Structure of large bosonic systems : the mean-field approximation and the quantum de Finetti theorem
|
|
- Dorthy Morris
- 5 years ago
- Views:
Transcription
1 Structure of large bosonic systems : the mean-field approximation and the quantum de Finetti theorem icolas Rougerie LPMMC CRS & Universite Grenoble 1 Mathematical horizons of quantum physics IMS, Singapore, September 2013 Joint work with Mathieu Lewin and Phan Tha nh am: arxiv:
2 1. Introduction : bosons in the mean-field regime 2. Trapped systems and the quantum de Finetti theorem 3. The need for a weak de Finetti theorem 4. Quantitative de Finetti in finite dimension 5. Localization in Fock space and quantum de Finetti 6. Validity of Hartree s theory in the general case: conclusion
3 The mean-field approximation interacting particles in d space dimensions, 1 : too many degrees of freedom for an efficient description Look for relevant approximations: simplest one = assume independent, identically distributed particles Classical mechanics: probability measure µ on (R d ). Ansatz: with ρ probability on R d. µ(x 1,..., x ) = ρ(x i ) Quantum mechanics: wave function Ψ L 2 (R d ) L 2 (R d ). Ansatz: Ψ(x 1,..., x ) = i=1 u(x i ) = u (x 1,..., x ), u L 2 (R d ). i=1 Ok for bosons. For fermions, rather a Slater determinant (minimal correlations allowed by the Pauli exclusion principle) General question: is that a sensible approximation in the limit?
4 Bosons in the mean-field regime H := j=1 T j k<l w kl. Acting on H := s H, -fold symmetric tensor product of H a separable Hilbert space. Example: H = L 2 (R d ) H Ψ L 2 (R d ) satisfying Ψ(x 1,..., x i,..., x j,..., x ) = Ψ(x 1,..., x j,..., x i,..., x ) i, j One-body part: T self-adjoint operator on H. Example: T = + V, V : R d R T j = xj + V (x j ). Two-body part: pair interaction w, symmetric operator on H 2. Example: w kl = w ( x k x l ), w : R d R, w( x) = w(x) Adequate assumptions to have H bounded below. Scaling factor to have a well-defined limit problem. Simplest possibility : mean-field scaling. Mean-field approximation for the ground state energy: does E() := inf Ψ, H Ψ inf u, H u when? Ψ H, Ψ =1 u H, u =1
5 Motivation For non interacting systems (w = 0) inf Ψ, H Ψ = Ψ H, Ψ =1 inf u H, u =1 all the particles in the same quantum state. u, H u on interacting bosons with temperature: for T < T c, most particles in the same quantum state. Bose-Einstein condensation. Experimental motivation: Bose-Einstein condensates of cold dilute atomic gases (first in 1995, 2001 obel prize Cornell-Wieman-Ketterle): Many particles in the same quantum state macroscopic quantum effects. Superfluidity, quantized vortices... [Coddington-etal-Cornell, PRA 70 (2004)]
6 Hartree s theory Restrict states to purely uncorrelated ansätze Ψ = u Linear -body problem on-linear 1-body problem: E H [u] = u, Tu H u u, w u u H = 1 u, H 2 u ( Example = u 2 + V u 2) + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R d Hartree ground state energy e H = inf { E H [u], u H = 1 } when minimizer u H exists, satisfies Hartree s nonlinear equation. Example: ( ) + V + u H 2 w u H = ε H u H Trivial energy upper bound by the variational principle E() e H. Expect the lower bound to also hold for very general systems E() lim = e H Expect also Bose-Einstein condensation (BEC) on the Hartree ground state when it is unique.
7 Trapped bosonic gases bosons in the full space with a (say regular) external potential H = j=1 ( ) ( j + ia(x j )) 2 + V (x j ) k<l w(x k x l ), rotating trapped Bose gas, gases with magnetic fields: A : R d R d a (say regular) vector potential. Simplest case V (x) when x trapping potential. Seiringer, Grech-Seiringer 11/12 : validity of Hartree for ŵ 0, A 0. ŵ 0 allows to bound from below the two-body Hamiltonian by a one-body Hamiltonian. More relevant for cold atoms physics: Contact interactions. Scale the interaction potential w with so that w δ 0. Gross-Pitaevskii limit (more subtle, correlations play a role): Dyson 57, Lieb-Seiringer-Yngvason 00-13, for w 0.
8 Bosonic atoms: Bosonic atoms and Boson stars Classical nucleus of charge Z, electrons without Pauli principle. Large almost neutral atom : Z large t = ( 1)/Z fixed. ( H := i 1 ) t x i 1 x k x l. j=1 1 k<l Validity of Hartree: Benguria-Lieb 83, using (formally) ŵ 0 via a Lieb-Oxford inequality. See also Kiessling 12. Boson stars: White dwarf with local neutrality and no Pauli principle (Lieb-Thirring 84). H = j=1 with 0 < κ < 2/π. (( 1 xj ) 1/2 1 ) κ 1 1 k<l Validity of Hartree: Lieb-Yau 87, uses that (formally) ŵ 0. 1 x k x l, This talk : Validity of Hartree s theory can be seen as a consequence of the structure of the set of bosonic states.
9 Related literature Fannes-Spohn-Verbeure 80, Raggio-Werner 89, Petz-Raggio-Verbeure similar in spirit to this talk, but very abstract. Hepp-Lieb 74: Dicke maser model. Gross-Pitaevskii limit (more subtle, correlations play a role): Lieb-Seiringer-Yngvason 00-13, for w 0. BECs in particular geometries, for high rotation: Lieb-Seiringer-Yngvason, Schnee-Yngvason, Bru-Correggi-Pickl-Yngvason... Beyond Hartree (Bogoliubov s theory): Lieb-Solovej 01-06, Cornean-Dereziński-Ziń 09, Seiringer 11, Grech-Seiringer 12, Lewin-am-Serfaty-Solovej 12, Dereziński-apiórkowski 13 Dynamics (-body Schrödinger flow preserves factorization). Hepp, Ginibre-Velo, Spohn, Erdös-Schlein-Yau, Bardos-Golse-Mauser, Rodniansky-Schlein, Pickl, Knowles-Pick, Fröhlich-Knowles-Schwarz, Machedon-Grillakis-Margetis, Chen-Pavlovic, Ammari-ier, Lewin-am-Schlein... and others.
10 1. Introduction : bosons in the mean-field regime 2. Trapped systems and the quantum de Finetti theorem 3. The need for a weak de Finetti theorem 4. Quantitative de Finetti in finite dimension 5. Localization in Fock space and quantum de Finetti 6. Validity of Hartree s theory in the general case: conclusion
11 H := Setting the stage j=1 T j k<l w kl. Assume T has compact resolvent. Example on a compact domain. (General) mixed states = positive, self-adjoint, trace-class operators on H Pure states = wave functions rank-one projectors To any Ψ H associate a density matrix, projector onto Ψ γ Ψ = Ψ Ψ S 1 (H ) Reduced density matrices (identify operators and kernels) γ (k) Ψ = Tr k+1 Ψ Ψ S 1 (H k ) i.e. for any bounded operators b 1,..., b k on H [ ] Tr H k γ (k) Ψ b1... b k = Tr H [ Ψ Ψ b 1... b k 1 k] Energy only depends on γ (2) Ψ : 1 Ψ, H Ψ = Tr H 1[T γ (1) Ψ ] Tr H 2[wγ(2) Ψ ] = 1 2 Tr H 2[H2γ(2) Ψ ] Is the infimum attained for γ (2) Ψ = u 2 u 2?
12 Representability of mixed states Spectral theorem: mixed states = convex combinations of pure states without loss E() = 1 { ( 2 inf Tr H 2 H 2γ (2)) }, γ (2) P (2) with the set of -representable 2-body mixed states { } P (2) = γ (2) S 1 (H 2 ) : 0 G S 1 (H ), Tr H G = 1, γ (2) = Tr 3 G How does P (2) look like? Representability Problem. Minimize a fixed energy functional on a -dependent set. Does the set have a non trivial limit? With a leap of faith (OK in the confined case), hope that E() lim = 1 { ( 2 inf Tr H 2 H 2γ (2)) }, γ (2) lim P(2) A decreasing sequence of sets P (2) +1 P(2), define the set of -representable 2-body mixed states P (2) := 2 P (2) = lim P(2) (for the strong S1 topology)
13 Representability and quantum de Finetti theorem Quantum analogue of the Hewitt-Savage theorem for symmetric probability measures. Theorem (Størmer 69, Hudson-Moody 75) An infinite hierarchy {γ (k) } k=0 of states, γ (k) acts on H k. Consistency condition Tr k+1 γ (k+1) = γ (k) γ (0) = 1 (implies Tr H k γ (k) = 1 for all k 0) Then there Exists a unique Borel probability measure µ on the sphere SH of H, invariant under the group action of S 1, such that γ (k) = SH u k u k dµ(u) for all k 0. Representability problem is solved for P (2) : { } P (2) = u 2 u 2 dµ(u), µ Borel probability measure SH Take a -body state, with large. Look at the k-body density matrix, for k small. It is almost a convex combination of Hartree states.
14 Validity of Hartree for confined systems lim 1 E() = 1 { 2 inf Tr H 2 (H 2γ (2)), γ (2) P (2)} = 1 ( ) 2 inf Tr H 2 H µ 2 u 2 u 2 dµ(u) = inf E H [u]dµ(u) e H SH µ SH Justify the exchange of lim and inf: T has compact resolvent strong S 1 convergence of reduced density matrices for a sequence of ground states Theorem (Lewin-am-R 13) Energy: lim 1 E() = e H. If (Ψ ) is any sequence of (approximate) minimizers, there is a probability measure µ on the set M of minimizers of e H (modulo a phase), such that lim j γ(k) Ψ j = dµ(u) u k u k strongly in S 1 (H k ) M Bose-Einstein condensation: if e H has a unique minimizer u H (up to phase) lim γ(k) Ψ = u k H u k H strongly in S 1 (H k ).
15 Examples Strategy related to earlier works Fannes-Spohn-Verbeure 80, Petz-Raggio-Verbeure 89, Raggio-Werner 89 cf the use of the classical de Finetti theorem in statistical mechanics: Messer-Spohn 82, Kiessling 93, Caglioti-Lions-Marchioro-Pulvirenti 93, Kiessling-Spohn 99 Homogeneous Bose gas: bosons in a periodic box, H = L 2 per([0, 1] d ) H := j=1 j k<l previously dealt with by Seiringer 11, with ŵ 0 w(x k x l ), Trapped Bose gas: bosons in the full space with a (say regular) trapping potential V (x) when x ( ) H = ( j + ia(x j )) 2 + V (x j ) j=1 1 k<l w(x k x l ), rotating trapped Bose gas, gases with magnetic fields: A : R d R d a (say regular) vector potential (Grech-Seiringer 12, with ŵ 0 and A = 0)
16 1. Introduction : bosons in the mean-field regime 2. Trapped systems and the quantum de Finetti theorem 3. The need for a weak de Finetti theorem 4. Quantitative de Finetti in finite dimension 5. Localization in Fock space and quantum de Finetti 6. Validity of Hartree s theory in the general case: conclusion
17 on confined case One-body potentials going to 0 at (cf bosonic atoms and boson stars) Typical example H V ( = xj + V (x j ) ) on L 2 (R d ) with j=1 1 k<l w(x k x l ) V, w L p + L 0, max(1, d/2) < p <, w(x) = w( x) V, w 0 at Can add magnetic fields, fractional Laplacians... only need stability. Hartree s functionals. Possible loss of mass at infinity ( EH V [u] = u 2 + V u 2) + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R d EH[u] 0 = u u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R { d eh V (λ) = inf EH V [u] } u 2 = λ. R d Large binding inequalities: for all 0 λ 1: e V H (1) e V H (λ) + e 0 H(1 λ)
18 Loss of compactness γ density matrices of (approximate) ground state, reduced density matrices γ (k) Weak- compactness in S 1 + diagonal argument: γ (k) γ(k) Tr H k [γ (k) B k] Tr H k [γ (k) B k ] for any compact operator B k on H k The trace is not weak- continuous in infinite dimension (i.e. the identity is not compact), the limiting hierarchy is not consistent. Only γ (k) Tr k+1 γ (k+1) Quantum de Finetti requires equality, hence strong S 1 convergence. Try to describe exhaustively the possible ways of losing compactness, i.e. in this context loss of mass at infinity One-body problems: concentration-compactness principle of Lions, 80 s (see also Lieb, Struwe, Brézis-Coron...) Many-body problem: geometric methods, cf Simon, Enss, Sigal... late 70 s. Main novelty: combine quantum de Finetti and geometric localization.
19 A weak quantum de Finetti theorem Theorem (Lewin-am-R 13) Let (γ ) be a sequence of mixed states, γ S 1 (H ) such that, for all k 1 γ (k) γ(k) in S 1 (H k ). Then there exists a unique Borel probability measure µ on the unit ball BH of H, invariant under the group action of S 1, such that for all k 0. γ (k) = BH dµ(u) u k u k Similar in spirit to results of Ammari-ier Sufficient to deal with weakly lower semi continuous systems General case: more is needed (see later)
20 Validity of Hartree for systems with no bound states at infinity Assume that the two-body potential has no bound states + w 0. energy is a weakly lower semi-continuous function of the density matrices and e 0 H = 0 Theorem (Lewin-am-R 13) Energy: lim 1 E V () = e V H (1). Take Ψ a sequence of approximate ground states in H. There exists a Borel probability measure µ supported on the set { M V = u H, u 1 } E V H [u] = eh V ( u 2 ) = eh V (1), such that γ (k) Ψ j M V u k u k dµ(u). If e V H (1) < e V H (λ), then µ is supported on SH and the limit is strong. If in addition e V H (1) admits a unique minimizer u H, there is complete Bose-Einstein condensation.
21 1. Introduction : bosons in the mean-field regime 2. Trapped systems and the quantum de Finetti theorem 3. The need for a weak de Finetti theorem 4. Quantitative de Finetti in finite dimension 5. Localization in Fock space and quantum de Finetti 6. Validity of Hartree s theory in the general case: conclusion
22 Quantitative versions of the de Finetti theorem? Pick a -body state Γ S(H ) for some Hilbert space H Try to construct a Γ of the form Γ = u u dµ (u) SH with µ a probability measure, such that Γ (k) Γ (k) C(, k) S 1 (H k ) C(, k) should at least satisfy C(, k) 0 when with k fixed Clearly implies the Størmer-Hudson-Moody theorem, with a (hopefully) constructive proof. Answer unknown for general Hilbert spaces. Maybe true, maybe not... Progress in finite dimension, mostly König-Renner 05 / Christandl-König-Mitchison-Renner 07. Sufficient to discuss pure states Γ = Ψ Ψ, Ψ H.
23 A coherent state decomposition by Schur s lemma By rotational invariance of the normalized uniform measure du on the unit sphere SH dim H du u u = 1 H where d = dim H and SH ( ) dim H + d 1 =. d 1 Generalized coherent state representation: continous partition of the identity Lower symbol of Ψ H in this representation: dµ low (u) := dim H u, Ψ 2 du. Looking for a de Finetti measure looking for an upper symbol, i.e. dµ up such that Ψ Ψ = u u dµ up (u). SH Lieb 73, Simon 80 : upper symbols (when they exist!) have a tendency to turn into lower symbols in semi-classical limits.
24 Making a guess for the upper symbol To get an idea, pretend that you do have an upper symbol to begin with Ψ Ψ = u u dµ up (u). How to obtain dµ up (u) as a function of Ψ? Compute the lower symbol of the upper symbol When SH dµ low (u) = dim H u, Ψ 2 du ( ) = dim H u, v 2 dµ up (v) du SH dµ low (u) dµ up (u) reasonable guess : for any state Ψ, pick the lower symbol as a candidate for the de Finetti measure: Γ := dim H SH u u u, Ψ 2 du.
25 The CKMR estimate Reduced density matrices of the original state Ψ H γ (k) := Tr k+1 Ψ Ψ Reduced density matrices of the de Finetti state = dµ (u) u k u k γ (k) SH dµ (u) = dim H u, Ψ 2 du. Theorem (Christandl-König-Mitchison-Renner 07) For every k = 1, 2,..., it holds Tr H k γ (k) γ(k) 4kd. has the desired property C(k, ) = kd/ 0 for and d, k fixed. efficient when d, i.e. much more particles than degrees of freedom. rather short proof, I present a variant (Lewin-am-R 13)
26 Alternative representation of the CKMR state Theorem (Lewin-am-R 13) γ (k) with the convention that γ (l) ( ( + d 1)! k = ( + k + d 1)! l=0 l )( k l ) γ (l) s 1 H k l s 1 H k l = σ S k (γ (l) ) σ(1),...,σ(l) (1 H k l) σ(l+1),...,σ(k). Then γ (k) with A, B 0 and γ(k) Obviously Tr A = Tr B. = (C(d, k, ) 1)γ(k) + B = A + B ( + d 1)!! C(d, k, ) = ( + k + d 1)! ( k)! < 1. Triangle inequality Tr γ (k) γ(k) TrA + TrB = 2 Tr A = 2(1 C(d, k, )) ot as good as before, but not so bad... 2k(d + 2k).
27 Expectations in Hartree vectors determine the state Let s prove the explicit formula. γ (k) = ( + d 1)! ( + k + d 1)! ( )( ) k k l l l=0 γ (l) First: Lemma If a trace class self-adjoint operator γ (k) on H k satisfies then γ (k) 0. u k, γ (k) u k = 0 for all u H, s 1 H k l. States are fully determined by their lower symbol in a coherent state representation. Well-known in very general settings : Klauder 64, Simon 80. to prove the formula, only need to compute v k, γ (k) v k.
28 Wick quantization Creation/annihilation operators: a (f ) : H k 1 H k a (f ) (f 1... f k ) = (k) 1/2 f f 1... f k (+ symetrization). a(f ) : H k+1 H k adjoint of a (f ) (+symetrization) a(f ) (f 1... f k+1 ) = (k + 1) 1/2 f, f 1 f 2... f k+1 Canonical Commutation Relations [a(f ), a(g)] = 0, [a (f ), a (g)] = 0, [a(f ), a (g)] = f, g H. Characterization of the density matrices: v k, γ (k) v k = ( k)! Ψ a (v) k a(v) k Ψ! a.k.a. normal order: creation on the left, annihilation on the right.
29 Anti-Wick quantization v k, γ (k) v k = dim H du u, Ψ 2 u k, v k 2 SH! = dim H du u (+k), a (v) k Ψ 2 ( + k)!! dim H a by Schur s lemma = (v) k 2 Ψ ( + k)! dim H +k SH! dim H = ( + k)! dim H Ψ, a(v) k a (v) k Ψ +k a.k.a. anti-normal order: creation on the right, annihilation on the left. Only need to commute creation and annihilation using the CCR repeatedly. Linked to ideas of Ammari-ier a(v) k and a (v) k almost commute for large, small k, fixed d. Lemma For any v SH and n a(v) n a (v) n = ( ) n n n! k k! a (v) k a(v) k. k=0
30 1. Introduction : bosons in the mean-field regime 2. Trapped systems and the quantum de Finetti theorem 3. The need for a weak de Finetti theorem 4. Quantitative de Finetti in finite dimension 5. Localization in Fock space and quantum de Finetti 6. Validity of Hartree s theory in the general case: conclusion
31 Theorem (Lewin-am-R 13) If for any k Then for any k. Simple to see that Weak quantum de Finetti again γ (k) γ(k) in S 1 (H k ), γ (k) = BH dµ(u) u k u k ow we will argue that weak def strong def finite dimension def weak def We use localization methods in Fock space to obtain an (almost) constructive proof. Størmer-Hudson-Moody prove a stronger def for general (not necessarily normal) states with stronger def weak def But there s more to the constructive proof using localization: description of particles lost at infinity, i.e. defect of compactness.
32 Geometric localization (1) Main Idea (Derezinski-Gérard 99, Ammari 04, Hainzl-Lewin-Solovej 09, Lewin 11...) Take a sequence of (mixed) -body states γ S 1 (H ), w.l.o.g. assume weak- convergence of reduced density matrices Take a compact self-adjoint 1-body operator A to convert weak convergence into stong convergence (example A = 1 B(0,R) ) Define a state G A by localizing the reduced density matrices of γ : ( ) (n) G A = A n γ (n) A n Actually, one needs to go to the truncated bosonic Fock space (localization can mean loss of particles): F = C H H 2... H State on F : positive trace-class self-adjoint operator on F.
33 Lemma (Lewin 11) Geometric localization (2) For any -body state γ S 1 (H ) and bounded self adjoint operator A on H with 0 A 2 1, there exists a unique state G A on F such that where G A,k S 1 (H k ) and ( ) (n) G A = A n γ (n) A n = ( ) 1 ( ) Tr n+1 k G,k A n k=n k n G A = G A,0 G A,1... G A, Fundamental relation (follows from an explicit formula) Tr H k G,k A 1 A = Tr H k G 2, k Think of A = 1 B(0,R) The probability of having k particles inside the ball B(0, R) equals the probability of having k particles outside the ball B(0, R).
34 Idea of proof for weak quantum de Finetti (1) A sequence of -body states ( large) γ with γ (n) γ(n), γ (n) Tr n+1γ (n+1) Want to convert weak convergence into strong convergence obtain a consistent hierarchy and apply strong quantum de Finetti Pick a finite dimensional projector P and localize the state, (in the very end, take P 1). Look at the localized reduced density matrices ( ) 1 ( ) P n γ (n) k P n = Tr n+1 k G,k P n n k=n k=n ( ) n k Tr n+1 kg,k P Small k: ( k ) n goes to 0 rather fast ignore these terms Large k: up to normalization G P,k is a k-body state on PH apply quantum de Finetti in the finite dimensional space PH (no compactness issue) when k.
35 Idea of proof for weak quantum de Finetti (2) P n γ (n) P n k ) n Tr n+1 kg P,k ( k k=n ( k k ( ) n k ) n Tr n+1 kg P,k u n u n dµ k (u) S(PH) In the limit, the sum becomes an integral over the unit ball of H, λ k/ P n γ (n) P n ) n k 1 0 λ n ( k S(PH) u n u n dµ k (u) S(PH) u n u n dµ λ (u) Measure on the ball in spherical coordinates. Add a δ at the origin to obtain a probability. Ultimately, remove localization P by taking P l 1.
36 Localization and the de Finetti Theorem By-product of the proof: information about particles lost at infinity Smooth partition of unity, χ 2 R + η 2 R = 1, χ R 1 BR localization in a ball For any continuous function f, with µ the de Finetti measure lim f k=0 ( ) k Tr H k G χ R,k = dµ(u) f ( χ R u 2 ) BH Interpretation: µ ({ u 2 = λ }) = the probability that a fraction λ of the particles does not escape to infinity Using the fundamental property of the geometric localization ( ) k lim f Tr H k G η R,k = dµ(u) f (1 χ R u 2 ) BH k=0 gives information on particles lost at infinity. Crucial for non weakly lower semi continuous systems (bound states at infinity) Use with (essentially) f ground state energy in the k-particle sector
37 1. Introduction : bosons in the mean-field regime 2. Trapped systems and the quantum de Finetti theorem 3. The need for a weak de Finetti theorem 4. Quantitative de Finetti in finite dimension 5. Localization in Fock space and quantum de Finetti 6. Validity of Hartree s theory in the general case: conclusion
38 Typical setting One-body potentials going to 0 at (cf bosonic atoms and boson stars) Typical example H V ( = xj + V (x j ) ) on L 2 (R d ) with j=1 1 k<l w(x k x l ) V, w L p + L 0, max(1, d/2) < p <, w(x) = w( x) V, w 0 at Can add magnetic fields, fractional Laplacians... Hartree s functionals. Possible loss of mass at infinity ( EH V [u] = u 2 + V u 2) + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R d EH[u] 0 = u u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R { d eh V (λ) = inf EH V [u] } u 2 = λ. R d Large binding inequalities: for all 0 λ 1: e V H (1) e V H (λ) + e 0 H(1 λ)
39 Validity of Hartree for general systems Theorem (Lewin-am-R 13) Energy: lim 1 E V () = e V H (1). Take Ψ a sequence of approximate ground states in H. There exists a Borel probability measure µ supported on the set { M V = u H, u 1 } EH V [u] = eh V ( u 2 ) = eh V (1) eh(1 0 u 2 ), such that γ (k) Ψ j M V u k u k dµ(u). If strict binding holds for Hartree: e V H (1) < e V H (λ) + e 0 H(1 λ). Then µ is supported on SH and the limit is strong. If in addition e V H (1) admits a unique minimizer u H, there is complete Bose-Einstein condensation γ (k) Ψ j u k H u k H strongly in S 1 (H k ).
40 Conclusion The validity of Hartree s theory in the mean-field scaling is a very general fact and follows from the structure of bosonic states. Main structure property: quantum de Finetti theorem, solving the representability problem in the limit. Explicit construction and estimates in finite dimension. General case using localization in Fock space. Outlook Quantitative estimates for the validity of Hartree s theory by this method? Extend the method to derive (static) non-linear Schrödinger equations by scaling the interaction potential? Discuss attractive interactions and many-body collapse? Positive (large) temperature?
Superfluidity & Bogoliubov Theory: Rigorous Results
Superfluidity & Bogoliubov Theory: Rigorous Results Mathieu LEWIN mathieu.lewin@math.cnrs.fr (CNRS & Université Paris-Dauphine) collaborations with P.T. Nam (Vienna), N. Rougerie (Grenoble), B. Schlein
More informationFrom many body quantum dynamics to nonlinear dispersive PDE, and back
From many body quantum dynamics to nonlinear dispersive PDE, and back ataša Pavlović (based on joint works with T. Chen, and with T. Chen and. Tzirakis) University of Texas at Austin June 18, 2013 SF-CBMS
More informationMean Field Limits for Interacting Bose Gases and the Cauchy Problem for Gross-Pitaevskii Hierarchies. Thomas Chen University of Texas at Austin
Mean Field Limits for Interacting Bose Gases and the Cauchy Problem for Gross-Pitaevskii Hierarchies Thomas Chen University of Texas at Austin Nataša Pavlović University of Texas at Austin Analysis and
More informationBose Gases, Bose Einstein Condensation, and the Bogoliubov Approximation
Bose Gases, Bose Einstein Condensation, and the Bogoliubov Approximation Robert Seiringer IST Austria Mathematical Horizons for Quantum Physics IMS Singapore, September 18, 2013 R. Seiringer Bose Gases,
More informationFrom quantum many body systems to nonlinear dispersive PDE, and back
From quantum many body systems to nonlinear dispersive PDE, and back Nataša Pavlović (University of Texas at Austin) Based on joint works with T. Chen, C. Hainzl, R. Seiringer, N. Tzirakis May 28, 2015
More informationDerivation of Mean-Field Dynamics for Fermions
IST Austria Mathematical Many Body Theory and its Applications Bilbao, June 17, 2016 Derivation = show that solution of microscopic eq. is close to solution of effective eq. (with fewer degrees of freedom)
More informationDerivation of the time dependent Gross-Pitaevskii equation for the dynamics of the Bose-Einstein condensate
Derivation of the time dependent Gross-Pitaevskii equation for the dynamics of the Bose-Einstein condensate László Erdős University of Munich Evian-les-Bains, 2007 June Joint work with B. Schlein and H.T.
More informationLow Regularity Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy
Low Regularity Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy Kenny Taliaferro (based on joint works with Y. Hong and Z. Xie) November 22, 2014 1 / 19 The Cubic Gross-Pitaevskii Hierarchy The
More informationBosonic quadratic Hamiltonians. diagonalization
and their diagonalization P. T. Nam 1 M. Napiórkowski 2 J. P. Solovej 3 1 Masaryk University Brno 2 University of Warsaw 3 University of Copenhagen Macroscopic Limits of Quantum Systems 70th Birthday of
More informationEffective dynamics of many-body quantum systems
Effective dynamics of many-body quantum systems László Erdős University of Munich Grenoble, May 30, 2006 A l occassion de soixantiéme anniversaire de Yves Colin de Verdiére Joint with B. Schlein and H.-T.
More informationEffective Dynamics of Solitons I M Sigal
Effective Dynamics of Solitons I M Sigal Porquerolles October, 008 Joint work with Jürg Fröhlich, Stephen Gustafson, Lars Jonsson, Gang Zhou and Walid Abou Salem Bose-Einstein Condensation Consider a system
More informationDiagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations 1
Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations 1 P. T. Nam 1 M. Napiórkowski 1 J. P. Solovej 2 1 Institute of Science and Technology Austria 2 Department of Mathematics,
More informationarxiv: v3 [math-ph] 8 Sep 2015
GROUD STATES OF LARGE BOSOIC SYSTEMS: THE GROSS-PITAEVSKII LIMIT REVISITED PHA THÀH AM, ICOLAS ROUGERIE, AD ROBERT SEIRIGER arxiv:1503.07061v3 [math-ph] 8 Sep 2015 Abstract. We study the ground state of
More informationOn the pair excitation function.
On the pair excitation function. Joint work with M. Grillakis based on our 2013 and 2016 papers Beyond mean field: On the role of pair excitations in the evolution of condensates, Journal of fixed point
More informationIncompressibility Estimates in the Laughlin Phase
Incompressibility Estimates in the Laughlin Phase Jakob Yngvason, University of Vienna with Nicolas Rougerie, University of Grenoble București, July 2, 2014 Jakob Yngvason (Uni Vienna) Incompressibility
More informationBose-Einstein condensation and limit theorems. Kay Kirkpatrick, UIUC
Bose-Einstein condensation and limit theorems Kay Kirkpatrick, UIUC 2015 Bose-Einstein condensation: from many quantum particles to a quantum superparticle Kay Kirkpatrick, UIUC/MSRI TexAMP 2015 The big
More informationA NEW PROOF OF EXISTENCE OF SOLUTIONS FOR FOCUSING AND DEFOCUSING GROSS-PITAEVSKII HIERARCHIES. 1. Introduction
A EW PROOF OF EXISTECE OF SOLUTIOS FOR FOCUSIG AD DEFOCUSIG GROSS-PITAEVSKII HIERARCHIES THOMAS CHE AD ATAŠA PAVLOVIĆ Abstract. We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in d
More informationDE FINETTI THEOREMS, MEAN-FIELD LIMITS AND BOSE-EINSTEIN CONDENSATION
DE FIETTI THEOREMS, MEA-FIELD LIMITS AD BOSE-EISTEI CODESATIO ICOLAS ROUGERIE Laboratoire de Physique et Modélisation des Milieux Condensés, Université Grenoble 1 & CRS. Lecture notes for a course at the
More informationLocal Density Approximation for the Almost-bosonic Anyon Gas. Michele Correggi
Local Density Approximation for the Almost-bosonic Anyon Gas Michele Correggi Università degli Studi Roma Tre www.cond-math.it QMATH13 Many-body Systems and Statistical Mechanics joint work with D. Lundholm
More informationCondensation of fermion pairs in a domain
Condensation of fermion pairs in a domain Marius Lemm (Caltech) joint with Rupert L. Frank and Barry Simon QMath 13, Georgia Tech, October 8, 2016 BCS states We consider a gas of spin 1/2 fermions, confined
More informationarxiv: v1 [math-ph] 15 Oct 2015
MEAN-FIELD LIMIT OF BOSE SYSTEMS: RIGOROUS RESULTS MATHIEU LEWIN arxiv:50.04407v [math-ph] 5 Oct 205 Abstract. We review recent results about the derivation of the Gross- Pitaevskii equation and of the
More informationGround-State Energies of Coulomb Systems and Reduced One-Particle Density Matrices
Ground-State Energies of Coulomb Systems and Reduced One-Particle Density Matrices Heinz Siedentop Chennai, August 16, 2010 I. Introduction Atomic Schrödinger operator H N,Z := N n=1 self-adjointly realized
More informationMean-Field Limits for Large Particle Systems Lecture 2: From Schrödinger to Hartree
for Large Particle Systems Lecture 2: From Schrödinger to Hartree CMLS, École polytechnique & CNRS, Université Paris-Saclay FRUMAM, Marseilles, March 13-15th 2017 A CRASH COURSE ON QUANTUM N-PARTICLE DYNAMICS
More informationMany body quantum dynamics and nonlinear dispersive PDE
Many body quantum dynamics and nonlinear dispersive PDE Nataša Pavlović (University of Texas at Austin) August 24-28, 2015 Introductory Workshop: Randomness and long time dynamics in nonlinear evolution
More informationFrom the N-body problem to the cubic NLS equation
From the N-body problem to the cubic NLS equation François Golse Université Paris 7 & Laboratoire J.-L. Lions golse@math.jussieu.fr Los Alamos CNLS, January 26th, 2005 Formal derivation by N.N. Bogolyubov
More informationEffective Potentials Generated by Field Interaction in the Quasi-Classical Limit. Michele Correggi
Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit Michele Correggi Dipartimento di Matematica Quantum Mean Field and Related Problems LAGA Université Paris 13 joint work
More informationBose-Einstein condensates in optical lattices: mathematical analysis and analytical approximate formulas
0.5 setgray0 0.5 setgray1 Bose-Einstein condensates in optical lattices: mathematical analysis and analytical approximate formulas IV EBED João Pessoa - 2011 Rolci Cipolatti Instituto de Matemática - UFRJ
More informationModeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas
Modeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas Weizhu Bao Department of Mathematics National University of Singapore Email: matbaowz@nus.edu.sg URL: http://www.math.nus.edu.sg/~bao
More informationMany-Body Quantum Mechanics
Quasi-Classical Limit for Quantum Particle-Field Systems Michele Correggi Many-Body Quantum Mechanics CRM, Montréal joint work with M. Falconi (Tübingen) and M. Olivieri (Roma 1) M. Correggi (Roma 1) Quasi-Classical
More informationSchrödinger-Klein-Gordon system as a classical limit of a scalar Quantum Field Theory.
Schrödinger-Klein-Gordon system as a classical limit of a scalar Quantum Field Theory. (in collaboration with Marco Falconi) Zied Ammari Rennes University, IRMAR BECAM, 28 October 2014 The Yukawa Theory
More informationIncompressibility Estimates in the Laughlin Phase
Incompressibility Estimates in the Laughlin Phase Jakob Yngvason, University of Vienna with Nicolas Rougerie, University of Grenoble ESI, September 8, 2014 Jakob Yngvason (Uni Vienna) Incompressibility
More informationFermionic coherent states in infinite dimensions
Fermionic coherent states in infinite dimensions Robert Oeckl Centro de Ciencias Matemáticas Universidad Nacional Autónoma de México Morelia, Mexico Coherent States and their Applications CIRM, Marseille,
More informationAlgebraic Theory of Entanglement
Algebraic Theory of (arxiv: 1205.2882) 1 (in collaboration with T.R. Govindarajan, A. Queiroz and A.F. Reyes-Lega) 1 Physics Department, Syracuse University, Syracuse, N.Y. and The Institute of Mathematical
More informationarxiv: v3 [math-ph] 21 May 2015
DERIATION OF NONLINEAR GIBBS MEASURES FROM MANY-BODY QUANTUM MECHANICS MATHIEU LEWIN, PHAN THÀNH NAM, AND NICOLAS ROUGERIE arxiv:1410.0335v3 [math-ph] 21 May 2015 Abstract. We prove that nonlinear Gibbs
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationDERIVATION OF THE CUBIC NLS AND GROSS-PITAEVSKII HIERARCHY FROM MANYBODY DYNAMICS IN d = 2, 3 BASED ON SPACETIME NORMS
DERIVATIO OF THE CUBIC LS AD GROSS-PITAEVSKII HIERARCHY FROM MAYBODY DYAMICS I d = 2, 3 BASED O SPACETIME ORMS THOMAS CHE AD ATAŠA PAVLOVIĆ Abstract. We derive the defocusing cubic Gross-Pitaevskii (GP)
More informationValidity of spin wave theory for the quantum Heisenberg model
Validity of spin wave theory for the quantum Heisenberg model Michele Correggi, Alessandro Giuliani, and Robert Seiringer Dipartimento di Matematica e Fisica, Università di Roma Tre, L.go S. L. Murialdo,
More informationGibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions d 3
Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions d 3 Jürg Fröhlich Antti Knowles 2 Benjamin Schlein 3 Vedran Sohinger 4 December, 207 Abstract We prove
More informationDerivation of Pekar s polaron from a microscopic model of quantum crystal
Derivation of Pekar s polaron from a microscopic model of quantum crystal Mathieu LEWIN Mathieu.Lewin@math.cnrs.fr (CNRS & University of Cergy-Pontoise) joint work with Nicolas Rougerie (Grenoble, France)
More informationFrom Bose-Einstein Condensation to Monster Waves: The Nonlinear Hartree Equation, and some of its Large Coupling Phenomena
From Bose-Einstein Condensation to Monster Waves: The Nonlinear Hartree Equation, and some of its Large Coupling Phenomena Walter H. Aschbacher (Ecole Polytechnique/TU München) Model Equations in Bose-Einstein
More informationOn the infimum of the excitation spectrum of a homogeneous Bose gas. H.D. Cornean, J. Dereziński,
On the infimum of the excitation spectrum of a homogeneous Bose gas H.D. Cornean, J. Dereziński, P. Ziń. 1 Homogeneous Bose gas n bosons on R d interacting with a 2-body potential v are ( described by
More informationarxiv: v2 [math-ph] 5 Oct 2018
GROUD STATE EERGY OF MIXTURE OF BOSE GASES ALESSADRO MICHELAGELI, PHA THÀH AM, AD ALESSADRO OLGIATI arxiv:1803.05413v [math-ph] 5 Oct 018 Abstract. We consider the asymptotic behavior of a system of multi-component
More informationCS286.2 Lecture 13: Quantum de Finetti Theorems
CS86. Lecture 13: Quantum de Finetti Theorems Scribe: Thom Bohdanowicz Before stating a quantum de Finetti theorem for density operators, we should define permutation invariance for quantum states. Let
More informationLOCALIZATION OF RELATIVE ENTROPY IN BOSE-EINSTEIN CONDENSATION OF TRAPPED INTERACTING BOSONS
LOCALIZATION OF RELATIVE ENTROPY IN BOSE-EINSTEIN CONDENSATION OF TRAPPED INTERACTING BOSONS LAURA M. MORATO 1 AND STEFANIA UGOLINI 2 1 Facoltà di Scienze, Università di Verona, Strada le Grazie, 37134
More informationThird Critical Speed for Rotating Bose-Einstein Condensates. Mathematical Foundations of Physics Daniele Dimonte, SISSA
Third Critical Speed for Rotating Bose-Einstein Condensates Mathematical Foundations of Physics Daniele Dimonte, SISSA 1 st November 2016 this presentation available on daniele.dimonte.it based on a joint
More informationGround State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory
Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou National Center of Theoretical Science December 19, 2015
More informationFOCUSING QUANTUM MANY-BODY DYNAMICS II: THE RIGOROUS DERIVATION OF THE 1D FOCUSING CUBIC NONLINEAR SCHRÖDINGER EQUATION FROM 3D
FOCUSING QUANTUM MANY-BODY DYNAMICS II: THE RIGOROUS DERIVATION OF THE 1D FOCUSING CUBIC NONLINEAR SCHRÖDINGER EQUATION FROM 3D XUWEN CHEN AND JUSTIN HOLMER Abstract. We consider the focusing 3D quantum
More information1 Quantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationNon-relativistic Quantum Electrodynamics
Rigorous Aspects of Relaxation to the Ground State Institut für Analysis, Dynamik und Modellierung October 25, 2010 Overview 1 Definition of the model Second quantization Non-relativistic QED 2 Existence
More information221B Lecture Notes Quantum Field Theory II (Fermi Systems)
1B Lecture Notes Quantum Field Theory II (Fermi Systems) 1 Statistical Mechanics of Fermions 1.1 Partition Function In the case of fermions, we had learnt that the field operator satisfies the anticommutation
More informationOn the relation between scaling properties of functionals and existence of constrained minimizers
On the relation between scaling properties of functionals and existence of constrained minimizers Jacopo Bellazzini Dipartimento di Matematica Applicata U. Dini Università di Pisa January 11, 2011 J. Bellazzini
More information2D Electrostatics and the Density of Quantum Fluids
2D Electrostatics and the Density of Quantum Fluids Jakob Yngvason, University of Vienna with Elliott H. Lieb, Princeton University and Nicolas Rougerie, University of Grenoble Yerevan, September 5, 2016
More informationNONEXISTENCE OF MINIMIZER FOR THOMAS-FERMI-DIRAC-VON WEIZSÄCKER MODEL. x y
NONEXISTENCE OF MINIMIZER FOR THOMAS-FERMI-DIRAC-VON WEIZSÄCKER MODEL JIANFENG LU AND FELIX OTTO In this paper, we study the following energy functional () E(ϕ) := ϕ 2 + F(ϕ 2 ) dx + D(ϕ 2,ϕ 2 ), R 3 where
More informationEIGENFUNCTIONS OF DIRAC OPERATORS AT THE THRESHOLD ENERGIES
EIGENFUNCTIONS OF DIRAC OPERATORS AT THE THRESHOLD ENERGIES TOMIO UMEDA Abstract. We show that the eigenspaces of the Dirac operator H = α (D A(x)) + mβ at the threshold energies ±m are coincide with the
More informationInteraction between atoms
Interaction between atoms MICHA SCHILLING HAUPTSEMINAR: PHYSIK DER KALTEN GASE INSTITUT FÜR THEORETISCHE PHYSIK III UNIVERSITÄT STUTTGART 23.04.2013 Outline 2 Scattering theory slow particles / s-wave
More informationMicroscopic Derivation of Ginzburg Landau Theory. Mathematics and Quantum Physics
Microscopic Derivation of Ginzburg Landau heory Robert Seiringer IS Austria Joint work with Rupert Frank, Christian Hainzl, and Jan Philip Solovej J. Amer. Math. Soc. 25 (2012), no. 3, 667 713 Mathematics
More informationRapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps. Michele Correggi. T. Rindler-Daller, J. Yngvason math-ph/
Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps Michele Correggi Erwin Schrödinger Institute, Vienna T. Rindler-Daller, J. Yngvason math-ph/0606058 in collaboration with preprint
More informationarxiv: v3 [math-ph] 23 Apr 2013
O THE RIGOROUS DERIVATIO OF THE 3D CUBIC OLIEAR SCHRÖDIGER EQUATIO WITH A QUADRATIC TRAP XUWE CHE Dedicated to Xuqing. arxiv:14.15v3 [math-ph] 3 Apr 13 Abstract. We consider the dynamics of the 3D -body
More informationarxiv: v1 [math-ph] 26 Oct 2017
ORM APPROXIMATIO FOR MAY-BODY QUATUM DYAMICS: FOCUSIG CASE I LOW DIMESIOS PHA THÀH AM AD MARCI APIÓRKOWSKI arxiv:70.09684v [math-ph] 26 Oct 207 Abstract. We study the norm approximation to the Schrödinger
More informationA complete criterion for convex-gaussian states detection
A complete criterion for convex-gaussian states detection Anna Vershynina Institute for Quantum Information, RWTH Aachen, Germany joint work with B. Terhal NSF/CBMS conference Quantum Spin Systems The
More informationDerivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems
Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems The Harvard community has made this article openly available. Please share how this access benefits you.
More informationABSTRACT. Professor Manoussos Grillakis Professor Matei Machedon Department of Mathematics
ABSTRACT Title of dissertation: QUANTITATIVE DERIVATION OF EFFECTIVE EVOLUTION EQUATIONS FOR THE DYNAMICS OF BOSE-EINSTEIN CONDENSATES Elif Kuz, Doctor of Philosophy, 2016 Dissertation directed by: Professor
More informationCORRELATION STRUCTURES, MANY-BODY SCATTERING PROCESSES AND THE DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY
CORRELATIO STRUCTURES, MAY-BODY SCATTERIG PROCESSES AD THE DERIVATIO OF THE GROSS-PITAEVSKII HIERARCHY XUWE CHE AD JUSTI HOLMER A. We consider the dynamics of bosons in three dimensions. We assume the
More informationBose-Einstein Condensate: A New state of matter
Bose-Einstein Condensate: A New state of matter KISHORE T. KAPALE June 24, 2003 BOSE-EINSTEIN CONDENSATE: A NEW STATE OF MATTER 1 Outline Introductory Concepts Bosons and Fermions Classical and Quantum
More informationMany-Body Problems and Quantum Field Theory
Philippe A. Martin Francois Rothen Many-Body Problems and Quantum Field Theory An Introduction Translated by Steven Goldfarb, Andrew Jordan and Samuel Leach Second Edition With 102 Figures, 7 Tables and
More informationHeat Flows, Geometric and Functional Inequalities
Heat Flows, Geometric and Functional Inequalities M. Ledoux Institut de Mathématiques de Toulouse, France heat flow and semigroup interpolations Duhamel formula (19th century) pde, probability, dynamics
More informationarxiv:cond-mat/ v1 [cond-mat.stat-mech] 2 Aug 2004
Ground state energy of a homogeneous Bose-Einstein condensate beyond Bogoliubov Christoph Weiss and André Eckardt Institut für Physik, Carl von Ossietzky Universität, D-6 Oldenburg, Germany (Dated: November
More informationhere, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional
15. Perturbations by compact operators In this chapter, we study the stability (or lack thereof) of various spectral properties under small perturbations. Here s the type of situation we have in mind:
More informationRealizing non-abelian statistics in quantum loop models
Realizing non-abelian statistics in quantum loop models Paul Fendley Experimental and theoretical successes have made us take a close look at quantum physics in two spatial dimensions. We have now found
More informationSupercell method for the computation of energies of crystals
Supercell method for the computation of energies of crystals David Gontier CEREMADE, Université Paris-Dauphine Warwick EPSRC Symposium: Density Functional Theory and Beyond: Analysis and Computation Warwick,
More informationQuantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More information1 Superfluidity and Bose Einstein Condensate
Physics 223b Lecture 4 Caltech, 04/11/18 1 Superfluidity and Bose Einstein Condensate 1.6 Superfluid phase: topological defect Besides such smooth gapless excitations, superfluid can also support a very
More informationBOGOLIUBOV TRANSFORMATIONS AND ENTANGLEMENT OF TWO FERMIONS
BOGOLIUBOV TRANSFORMATIONS AND ENTANGLEMENT OF TWO FERMIONS P. Caban, K. Podlaski, J. Rembieliński, K. A. Smoliński and Z. Walczak Department of Theoretical Physics, University of Lodz Pomorska 149/153,
More informationSecond Quantization: Quantum Fields
Second Quantization: Quantum Fields Bosons and Fermions Let X j stand for the coordinate and spin subscript (if any) of the j-th particle, so that the vector of state Ψ of N particles has the form Ψ Ψ(X
More informationQuantum Marginal Problems
Quantum Marginal Problems David Gross (Colgone) Joint with: Christandl, Doran, Lopes, Schilling, Walter Outline Overview: Marginal problems Overview: Entanglement Main Theme: Entanglement Polytopes Shortly:
More information221B Lecture Notes Quantum Field Theory II (Fermi Systems)
1B Lecture Notes Quantum Field Theory II (Fermi Systems) 1 Statistical Mechanics of Fermions 1.1 Partition Function In the case of fermions, we had learnt that the field operator satisfies the anticommutation
More informationPRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in
LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific
More informationNumerical simulation of Bose-Einstein Condensates based on Gross-Pitaevskii Equations
1/38 Numerical simulation of Bose-Einstein Condensates based on Gross-Pitaevskii Equations Xavier ANTOINE Institut Elie Cartan de Lorraine (IECL) & Inria-SPHINX Team Université de Lorraine, France Funded
More informationUniform in N estimates for a Bosonic system of Hartree-Fock-Bogoliubov type. Joint work with M. Grillakis
Uniform in N estimates for a Bosonic system of Hartree-Fock-Bogoliubov type Joint work with M. Grillakis 1 Overview of the problem: Approximate symmetric solutions to the manybody problem 1 i t ψ N(t,
More informationRecent developments on the global behavior to critical nonlinear dispersive equations. Carlos E. Kenig
Recent developments on the global behavior to critical nonlinear dispersive equations Carlos E. Kenig In the last 25 years or so, there has been considerable interest in the study of non-linear partial
More informationUltra-cold gases. Alessio Recati. CNR INFM BEC Center/ Dip. Fisica, Univ. di Trento (I) & Dep. Physik, TUM (D) TRENTO
Ultra-cold gases Alessio Recati CNR INFM BEC Center/ Dip. Fisica, Univ. di Trento (I) & Dep. Physik, TUM (D) TRENTO Lectures L. 1) Introduction to ultracold gases Bosonic atoms: - From weak to strong interacting
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationNo-hair and uniqueness results for analogue black holes
No-hair and uniqueness results for analogue black holes LPT Orsay, France April 25, 2016 [FM, Renaud Parentani, and Robin Zegers, PRD93 065039] Outline Introduction 1 Introduction 2 3 Introduction Hawking
More informationSemicircle law on short scales and delocalization for Wigner random matrices
Semicircle law on short scales and delocalization for Wigner random matrices László Erdős University of Munich Weizmann Institute, December 2007 Joint work with H.T. Yau (Harvard), B. Schlein (Munich)
More informationTHE QUINTIC NLS AS THE MEAN FIELD LIMIT OF A BOSON GAS WITH THREE-BODY INTERACTIONS. 1. Introduction
THE QUINTIC NLS AS THE MEAN FIELD LIMIT OF A BOSON GAS WITH THREE-BODY INTERACTIONS THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We investigate the dynamics of a boson gas with three-body interactions in
More informationAccuracy of the time-dependent Hartree-Fock approximation
Accuracy of the time-dependent Hartree-Fock approximation Claude BARDOS, François GOLSE, Alex D. GOTTLIEB and Norbert J. MAUSER Abstract This article examines the time-dependent Hartree-Fock (TDHF) approximation
More informationINTERACTING BOSE GAS AND QUANTUM DEPLETION
922 INTERACTING BOSE GAS AND QUANTUM DEPLETION Chelagat, I., *Tanui, P.K., Khanna, K.M.,Tonui, J.K., Murunga G.S.W., Chelimo L.S.,Sirma K. K., Cheruiyot W.K. &Masinde F. W. Department of Physics, University
More informationBRST and Dirac Cohomology
BRST and Dirac Cohomology Peter Woit Columbia University Dartmouth Math Dept., October 23, 2008 Peter Woit (Columbia University) BRST and Dirac Cohomology October 2008 1 / 23 Outline 1 Introduction 2 Representation
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationBOGOLIUBOV SPECTRUM OF INTERACTING BOSE GASES
BOGOLIUBOV SPECTRUM OF ITERACTIG BOSE GASES MATHIEU LEWI, PHA THÀH AM, SYLVIA SERFATY, AD JA PHILIP SOLOVEJ Abstract. We study the large- limit of a system of bosons interacting with a potential of intensity
More informationLocal exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics
Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Douglas Lundholm IHES / IHP / CRMIN joint work with Jan Philip Solovej, University of Copenhagen Variational &
More informationEFFECT OF A LOCALLY REPULSIVE INTERACTION ON S WAVE SUPERCONDUCTORS. (Version )
EFFECT OF A LOCALLY REPLSIVE ITERACTIO O S WAVE SPERCODCTORS J.-B. Bru 1 and W. de Siqueira Pedra 2 Version 23.09.2009 Abstract. The thermodynamic impact of the Coulomb repulsion on s wave superconductors
More informationINSTITUT FOURIER. Quantum correlations and Geometry. Dominique Spehner
i f INSTITUT FOURIER Quantum correlations and Geometry Dominique Spehner Institut Fourier et Laboratoire de Physique et Modélisation des Milieux Condensés, Grenoble Outlines Entangled and non-classical
More informationPreface Introduction to the electron liquid
Table of Preface page xvii 1 Introduction to the electron liquid 1 1.1 A tale of many electrons 1 1.2 Where the electrons roam: physical realizations of the electron liquid 5 1.2.1 Three dimensions 5 1.2.2
More informationNonlinear problems with lack of compactness in Critical Point Theory
Nonlinear problems with lack of compactness in Critical Point Theory Carlo Mercuri CASA Day Eindhoven, 11th April 2012 Critical points Many linear and nonlinear PDE s have the form P(u) = 0, u X. (1) Here
More informationwhere P a is a projector to the eigenspace of A corresponding to a. 4. Time evolution of states is governed by the Schrödinger equation
1 Content of the course Quantum Field Theory by M. Srednicki, Part 1. Combining QM and relativity We are going to keep all axioms of QM: 1. states are vectors (or rather rays) in Hilbert space.. observables
More informationThe Gross-Pitaevskii Equation and the Hydrodynamic Expansion of BECs
The Gross-Pitaevskii Equation and the Hydrodynamic Expansion of BECs RHI seminar Pascal Büscher i ( t Φ (r, t) = 2 2 ) 2m + V ext(r) + g Φ (r, t) 2 Φ (r, t) 27 Nov 2008 RHI seminar Pascal Büscher 1 (Stamper-Kurn
More informationA PRIORI ESTIMATES FOR MANY-BODY HAMILTONIAN EVOLUTION OF INTERACTING BOSON SYSTEM
Journal of Hyperbolic Differential Equations Vol. 5, No. 4 (2008) 857 883 c World Scientific Publishing Company A PRIORI ESTIMATES FOR MANY-BODY HAMILTONIAN EVOLUTION OF INTERACTING BOSON SYSTEM MANOUSSOS
More informationQuantum Symmetries in Free Probability Theory. Roland Speicher Queen s University Kingston, Canada
Quantum Symmetries in Free Probability Theory Roland Speicher Queen s University Kingston, Canada Quantum Groups are generalizations of groups G (actually, of C(G)) are supposed to describe non-classical
More informationWave function methods for the electronic Schrödinger equation
Wave function methods for the electronic Schrödinger equation Zürich 2008 DFG Reseach Center Matheon: Mathematics in Key Technologies A7: Numerical Discretization Methods in Quantum Chemistry DFG Priority
More information