
N→∞ |ϕ0ϕ0|
where ϕ0(x) = 1 ∀x ∈ Λ.
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 7/22
SLIDE 16 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
The Gross-Pitaevskii regime
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- [Lieb-Seiringer-Yngvason,‘00] The ground state energy of HN is
EN = 4πa N + o(N) Note that
HNϕ⊗N
V (0) 2
≫ 4πaN [Lieb-Seiringer,‘02] The one particle reduced density γ(1)
ψN associated to the
ground state vector of HN is such that in trace norm γ(1)
ψN −
− − − →
N→∞ |ϕ0ϕ0|
where ϕ0(x) = 1 ∀x ∈ Λ. 8πa =
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 7/22
SLIDE 17 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
The Gross-Pitaevskii regime
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- [Lieb-Seiringer-Yngvason,‘00] The ground state energy of HN is
EN = 4πa N + o(N) Note that
HNϕ⊗N
V (0) 2
≫ 4πaN [Lieb-Seiringer,‘02] The one particle reduced density γ(1)
ψN associated to the
ground state vector of HN is such that in trace norm γ(1)
ψN −
− − − →
N→∞ |ϕ0ϕ0|
where ϕ0(x) = 1 ∀x ∈ Λ. Same result for ψN : limN→∞
[Lieb-Seiringer,’06],[Nam-Rougerie-Seiringer’16]. 8πa =
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 7/22
SLIDE 18 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Optimal rate for Bose-Einstein Condensation
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- Theorem 1 [Boccato-Brennecke-C.-Schlein, ’19] Let V ∈ L3(R3) positive,
spherically symmetric and compactly supported. Let ψN ∈ L2
s(ΛN) be a
sequence with ψN = 1 and ψN, HNψN ≤ 4πaN + ζ for a ζ > 0. Then 1 − ϕ0, γ(1)
ψN ϕ0 ≤ C(ζ + 1)
N for all N ∈ N large enough and ϕ0 = 1.
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 8/22
SLIDE 19 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Optimal rate for Bose-Einstein Condensation
Consider N bosons in the box Λ = [− 1
2, 1 2]3, with periodic b.c. and
HN = −
N
∆xi +
N
N2V
- N(xi − xj)
- Theorem 1 [Boccato-Brennecke-C.-Schlein, ’19] Let V ∈ L3(R3) positive,
spherically symmetric and compactly supported. Let ψN ∈ L2
s(ΛN) be a
sequence with ψN = 1 and ψN, HNψN ≤ 4πaN + ζ for a ζ > 0. Then 1 − ϕ0, γ(1)
ψN ϕ0 ≤ C(ζ + 1)
N for all N ∈ N large enough and ϕ0 = 1. We also show that the ground state energy satisfies EN = 4πaN + O(1)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 8/22
SLIDE 20 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Bogoliubov theory in the Gross-Pitaevskii regime
Theorem 2 [Boccato-Brennecke-C.-Schlein, ‘19] Let V ∈ L3(R3), positive, spherically symmetric and compactly supported. Then
EN = 4πa(N−1) + eΛa2 − 1 2
+
- p2 + 8πa −
- |p|4 + 16πap2 − (8πa)2
2p2
4 )
where Λ∗
+ = 2πZ3 \ {0} and
eΛ = 2 − lim
M→∞
|p1|,|p2|,|p3|≤M
cos(|p|) p2
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 9/22
SLIDE 21 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Bogoliubov theory in the Gross-Pitaevskii regime
Theorem 2 [Boccato-Brennecke-C.-Schlein, ‘19] Let V ∈ L3(R3), positive, spherically symmetric and compactly supported. Then
EN = 4πa(N−1) + eΛa2 − 1 2
+
- p2 + 8πa −
- |p|4 + 16πap2 − (8πa)2
2p2
4 )
where Λ∗
+ = 2πZ3 \ {0} and
eΛ = 2 − lim
M→∞
|p1|,|p2|,|p3|≤M
cos(|p|) p2 (1) For potentials κV with κ ≪ 1 one has 4πa(N−1) + eΛa2 = 4πaN(N−1) with
8πaN = κ V (0) − 1 2N
+
κ2 V 2(p1/N) 2p2
1
+ . . .
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 9/22
SLIDE 22 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Bogoliubov theory in the Gross-Pitaevskii regime
Theorem 2 [Boccato-Brennecke-C.-Schlein, ‘19] Let V ∈ L3(R3), positive, spherically symmetric and compactly supported. Then
EN = 4πa(N−1) + eΛa2 − 1 2
+
- p2 + 8πa −
- |p|4 + 16πap2 − (8πa)2
2p2
4 )
where Λ∗
+ = 2πZ3 \ {0} and
eΛ = 2 − lim
M→∞
|p1|,|p2|,|p3|≤M
cos(|p|) p2 (2) Replace V by VR(x) = R−2V (x/R) with scattering length aR = aR, and compare eΛa2R2 vs − 1
2
+[ ... ] ∼ a5/2R5/2 The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 9/22
SLIDE 23 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Bogoliubov theory in the Gross-Pitaevskii regime
Theorem 2 [Boccato-Brennecke-C.-Schlein, ‘19] Let V ∈ L3(R3), positive, spherically symmetric and compactly supported. Then
EN = 4πa(N−1) + eΛa2 − 1 2
+
- p2 + 8πa −
- |p|4 + 16πap2 − (8πa)2
2p2
4 )
where Λ∗
+ = 2πZ3 \ {0} and
eΛ = 2 − lim
M→∞
|p1|,|p2|,|p3|≤M
cos(|p|) p2 The spectrum of HN − EN below an energy ζ consists of eigenvalues
+
np
- |p|4 + 16πa|p|2 + O(N−1/4(1 + ζ3)) ,
np ∈ N
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 9/22
SLIDE 24 Intro & results Proofs Bose-Einstein condensation The Gross-Pitaevskii regime
Bogoliubov theory in the Gross-Pitaevskii regime
Theorem 2 [Boccato-Brennecke-C.-Schlein, ‘19] Let V ∈ L3(R3), positive, spherically symmetric and compactly supported. Then
EN = 4πa(N−1) + eΛa2 − 1 2
+
- p2 + 8πa −
- |p|4 + 16πap2 − (8πa)2
2p2
4 )
where Λ∗
+ = 2πZ3 \ {0} and
eΛ = 2 − lim
M→∞
|p1|,|p2|,|p3|≤M
cos(|p|) p2 The spectrum of HN − EN below an energy ζ consists of eigenvalues
+
np
- |p|4 + 16πa|p|2 + O(N−1/4(1 + ζ3)) ,
np ∈ N Previous result for trapped bosons:
p∈Λ∗
+ np
V (0)|p|2
mean field scaling
1 N V (x): [Seiringer ’11], [Grech-Seiringer ’13],
[Lewin-Nam-Serfaty-Solovej ’14], [Derezinski-Napiorkovski ’14], [Pizzo ’16] singular interactions: [Boccato-Brennecke-C.-Schlein ’17]
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 9/22
SLIDE 25
Proof of Theorem 1 Bose-Einstein condensation with optimal rate
SLIDE 26 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
A useful tool: the Fock space
The many-particle system is represented by Ψ = {ψn}n≥0 ∈ F = ⊕n≥0L2
s(Λn)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 11/22
SLIDE 27 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
A useful tool: the Fock space
The many-particle system is represented by Ψ = {ψn}n≥0 ∈ F = ⊕n≥0L2
s(Λn)
Creation and annihilation operators: for p ∈ 2πZ3 we introduce operators a∗
p and ap, creating and annihilating a particle with momentum p:
(a∗
pΨ)(n)(x1, . . . , xn) =
1 √n
n
eipxj Ψ(n−1)(x1, . . . , xj−1, xj+1, . . . , xn) (apΨ)(n)(x1, . . . , xn) = √ n + 1
e−ipx Ψ(n+1)(x, x1, . . . , xn) dx For any p, q ∈ 2πZ3 we have [ap, a∗
q] = δp,q ,
[ap, aq] = [a∗
p, a∗ q] = 0
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 11/22
SLIDE 28 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
A useful tool: the Fock space
The many-particle system is represented by Ψ = {ψn}n≥0 ∈ F = ⊕n≥0L2
s(Λn)
Creation and annihilation operators: for p ∈ 2πZ3 we introduce operators a∗
p and ap, creating and annihilating a particle with momentum p:
(a∗
pΨ)(n)(x1, . . . , xn) =
1 √n
n
eipxj Ψ(n−1)(x1, . . . , xj−1, xj+1, . . . , xn) (apΨ)(n)(x1, . . . , xn) = √ n + 1
e−ipx Ψ(n+1)(x, x1, . . . , xn) dx For any p, q ∈ 2πZ3 we have [ap, a∗
q] = δp,q ,
[ap, aq] = [a∗
p, a∗ q] = 0
Number of particles operator: N =
p∈2πZ3 a∗ pap
Fock space Hamiltonian: HN =
p2a∗
pap + 1
2N
p+ra∗ q−rapaq
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 11/22
SLIDE 29 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Our setting: the Fock space of excitations
For ψN ∈ L2
s(ΛN) and ϕ0 ∈ L2(Λ)
[Lewin-Nam-Serfaty-Solovej ‘12] ψN = α0 ϕ⊗N + α1 ⊗s ϕ⊗N−1 + . . . + αj ⊗s ϕ⊗N−j + . . . + αN , where αj ∈ L2(Λ)⊗sj and αj ⊥ ϕ0 ; ϕ0(x) = 1 for all x ∈ Λ.
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 12/22
SLIDE 30 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Our setting: the Fock space of excitations
For ψN ∈ L2
s(ΛN) and ϕ0 ∈ L2(Λ)
[Lewin-Nam-Serfaty-Solovej ‘12] ψN = α0 ϕ⊗N + α1 ⊗s ϕ⊗N−1 + . . . + αj ⊗s ϕ⊗N−j + . . . + αN , where αj ∈ L2(Λ)⊗sj and αj ⊥ ϕ0 ; ϕ0(x) = 1 for all x ∈ Λ. Unitary map: UN(ϕ0) : L2
s(ΛN) −
→ F ≤N
+
:=
N
L2
⊥ϕ0(Λ)⊗sn
ψN − → {α0, α1, . . . , αN, 0, 0, . . .}
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 12/22
SLIDE 31 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Our setting: the Fock space of excitations
For ψN ∈ L2
s(ΛN) and ϕ0 ∈ L2(Λ)
[Lewin-Nam-Serfaty-Solovej ‘12] ψN = α0 ϕ⊗N + α1 ⊗s ϕ⊗N−1 + . . . + αj ⊗s ϕ⊗N−j + . . . + αN , where αj ∈ L2(Λ)⊗sj and αj ⊥ ϕ0 ; ϕ0(x) = 1 for all x ∈ Λ. Unitary map: UN(ϕ0) : L2
s(ΛN) −
→ F ≤N
+
:=
N
L2
⊥ϕ0(Λ)⊗sn
ψN − → {α0, α1, . . . , αN, 0, 0, . . .} Conjugation with UN allows to focus on excitations: UN a∗
0a0 U∗ N = N − N+
UN a∗
0ap U∗ N =
UN a∗
pa0 U∗ N = a∗ p
UN a∗
paq U∗ N = a∗ paq
N+ =
a∗
pap
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 12/22
SLIDE 32 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The excitation Hamiltonian
HN =
p∈Λ∗ p2a∗ pap + 1 2N
V (r/N)a∗
p+ra∗ q−rapaq ,
Λ∗ = 2πZ3 We define LN = UNHNU∗
N : F≤N +
→ F ≤N
+
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 13/22
SLIDE 33 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The excitation Hamiltonian
HN =
p∈Λ∗ p2a∗ pap + 1 2N
V (r/N)a∗
p+ra∗ q−rapaq ,
Λ∗ = 2πZ3 We define LN = UNHNU∗
N : F≤N +
→ F ≤N
+
LN =
N−1 2
2
V (0)
N − N 2
+
N
+
p2a∗
pap +
+
N
pap
+ 1 2
+
p (N−N+)(N−1−N+) N2
a∗
−p + h.c.
1 √ N
+:p+q=0
p+qa∗ −paq
N + h.c.
2N
+,r∈Λ∗:r=−p,−q
p+ra∗ qapaq+r
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 13/22
SLIDE 34 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The excitation Hamiltonian
HN =
p∈Λ∗ p2a∗ pap + 1 2N
V (r/N)a∗
p+ra∗ q−rapaq ,
Λ∗ = 2πZ3 We define LN = UNHNU∗
N : F≤N +
→ F ≤N
+
LN =
N−1 2
2
V (0)
N − N 2
+
N
+
p2a∗
pap +
+
N
pap
+ 1 2
+
p (N−N+)(N−1−N+) N2
a∗
−p + h.c.
1 √ N
+:p+q=0
p+qa∗ −paq
N + h.c.
2N
+,r∈Λ∗:r=−p,−q
p+ra∗ qapaq+r
Our goal: show that LN − 4πaN ≥ cN+ − C for c, C > 0
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 13/22
SLIDE 35 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Include correlations between condensate and excitations
We include correlations in F ≤N
+
defining, T(η) = exp 1 2
ηp
pb∗ −p − bpb−p
+
→ F ≤N
+
S(η) = exp 1 √ N
ηr
r+va∗ −rav − h.c.
+
→ F ≤N
+
with ηp = − 1
N2
and b∗
p := a∗ p
N , bp :=
N ap : F≤N
+
− → F ≤N
+
U∗
Nb∗ pUN = a∗ p
a0 √ N , U∗
NbpUN = a∗
√ N ap : L2(ΛN) − → L2(ΛN) The operators b∗
p and bp create and annihilate excitations, but do not change
the total number of particles.
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 14/22
SLIDE 36 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Include correlations between condensate and excitations
We include correlations in F ≤N
+
defining, T(η) = exp 1 2
ηp
pb∗ −p − bpb−p
+
→ F ≤N
+
S(η) = exp 1 √ N
ηr
r+va∗ −rav − h.c.
+
→ F ≤N
+
with ηp = − 1
N2
and b∗
p := a∗ p
N , bp :=
N ap : F≤N
+
− → F ≤N
+
U∗
Nb∗ pUN = a∗ p
a0 √ N , U∗
NbpUN = a∗
√ N ap : L2(ΛN) − → L2(ΛN) The operators b∗
p and bp create and annihilate excitations, but do not change
the total number of particles. |ηp| ≤ C e−|p|/N p2 η2 ≤ C, ηH1 ≤ C √ N
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 14/22
SLIDE 37 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Include correlations between condensate and excitations
We define RN = S∗(η)T ∗(η)UN HN U∗
NT(η)S(η) : F ≤N +
→ F≤N
+
Then RN = 4πaN +
:=K
+
p2a∗
pap + 8πa
a∗
pap + 4πa
pb∗ −p
√ N
+
p+qa∗ −paq + h.c.
2N
+,r∈Λ∗:
r=−p,−q
p+ra∗ qapaq+r
+ ˜ EN with ± ˜ EN ≤
C µα (K + VN) + Cµβ , for some α, β > 0.
RN is almost excitation Hamiltonian for mean field potential 8πa χ(|p| ≤ µ)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 15/22
SLIDE 38 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Include correlations between condensate and excitations
We define RN = S∗(η)T ∗(η)UN HN U∗
NT(η)S(η) : F ≤N +
→ F≤N
+
There exists α > 0 s.t. RN ≥ UN 1 N
ν(xi − xj)U∗
N
+ (K + VN)(1 − Cµ−α) − 4πa N
p,q∈Λ∗
+
p+ra∗ qapaq+r − C
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 16/22
SLIDE 39 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Include correlations between condensate and excitations
We define RN = S∗(η)T ∗(η)UN HN U∗
NT(η)S(η) : F ≤N +
→ F≤N
+
There exists α > 0 s.t. RN ≥ UN 1 N
ν(xi − xj)U∗
N
+ (K + VN)(1 − Cµ−α) − 4πa N
p,q∈Λ∗
+
p+ra∗ qapaq+r − C
With results for mean-field interactions [Seiringer ’11] 1 N
ν(xi − xj) ≥ N 2 ν(0) − ν(0) = 4πaN − Cµ3 Hence RN − 4πaN ≥ 1 2(K + VN) − Cµ3N 2
+/N − Cµ3
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 16/22
SLIDE 40 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Include correlations between condensate and excitations
We define RN = S∗(η)T ∗(η)UN HN U∗
NT(η)S(η) : F ≤N +
→ F≤N
+
There exists α > 0 s.t. RN ≥ UN 1 N
ν(xi − xj)U∗
N
+ (K + VN)(1 − Cµ−α) − 4πa N
p,q∈Λ∗
+
p+ra∗ qapaq+r − C
With results for mean-field interactions [Seiringer ’11] 1 N
ν(xi − xj) ≥ N 2 ν(0) − ν(0) = 4πaN − Cµ3 Hence RN − 4πaN ≥ 1 2(K + VN) − Cµ3N 2
+/N − Cµ3
≥ cN+ − C by localization techniques [Lewin-Nam-Serfaty-Solovej ’14]
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 16/22
SLIDE 41 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Bounds on excitation vectors
By conjugating LN = UNHNU∗
N with suitable unitary maps we obtain
RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2
s(ΛN) and ξN = S∗(η)T ∗(η)UNψN. Then
def =
hp ≤ 4πaN + ζ
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 17/22
SLIDE 42 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Bounds on excitation vectors
By conjugating LN = UNHNU∗
N with suitable unitary maps we obtain
RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2
s(ΛN) and ξN = S∗(η)T ∗(η)UNψN. Then
4πaN + c
def =
hp ≤ 4πaN + ζ
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 17/22
SLIDE 43 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Bounds on excitation vectors
By conjugating LN = UNHNU∗
N with suitable unitary maps we obtain
RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2
s(ΛN) and ξN = S∗(η)T ∗(η)UNψN. Then
4πaN + c
def =
hp ≤ 4πaN + ζ ⇒ ξN, N+ξN ≤ C(ζ + 1)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 17/22
SLIDE 44 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Bounds on excitation vectors
By conjugating LN = UNHNU∗
N with suitable unitary maps we obtain
RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2
s(ΛN) and ξN = S∗(η)T ∗(η)UNψN. Then
4πaN + c
def =
hp ≤ 4πaN + ζ ⇒ ξN, N+ξN ≤ C(ζ + 1) Stronger apriori bounds: if ψN = χ(HN ≤ 4πaN + ζ)ψN we find ξN, (K + VN + 1)(N+ + 1)kξN ≤ C(1 + ζ)k for any k ∈ N.
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 17/22
SLIDE 45
Proof of Theorem 2 Ground state energy and low energy spectrum
SLIDE 46 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The excitation Hamiltonian
HN =
p∈Λ∗ p2a∗ pap + 1 2N
V (r/N)a∗
p+ra∗ q−rapaq ,
Λ∗ = 2πZ3 We define LN = UNHNU∗
N : F≤N +
→ F ≤N
+
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 19/22
SLIDE 47 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The excitation Hamiltonian
HN =
p∈Λ∗ p2a∗ pap + 1 2N
V (r/N)a∗
p+ra∗ q−rapaq ,
Λ∗ = 2πZ3 We define LN = UNHNU∗
N : F≤N +
→ F ≤N
+
LN =
N−1 2
2
V (0)
N − N 2
+
N
+
p2a∗
pap +
+
N
pap
+ 1 2
+
p (N−N+)(N−1−N+) N2
a∗
−p + h.c.
1 √ N
+:p+q=0
p+qa∗ −paq
N + h.c.
2N
+,r∈Λ∗:r=−p,−q
p+ra∗ qapaq+r
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 19/22
SLIDE 48 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The excitation Hamiltonian
HN =
p∈Λ∗ p2a∗ pap + 1 2N
V (r/N)a∗
p+ra∗ q−rapaq ,
Λ∗ = 2πZ3 We define LN = UNHNU∗
N : F≤N +
→ F ≤N
+
LN =
N−1 2
2
V (0)
N − N 2
+
N
+
p2a∗
pap +
+
N
pap
+ 1 2
+
p (N−N+)(N−1−N+) N2
a∗
−p + h.c.
1 √ N
+:p+q=0
p+qa∗ −paq
N + h.c.
2N
+,r∈Λ∗:r=−p,−q
p+ra∗ qapaq+r
Key fact: cubic and quartic terms cannot be neglected on low energy states
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 19/22
SLIDE 49 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The renormalized excitation Hamiltonian
We exhibit an excitation Hamiltonian which is quadratic up to error terms which are small on low energy states: JN = ˜ S∗(η) ˜ T ∗(η) UNHNU∗
N ˜
T(η) ˜ S(η) = CJN + QJN
low energy spectrum
+ VN + EJN where ± EJN ≤ C N−1/4(N+ + 1)2(HN + 1)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 20/22
SLIDE 50 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The renormalized excitation Hamiltonian
We exhibit an excitation Hamiltonian which is quadratic up to error terms which are small on low energy states: JN = ˜ S∗(η) ˜ T ∗(η) UNHNU∗
N ˜
T(η) ˜ S(η) = CJN + QJN
low energy spectrum
+ VN + EJN where ± EJN ≤ C N−1/4(N+ + 1)2(HN + 1) and QJN =
+
pbp + 1
2Gp( b∗
pb∗ −p + bpb−p )
Fp = p2(sinh2ηp + cosh2ηp) + V (·/N) ⋆ fN
Gp = 2p2 sinhηp coshηp + V (·/N) ⋆ fN
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 20/22
SLIDE 51 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
The renormalized excitation Hamiltonian
We exhibit an excitation Hamiltonian which is quadratic up to error terms which are small on low energy states: JN = ˜ S∗(η) ˜ T ∗(η) UNHNU∗
N ˜
T(η) ˜ S(η) = CJN + QJN
low energy spectrum
+ VN + EJN where ± EJN ≤ C N−1/4(N+ + 1)2(HN + 1) and QJN =
+
pbp + 1
2Gp( b∗
pb∗ −p + bpb−p )
Fp = p2(sinh2ηp + cosh2ηp) + V (·/N) ⋆ fN
- p (sinhηp + coshηp)2 ≃ p2
Gp = 2p2 sinhηp coshηp + V (·/N) ⋆ fN
p2 The operator QJN may be diagonalized using ˜ T(τ) = exp 1 2
+
τp(b∗
pb∗ −p − bpb−p)
tanh(2τp) = −Gp Fp |τp| ≃ |p|−4
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 20/22
SLIDE 52 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Further predictions
◮ Condensate depletion: the expected number of excitations of the condensate, in the ground state ψN N
ψN ϕ0
+
2
γ(1)
N
denotes the one-particle reduced density associated with ψN. ◮ Approximation of eigenvectors: if ψN denotes a ground state vector of HN, and θ1, θ2 are the first two eigenvalues of HN
N ˜
T(η) ˜ S(η) ˜ T(τ)Ω
C θ2 − θ1 N−1/4 for a phase ω ∈ [0; 2π)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 21/22
SLIDE 53 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Conclusions
◮ Conjugating the excitation Hamiltonian with suitable unitary maps modeling the particle correlations we are able to : show uniform bounds on the number and energy of excitations; build a renormalized quadratic excitation Hamiltonian, by extracting the large contributions to the energy neglected in Bogoliubov approximation.
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 22/22
SLIDE 54 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Conclusions
◮ Conjugating the excitation Hamiltonian with suitable unitary maps modeling the particle correlations we are able to : show uniform bounds on the number and energy of excitations; build a renormalized quadratic excitation Hamiltonian, by extracting the large contributions to the energy neglected in Bogoliubov approximation. ◮ The results extend to non-translation-invariant bosonic systems trapped by confining external fields [Brennecke-Schlein-Schraven, in preparation]
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 22/22
SLIDE 55 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Conclusions
◮ Conjugating the excitation Hamiltonian with suitable unitary maps modeling the particle correlations we are able to : show uniform bounds on the number and energy of excitations; build a renormalized quadratic excitation Hamiltonian, by extracting the large contributions to the energy neglected in Bogoliubov approximation. ◮ The results extend to non-translation-invariant bosonic systems trapped by confining external fields [Brennecke-Schlein-Schraven, in preparation] ◮ The same tools allow to investigate the equilibrium properties
- f two dimensional bosons interacting through singular potentials
(work in progress with C. Caraci & B. Schlein)
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 22/22
SLIDE 56 Intro & results Proofs Optimal rate for condensation Ground state energy and low energy spectrum
Conclusions
◮ Conjugating the excitation Hamiltonian with suitable unitary maps modeling the particle correlations we are able to : show uniform bounds on the number and energy of excitations; build a renormalized quadratic excitation Hamiltonian, by extracting the large contributions to the energy neglected in Bogoliubov approximation. ◮ The results extend to non-translation-invariant bosonic systems trapped by confining external fields [Brennecke-Schlein-Schraven, in preparation] ◮ The same tools allow to investigate the equilibrium properties
- f two dimensional bosons interacting through singular potentials
(work in progress with C. Caraci & B. Schlein) ◮ · · · ◮ Beyond scaling limits?
The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
CIRM - October 21, 2019 22/22
SLIDE 57 Finite size effect
Replace V by VR(x) = R−2V (x/R) with scattering length aR = aR. For large R the order one contributions to the ground state energy scale as eΛa2R2 and −1 2
- p∈2πZ3\{0}
- p2 + 8πaR −
- |p|4 + 16πaRp2 − (8πaR)2
p2
2
√ R Z3\{0}
- p2 + 8πa −
- |p|4 + 16πap2 − (8πa)2
p2
R5/2 2(2π)3
- R3
- p2 + 8πa −
- |p|4 + 16πap2 − (8πa)2
p2
= 4πR5/2(16πa)5/2 15(2π)3 = 4πa · 128 15√π a3/2R5/2 Letting R → ∞ (independently of N), i.e. making the effective density larger, the finite volume correction becomes subleading, w.r.t. Bogoliubov sum.
SLIDE 58 Action of the quadratic conjugation
With T(η) = exp 1
2
pb∗ −p − bpb−p)
GN = T ∗LNT = T ∗UNHNU∗
NT : F≤N +
→ F ≤N
+
With K = p2a∗
pap (kinetic energy) and VN = 1 2N
V (r/N)a∗
p+ra∗ qaq+rap
(potential energy), we have T ∗KT ≃ K +
p +
pa∗ −p + a−pap
2N V (r/N)ηq+rηq + 1 2N V (r/N)ηr+p
pa∗ −p + apa−p
T ∗ 1 2 V (p/N)
pa∗ −p + apa−p
≃ 1 2 V (p/N)ηp + 1 2 V (p/N)
pa∗ −p + apa−p
- to get rid of off-diagonal quadratic term.
SLIDE 59
Action of the cubic conjugation
Renormalized excitation Hamiltonians: for α, β > 0
GN = CGN + QGN + CN + VN + EGN , ± EGN ≤ Cµ−αHN + Cµβ RN = CRN + QRN + VN + ERN , ± ERN ≤ Cµ−αHN + Cµβ Expanding to second order, we find RN := e−A GN eA ≃ GN + [GN, A] + 1 2[[GN, A], A] + . . . ◮ [K, A] is cubic in creation and annihilation operators [VN, A] is quintic in creation and annihilation operators, and contains terms which are not in normal order. Restoring normal order generates an additional cubic term. We choose A so that the sum of these cubic terms cancels the large contribution contained in e−ACNeA ◮ The commutators [CN, A] and [[HN, A], A] produce constant and quadratic contributions that transform {CGN , QGN } into {CRN , QRN }.
SLIDE 60 Localization techniques
With results for mean-field interactions [Seiringer ’11]
RN ≥ UN
ν(xi − xj)U∗
N + (K + VN)(1 − C
µα ) − 4πa N
p,q∈Λ∗
+
p+ra∗ qapaq+r − C
≥ 4πaN + 1 2 (K + VN) − Cµ3N 2
+/N − Cµ3
Localization technique: [Lieb-Solovej, ’04], [Lewin-Nam-Serfaty-Solovej, ’12] RN ≥ fMRNfM + gMRNgM − C N/M2 fM =
if N+ ≤ M f 2
M + g 2 M= 1
if N+ > 2M Pick M = εN, for sufficiently small ε > 0. Then fMRNfM ≥ f 2
M 4πaN + Cf 2 M N+ − Cf 2 M
On gM use [Lieb-Seiringer, ’06], [Nam-Rougerie-Seiringer, ’16] gMRNgM ≥ g 2
M
M
SLIDE 61 Localization techniques
With results for mean-field interactions [Seiringer ’11]
RN ≥ UN
ν(xi − xj)U∗
N + (K + VN)(1 − C
µα ) − 4πa N
p,q∈Λ∗
+
p+ra∗ qapaq+r − C
≥ 4πaN + 1 2 (K + VN) − Cµ3N 2
+/N − Cµ3 ≥ 4πaN + cN+ − C
Localization technique: [Lieb-Solovej, ’04], [Lewin-Nam-Serfaty-Solovej, ’12] RN ≥ fMRNfM + gMRNgM − C N/M2 fM =
if N+ ≤ M f 2
M + g 2 M= 1
if N+ > 2M Pick M = εN, for sufficiently small ε > 0. Then fMRNfM ≥ f 2
M 4πaN + Cf 2 M N+ − Cf 2 M
On gM use [Lieb-Seiringer, ’06], [Nam-Rougerie-Seiringer, ’16] gMRNgM ≥ g 2
M
M
SLIDE 62 Bose-Einstein condensation
We obtained RN ≥ 4πaN + cN+ − C . Let ψN ∈ L2
s(ΛN) with ψN, HNψN ≤ 4πaN + ζ
Then, the excitation vector ξN = S∗T ∗UNψN is such that 4πaN + ζ ≥ ξN, RNξN ⇒ ξN, N+ξN ≤ C(ζ + 1) This bound implies BEC since 1 − ϕ0, γNϕ0 = 1 − 1 N ψN, a∗
0a0ψN
= 1 − 1 N TSξN, UNa∗
0a0U∗ NTSξN
= 1 − 1 N TSξN, (N − N+)TSξN = 1 N ξN, T ∗S∗N+TS ξN ≤ C N ξN, N+ ξN ≤ C(ζ + 1) N
SLIDE 63 Proof of Bogoliubov theory: unitary operators
To prove Bogoliubov prediction we use ˜ T(η) = exp 1 2
+
ηp
pb∗ −p − bpb−p
˜ S(η) = exp 1 √ N
ηr
r+va∗ −r (cosh(η)vav + sinh(η)va∗ −v
PL = {p ∈ Λ∗
+ : |p| ≤ N1/2}
PH = Λ∗
+/PL
(16πa)1/2 N N1/2
(1/R)
|p|
free particle regime linear spectrum
PL PH Then, the excitation hamiltonian JN = ˜ S∗(η) ˜ T ∗(η)UN HN U∗
N ˜
T(η) ˜ S(η) can be decomposed as JN = CJN + QJN
low energy spectrum
+ VN + EJN , ± EJN ≤ C N−1/4(HN + N 2
+ + 1)(N+ + 1)
SLIDE 64 Proof of Bogoliubov spectrum: diagonalization
Let MN = T ∗(τ)JN T(τ) : F ≤N
+
→ F ≤N
+ , then
MN = EN +
p∈Λ∗
+
pap + EMN
with EN = 4πa(N − 1) + eΛa2 − 1
2
+
- p2 + 8πa −
- |p|4 + +16πa|p|2 + (8πa)2
2p2
EMN ≤ CN−1/4(HN + N 2
+ + 1)(N+ + 1).
Finally, we use of the min-max principle to compare the eigenvalues λm of MN − EN (i.e. the eigenvalues of HN − EN) with the eigenvalues ˜ λm of DN =
p∈Λ∗
+
pap ,
showing that below an energy ζ |λm − ˜ λm| ≤ C N−1(1 + ζ3)
SLIDE 65 Superfluidity
The dispersion relation of excitations is linear for small momenta E(p) =
√ 16πa |p|
A classical particle with mass m and momentum k can only excite the Bose gas if k2 2m = (k − p)2 2m + E(p) (2) ⇒ there is a critical velocity below which a test particle entering the Bose gas can travel without friction k m > vcritical = inf
p
1 |p| p2 2m + E(p)
√ 16πa (3)
