Many-Body Quantum Mechanics

Transcription

1 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 Limit CRM 14/9/ / 22

2 Outline 1 Introduction: Physical motivations: effective models of radiation-matter interaction; Mathematical formulation: quasi-classical limit. 2 Main results [CF,CFO1]: derivation of effective potentials & ground state energy convergence. 3 Perspectives: dynamics in the quasi-classical limit for suitable coherent initial states = derivation of time-dependent point interaction models [CCFO]; generic initial states [CFO2]. Main References [CF] M.C., M. Falconi, Ann. H. Poincaré 19 (2018), ; [CFO1] M.C., M. Falconi, M. Olivieri, preprint arxiv: [math-ph] (2017), J. Spectr. Theory to appear; [CFO2] M.C., M. Falconi, M. Olivieri in preparation; [CCFO] R. Carlone, M.C., M. Falconi, M. Olivieri in preparation. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

3 Outline 1 Introduction: Physical motivations: effective models of radiation-matter interaction; Mathematical formulation: quasi-classical limit. 2 Main results [CF,CFO1]: derivation of effective potentials & ground state energy convergence. 3 Perspectives: dynamics in the quasi-classical limit for suitable coherent initial states = derivation of time-dependent point interaction models [CCFO]; generic initial states [CFO2]. Main References [CF] M.C., M. Falconi, Ann. H. Poincaré 19 (2018), ; [CFO1] M.C., M. Falconi, M. Olivieri, preprint arxiv: [math-ph] (2017), J. Spectr. Theory to appear; [CFO2] M.C., M. Falconi, M. Olivieri in preparation; [CCFO] R. Carlone, M.C., M. Falconi, M. Olivieri in preparation. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

4 Outline 1 Introduction: Physical motivations: effective models of radiation-matter interaction; Mathematical formulation: quasi-classical limit. 2 Main results [CF,CFO1]: derivation of effective potentials & ground state energy convergence. 3 Perspectives: dynamics in the quasi-classical limit for suitable coherent initial states = derivation of time-dependent point interaction models [CCFO]; generic initial states [CFO2]. Main References [CF] M.C., M. Falconi, Ann. H. Poincaré 19 (2018), ; [CFO1] M.C., M. Falconi, M. Olivieri, preprint arxiv: [math-ph] (2017), J. Spectr. Theory to appear; [CFO2] M.C., M. Falconi, M. Olivieri in preparation; [CCFO] R. Carlone, M.C., M. Falconi, M. Olivieri in preparation. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

5 ❶ Introduction Physical Motivations (I) Magneto-optical traps In the experiments cold atomic systems are usually produced using magneto-optical traps; Atoms are cooled down by using suitably tuned lasers (Doppler cooling) and trapped by means of non-uniform magnetic fields (particle-field interaction); The effective models describe non-relativistic quantum particles with trapping potentials, e.g., V (x) = x 2 or x 4 etc. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

6 ❶ Introduction Physical Motivations (I) Magneto-optical traps In the experiments cold atomic systems are usually produced using magneto-optical traps; Atoms are cooled down by using suitably tuned lasers (Doppler cooling) and trapped by means of non-uniform magnetic fields (particle-field interaction); The effective models describe non-relativistic quantum particles with trapping potentials, e.g., V (x) = x 2 or x 4 etc. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

7 ❶ Introduction Physical Motivations (I) Magneto-optical traps In the experiments cold atomic systems are usually produced using magneto-optical traps; Atoms are cooled down by using suitably tuned lasers (Doppler cooling) and trapped by means of non-uniform magnetic fields (particle-field interaction); The effective models describe non-relativistic quantum particles with trapping potentials, e.g., V (x) = x 2 or x 4 etc. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

8 ❶ Introduction Physical Motivations (II) Optical lattices Another typical setting for cold atom experiments is given by optical lattices; Optical lattices are generated by means of the interference of counter-propagating laser beams suitably coupled to the atoms (particle-field interaction); The effective models describe non-relativistic quantum particles in periodic potentials, e.g., of the form V (x) = j V 0 sin 2 (k j x j ). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

9 ❶ Introduction Physical Motivations (II) Optical lattices Another typical setting for cold atom experiments is given by optical lattices; Optical lattices are generated by means of the interference of counter-propagating laser beams suitably coupled to the atoms (particle-field interaction); The effective models describe non-relativistic quantum particles in periodic potentials, e.g., of the form V (x) = j V 0 sin 2 (k j x j ). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

10 ❶ Introduction Physical Motivations (II) Optical lattices Another typical setting for cold atom experiments is given by optical lattices; Optical lattices are generated by means of the interference of counter-propagating laser beams suitably coupled to the atoms (particle-field interaction); The effective models describe non-relativistic quantum particles in periodic potentials, e.g., of the form V (x) = j V 0 sin 2 (k j x j ). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

11 ❶ Introduction Physical Motivations (III) Quantum particle-field interaction More in general when a quantum particle system interact with an intense electro-magnetic field, e.g., a laser, the field generated by a strong magnet, etc., the full system is described by the first-quantized magnetic Schrödinger operator (MSO) N ( i j A(x j )) 2 + V (x 1,..., x N ), j=1 where A is the magnetic vector potential and V the electric and interaction potential. The quantum nature of radiation is discarded and the field replaced with its classical counterpart. Can this replacement be justified rigorously in a suitable regime? M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

12 ❶ Introduction Physical Motivations (III) Quantum particle-field interaction More in general when a quantum particle system interact with an intense electro-magnetic field, e.g., a laser, the field generated by a strong magnet, etc., the full system is described by the first-quantized magnetic Schrödinger operator (MSO) N ( i j A(x j )) 2 + V (x 1,..., x N ), j=1 where A is the magnetic vector potential and V the electric and interaction potential. The quantum nature of radiation is discarded and the field replaced with its classical counterpart. Can this replacement be justified rigorously in a suitable regime? M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

13 ❶ Introduction Physical Motivations (III) Quantum particle-field interaction More in general when a quantum particle system interact with an intense electro-magnetic field, e.g., a laser, the field generated by a strong magnet, etc., the full system is described by the first-quantized magnetic Schrödinger operator (MSO) N ( i j A(x j )) 2 + V (x 1,..., x N ), j=1 where A is the magnetic vector potential and V the electric and interaction potential. The quantum nature of radiation is discarded and the field replaced with its classical counterpart. Can this replacement be justified rigorously in a suitable regime? M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

14 ❶ Introduction Physical Motivations (III) Quantum particle-field interaction More in general when a quantum particle system interact with an intense electro-magnetic field, e.g., a laser, the field generated by a strong magnet, etc., the full system is described by the first-quantized magnetic Schrödinger operator (MSO) N ( i j A(x j )) 2 + V (x 1,..., x N ), j=1 where A is the magnetic vector potential and V the electric and interaction potential. The quantum nature of radiation is discarded and the field replaced with its classical counterpart. Can this replacement be justified rigorously in a suitable regime? M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

15 ❶ Introduction Particle-Field Models (I) We want to describe N non-relativistic quantum particles (cold atoms) interacting with a bosonic quantum field (lasers or magnetic fields). Hamiltonian Space of states: H = L 2 (R Nd ) Γ sym (H): L 2 (R dn ) describes the particle degrees of freedom; H is the field one-excitation space (H = l 2 (Z d ), L 2 (R d ) or L 2 (R 3 ) 2 ). The full Hamiltonian is N H = K(x j ) + U(x 1,..., x N ) dγ(ω), j=1 K is a one-particle operator containing the kinetic energy of the particles and the particle-field coupling; U(x 1,..., x N ) describes the particle interaction and/or the trapping; dγ(ω) is the field energy. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

16 ❶ Introduction Particle-Field Models (I) We want to describe N non-relativistic quantum particles (cold atoms) interacting with a bosonic quantum field (lasers or magnetic fields). Hamiltonian Space of states: H = L 2 (R Nd ) Γ sym (H): L 2 (R dn ) describes the particle degrees of freedom; H is the field one-excitation space (H = l 2 (Z d ), L 2 (R d ) or L 2 (R 3 ) 2 ). The full Hamiltonian is N H = K(x j ) + U(x 1,..., x N ) dγ(ω), j=1 K is a one-particle operator containing the kinetic energy of the particles and the particle-field coupling; U(x 1,..., x N ) describes the particle interaction and/or the trapping; dγ(ω) is the field energy. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

17 ❶ Introduction Particle-Field Models (II) H = L 2 (R Nd ) Γ sym (H), H = K(x j ) + U(x 1,..., x N ) + dγ(ω) One-Particle Operator The one-particle operator K(x) contains the particle kinetic energy and the particle-field interaction. It can have the following form: ❶ & ❷ Nelson-like models (linear coupling) K(x) = + A(x) ❸ Pauli-Fierz model K(x) = ( i + A(x)) 2 A(x) = a (λ) + a(λ) with λ H (resp. A(x)) is a scalar (resp. vectorial) interaction between the field and the particles. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

18 ❶ Introduction Particle-Field Models (II) H = L 2 (R Nd ) Γ sym (H), H = K(x j ) + U(x 1,..., x N ) + dγ(ω) One-Particle Operator The one-particle operator K(x) contains the particle kinetic energy and the particle-field interaction. It can have the following form: ❶ & ❷ Nelson-like models (linear coupling) K(x) = + A(x) ❸ Pauli-Fierz model K(x) = ( i + A(x)) 2 A(x) = a (λ) + a(λ) with λ H (resp. A(x)) is a scalar (resp. vectorial) interaction between the field and the particles. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

19 ❶ Introduction Particle-Field Models (III) H = L 2 (R Nd ) Γ sym (H), H = K(x j ) + U(x 1,..., x N ) + dγ(ω) Particle-field interaction The particle-field interaction is linear (λ H is the form factor) ❶ A(x) = n Z d (λ n a ne ikn x + λ n a n e ikn x) ; ❷ A(x) = dk R d ❸ A(x) = 2 γ=1 ( ) λ(k)a (k)e ik x + h.c. ; R 3 dk ( ) λ(k; x)a γ(k) + h.c. e γ (k), where e γ (k) are the polarization vectors, i.e., {ˆk, e 1, e 2 } is a basis of R 3. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

20 ❶ Introduction Particle-Field Models (III) H = L 2 (R Nd ) Γ sym (H), H = K(x j ) + U(x 1,..., x N ) + dγ(ω) Particle-field interaction The particle-field interaction is linear (λ H is the form factor) ❶ A(x) = n Z d (λ n a ne ikn x + λ n a n e ikn x) ; ❷ A(x) = dk R d ❸ A(x) = 2 γ=1 ( ) λ(k)a (k)e ik x + h.c. ; R 3 dk ( ) λ(k; x)a γ(k) + h.c. e γ (k), where e γ (k) are the polarization vectors, i.e., {ˆk, e 1, e 2 } is a basis of R 3. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

21 ❶ Introduction Particle-Field Models (IV) H = L 2 (R Nd ) Γ sym (H), H = K(x j ) + U(x 1,..., x N ) + dγ(ω) Field free energy The field energy with dispersion ω is ❶ dγ(ω) = n Z d k n a na n (discrete modes); ❷ dγ(ω) = dk ω(k)a (k)a(k) (Nelson model or polaron*); R d 2 ❸ dγ(ω) = dk ω(k)a γ(k)a γ (k) (Pauli-Fierz model). R 3 γ=1 * polaron: special Nelson model with λ(k) = k 1 and ω(k) = 1. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

22 ❶ Introduction Particle-Field Models (IV) H = L 2 (R Nd ) Γ sym (H), H = K(x j ) + U(x 1,..., x N ) + dγ(ω) Field free energy The field energy with dispersion ω is ❶ dγ(ω) = n Z d k n a na n (discrete modes); ❷ dγ(ω) = dk ω(k)a (k)a(k) (Nelson model or polaron*); R d 2 ❸ dγ(ω) = dk ω(k)a γ(k)a γ (k) (Pauli-Fierz model). R 3 γ=1 * polaron: special Nelson model with λ(k) = k 1 and ω(k) = 1. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

23 ❶ Introduction Quasi-classical Limit In physics effective models the fields are replaced by macroscopic non-relativistic potentials. This approximation is justified when the field is very intense and the number of excitations is very large, i.e., N = a a Hence 1 = [a, a ] N and the field quantum nature can be discarded. The field becomes classical, while the particles are still quantum. Quasi-classical limit The quasi-classical limit can be concretely realized (after rescaling) by setting [Ginibre, Nironi, Velo 06] [ a(k), a (k ) ] = εδ(k k ), with ε 0. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

24 ❶ Introduction Quasi-classical Limit In physics effective models the fields are replaced by macroscopic non-relativistic potentials. This approximation is justified when the field is very intense and the number of excitations is very large, i.e., N = a a Hence 1 = [a, a ] N and the field quantum nature can be discarded. The field becomes classical, while the particles are still quantum. Quasi-classical limit The quasi-classical limit can be concretely realized (after rescaling) by setting [Ginibre, Nironi, Velo 06] [ a(k), a (k ) ] = εδ(k k ), with ε 0. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

25 ❶ Introduction Quasi-classical Limit In physics effective models the fields are replaced by macroscopic non-relativistic potentials. This approximation is justified when the field is very intense and the number of excitations is very large, i.e., N = a a Hence 1 = [a, a ] N and the field quantum nature can be discarded. The field becomes classical, while the particles are still quantum. Quasi-classical limit The quasi-classical limit can be concretely realized (after rescaling) by setting [Ginibre, Nironi, Velo 06] [ a(k), a (k ) ] = εδ(k k ), with ε 0. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

26 ❶ Introduction Effective Models To identify the effective model for the particles, we trace out the field degrees of freedom from the full space H = L 2 (R Nd ) Γ sym (H) This is obtained by taking the partial trace of quantum expectations and so obtaining an effective Schrödinger operator H ε on L 2 (R Nd ). Partial trace For any normalized state of the field Ψ ε Γ sym (H), we define H ε := Ψ ε H Ψ ε Γsym(H) c ε, c ε := Ψ ε dγ(ω) Ψ ε dγ(ω), as the operator of the quadratic form Q ε [ψ] = ψ Ψ ε H ψ Ψ ε H. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

27 ❶ Introduction Effective Models To identify the effective model for the particles, we trace out the field degrees of freedom from the full space H = L 2 (R Nd ) Γ sym (H) This is obtained by taking the partial trace of quantum expectations and so obtaining an effective Schrödinger operator H ε on L 2 (R Nd ). Partial trace For any normalized state of the field Ψ ε Γ sym (H), we define H ε := Ψ ε H Ψ ε Γsym(H) c ε, c ε := Ψ ε dγ(ω) Ψ ε dγ(ω), as the operator of the quadratic form Q ε [ψ] = ψ Ψ ε H ψ Ψ ε H. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

28 ❷ Main Results Assumptions H = L 2 (R Nd ) Γ sym(h), H = ( i j + A(x j)) 2 + U(x 1,..., x N ) + dγ(ω) H is self-adjoint on D( + U + ) D(dΓ(ω)). Assumptions Set for concreteness H = C 2 L 2 (R 3, dν): ω(k) is strictly positive ν-a.e. and ω 1 is a densely defined operator on L 2 (R 3, dν); The form factor λ in 2 A(x) = dk R 3 γ=1 ( ) λ(k; x)a γ(k) + h.c. e γ (k) belongs to L (R 3 ; D ω ), D ω = D(ω + ω 1 ), and (Coulomb gauge) λ(k; x) e γ (k) = 0, γ = 1, 2. The potential is such that U L 2 (R dn ; R + ) + K (R dn ). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

29 ❷ Main Results Assumptions H = L 2 (R Nd ) Γ sym(h), H = ( i j + A(x j)) 2 + U(x 1,..., x N ) + dγ(ω) H is self-adjoint on D( + U + ) D(dΓ(ω)). Assumptions Set for concreteness H = C 2 L 2 (R 3, dν): ω(k) is strictly positive ν-a.e. and ω 1 is a densely defined operator on L 2 (R 3, dν); The form factor λ in 2 A(x) = dk R 3 γ=1 ( ) λ(k; x)a γ(k) + h.c. e γ (k) belongs to L (R 3 ; D ω ), D ω = D(ω + ω 1 ), and (Coulomb gauge) λ(k; x) e γ (k) = 0, γ = 1, 2. The potential is such that U L 2 (R dn ; R + ) + K (R dn ). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

30 ❷ Main Results Assumptions H = L 2 (R Nd ) Γ sym(h), H = ( i j + A(x j)) 2 + U(x 1,..., x N ) + dγ(ω) H is self-adjoint on D( + U + ) D(dΓ(ω)). Assumptions Set for concreteness H = C 2 L 2 (R 3, dν): ω(k) is strictly positive ν-a.e. and ω 1 is a densely defined operator on L 2 (R 3, dν); The form factor λ in 2 A(x) = dk R 3 γ=1 ( ) λ(k; x)a γ(k) + h.c. e γ (k) belongs to L (R 3 ; D ω ), D ω = D(ω + ω 1 ), and (Coulomb gauge) λ(k; x) e γ (k) = 0, γ = 1, 2. The potential is such that U L 2 (R dn ; R + ) + K (R dn ). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

31 ❷ Main Results Assumptions H = L 2 (R Nd ) Γ sym(h), H = ( i j + A(x j)) 2 + U(x 1,..., x N ) + dγ(ω) H is self-adjoint on D( + U + ) D(dΓ(ω)). Assumptions Set for concreteness H = C 2 L 2 (R 3, dν): ω(k) is strictly positive ν-a.e. and ω 1 is a densely defined operator on L 2 (R 3, dν); The form factor λ in 2 A(x) = dk R 3 γ=1 ( ) λ(k; x)a γ(k) + h.c. e γ (k) belongs to L (R 3 ; D ω ), D ω = D(ω + ω 1 ), and (Coulomb gauge) λ(k; x) e γ (k) = 0, γ = 1, 2. The potential is such that U L 2 (R dn ; R + ) + K (R dn ). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

32 ❷ Main Results Wigner Measures (I) A key ingredient of our proofs is the semiclassical analysis of infinite dimensional systems [Ammari, Nier 08, Falconi 16]. The main result is that, under suitable conditions on the field state Ψ ε, Ψ ε Ψ ε Γsym(H) dµ, µ M (H). ε 0 H Theorem (Classical limit [Ammari, Nier 08]) Assume δ 1 such that dγ(1 + ω) δ Ψ ε C δ < +. Then a subsequence {ε k } k N and µ M (H), s.t. for any g H dγ(ω) Ψε dµ(z) ωz 2 k H, (a + a)(g) Ψ ε k H 2R dµ(z) z g H H M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

33 ❷ Main Results Wigner Measures (I) A key ingredient of our proofs is the semiclassical analysis of infinite dimensional systems [Ammari, Nier 08, Falconi 16]. The main result is that, under suitable conditions on the field state Ψ ε, Ψ ε Ψ ε Γsym(H) dµ, µ M (H). ε 0 H Theorem (Classical limit [Ammari, Nier 08]) Assume δ 1 such that dγ(1 + ω) δ Ψ ε C δ < +. Then a subsequence {ε k } k N and µ M (H), s.t. for any g H dγ(ω) Ψε dµ(z) ωz 2 k H, (a + a)(g) Ψ ε k H 2R dµ(z) z g H H M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

34 ❷ Main Results Wigner Measures (I) A key ingredient of our proofs is the semiclassical analysis of infinite dimensional systems [Ammari, Nier 08, Falconi 16]. The main result is that, under suitable conditions on the field state Ψ ε, Ψ ε Ψ ε Γsym(H) dµ, µ M (H). ε 0 H Theorem (Classical limit [Ammari, Nier 08]) Assume δ 1 such that dγ(1 + ω) δ Ψ ε C δ < +. Then a subsequence {ε k } k N and µ M (H), s.t. for any g H dγ(ω) Ψε dµ(z) ωz 2 k H, (a + a)(g) Ψ ε k H 2R dµ(z) z g H H M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

35 ❷ Main Results Wigner Measures (II) The expectation value of any observable F (a, a) converges to its classical counterpart, i.e., if Ψ ε µ in the sense above, then Ψε F (a, a) Ψ ε H ε 0 H dµ(z) f (z, z), where f is the semiclassical symbol associated to F. The limit measure µ might depend on the subsequence, but once it is chosen all the results hold true on the same subsequence. In general, if only dγ(ω) δ Ψε C δ, then the Wigner measure µ is concentrated [Falconi 16] on H ω := L 2 (R 3, ωdν) which is different from H = L 2 (R 3, dν). Moreover, for any 0 δ δ dγ(ω) δ dµ(z) ωz 2 δ Ψ ε ε 0 H < +, H ω M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

36 ❷ Main Results Wigner Measures (II) The expectation value of any observable F (a, a) converges to its classical counterpart, i.e., if Ψ ε µ in the sense above, then Ψε F (a, a) Ψ ε H ε 0 H dµ(z) f (z, z), where f is the semiclassical symbol associated to F. The limit measure µ might depend on the subsequence, but once it is chosen all the results hold true on the same subsequence. In general, if only dγ(ω) δ Ψε C δ, then the Wigner measure µ is concentrated [Falconi 16] on H ω := L 2 (R 3, ωdν) which is different from H = L 2 (R 3, dν). Moreover, for any 0 δ δ dγ(ω) δ dµ(z) ωz 2 δ Ψ ε ε 0 H < +, H ω M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

37 ❷ Main Results Convergence of the Hamiltonian Theorem (H eff [MC, Falconi, Olivieri 17]) In case ❸ (Pauli-Fierz model), up to subsequences, H ε res H eff (µ) = ε 0 N j=1 { ( i j A µ (x j ) ) 2 + Wµ (x j )} + U(x 1,...), where µ is the classical measure associated to Ψ ε. Furthermore, H eff is self-adjoint on D( + U + ) and (setting ê C 4 := (e 1, e 2 ) and recalling that H = C 2 L 2 (R 3, dν)) A µ (x) = 2R dµ(z) z λ( ; x)ê( ) H H ω [ W µ(x) = 4 dµ(z) ( ( ) ) 2 ] 2 R z λ( ; x)ê( ) H R dµ(z) z λ( ; x)ê( ) H H ω H ω M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

38 ❷ Main Results Convergence of the Hamiltonian Theorem (H eff [MC, Falconi, Olivieri 17]) In case ❸ (Pauli-Fierz model), up to subsequences, H ε res H eff (µ) = ε 0 N j=1 { ( i j A µ (x j ) ) 2 + Wµ (x j )} + U(x 1,...), where µ is the classical measure associated to Ψ ε. Furthermore, H eff is self-adjoint on D( + U + ) and (setting ê C 4 := (e 1, e 2 ) and recalling that H = C 2 L 2 (R 3, dν)) A µ (x) = 2R dµ(z) z λ( ; x)ê( ) H H ω [ W µ(x) = 4 dµ(z) ( ( ) ) 2 ] 2 R z λ( ; x)ê( ) H R dµ(z) z λ( ; x)ê( ) H H ω H ω M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

39 ❷ Main Results Convergence: Remarks The electric potential W µ is a sort of variance w.r.t. to µ and vanishes when µ is a Dirac delta (coherent states). For any µ M (H ω ), W µ 0. The form factors λ j, j = 1,..., N might be different = each particle would feel a different magnetic and electric potential. The particles can have internal degrees of freedom (e.g., spin). We can allow for an additional spin-field coupling B(x) σ j, where the field B is second-quantized and σ j is the spin of the j-th particle = as ε 0, the additional terms B µ (x j ) σ j are generated. Under the assumptions made, A µ (x) and W µ (x) are both continuous and vanishing at, i.e., both rougher and stronger fields are excluded... M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

40 ❷ Main Results Convergence: Remarks The electric potential W µ is a sort of variance w.r.t. to µ and vanishes when µ is a Dirac delta (coherent states). For any µ M (H ω ), W µ 0. The form factors λ j, j = 1,..., N might be different = each particle would feel a different magnetic and electric potential. The particles can have internal degrees of freedom (e.g., spin). We can allow for an additional spin-field coupling B(x) σ j, where the field B is second-quantized and σ j is the spin of the j-th particle = as ε 0, the additional terms B µ (x j ) σ j are generated. Under the assumptions made, A µ (x) and W µ (x) are both continuous and vanishing at, i.e., both rougher and stronger fields are excluded... M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

41 ❷ Main Results Convergence: Remarks The electric potential W µ is a sort of variance w.r.t. to µ and vanishes when µ is a Dirac delta (coherent states). For any µ M (H ω ), W µ 0. The form factors λ j, j = 1,..., N might be different = each particle would feel a different magnetic and electric potential. The particles can have internal degrees of freedom (e.g., spin). We can allow for an additional spin-field coupling B(x) σ j, where the field B is second-quantized and σ j is the spin of the j-th particle = as ε 0, the additional terms B µ (x j ) σ j are generated. Under the assumptions made, A µ (x) and W µ (x) are both continuous and vanishing at, i.e., both rougher and stronger fields are excluded... M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

42 ❷ Main Results Convergence: Remarks The electric potential W µ is a sort of variance w.r.t. to µ and vanishes when µ is a Dirac delta (coherent states). For any µ M (H ω ), W µ 0. The form factors λ j, j = 1,..., N might be different = each particle would feel a different magnetic and electric potential. The particles can have internal degrees of freedom (e.g., spin). We can allow for an additional spin-field coupling B(x) σ j, where the field B is second-quantized and σ j is the spin of the j-th particle = as ε 0, the additional terms B µ (x j ) σ j are generated. Under the assumptions made, A µ (x) and W µ (x) are both continuous and vanishing at, i.e., both rougher and stronger fields are excluded... M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

43 ❷ Main Results Convergence: Other Models ❶ (discrete modes) one can reproduce optical lattice potentials, i.e., µ M (l 2 (Z 3 )) and λ l 2 (Z 3 ), such that one obtains the effective electric potential V µ (x) = n Z d (a n cos(k n x) + b n sin(k n x)). ❷ (Nelson model) for any µ M (L 2 (R d )) and λ L 2 (R d ), one obtains the effective electric potential V µ (x) = (2π) 3/2 dµ(z) F 1( zλ ) (x), L 2 (R 3 ) which is continuous and vanishing at. ❷ (polaron) for any µ M (L 2 (R 3 )), the effective potential might not be bounded but is Kato infinitesimally small w.r.t.. ❷ No trapping potentials can be generated in this setting... M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

44 ❷ Main Results Convergence: Other Models ❶ (discrete modes) one can reproduce optical lattice potentials, i.e., µ M (l 2 (Z 3 )) and λ l 2 (Z 3 ), such that one obtains the effective electric potential V µ (x) = n Z d (a n cos(k n x) + b n sin(k n x)). ❷ (Nelson model) for any µ M (L 2 (R d )) and λ L 2 (R d ), one obtains the effective electric potential V µ (x) = (2π) 3/2 dµ(z) F 1( zλ ) (x), L 2 (R 3 ) which is continuous and vanishing at. ❷ (polaron) for any µ M (L 2 (R 3 )), the effective potential might not be bounded but is Kato infinitesimally small w.r.t.. ❷ No trapping potentials can be generated in this setting... M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

45 ❷ Main Results Convergence: Other Models ❶ (discrete modes) one can reproduce optical lattice potentials, i.e., µ M (l 2 (Z 3 )) and λ l 2 (Z 3 ), such that one obtains the effective electric potential V µ (x) = n Z d (a n cos(k n x) + b n sin(k n x)). ❷ (Nelson model) for any µ M (L 2 (R d )) and λ L 2 (R d ), one obtains the effective electric potential V µ (x) = (2π) 3/2 dµ(z) F 1( zλ ) (x), L 2 (R 3 ) which is continuous and vanishing at. ❷ (polaron) for any µ M (L 2 (R 3 )), the effective potential might not be bounded but is Kato infinitesimally small w.r.t.. ❷ No trapping potentials can be generated in this setting... M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

46 ❷ Main Results Convergence: Generic Potentials Let H A,V = N j=1( i j A(x j ) ) 2 + V (x1,..., x N ) be a magnetic Schrödinger operator with V L 1 loc ( R 3N ; R +) + K ( R 3N ) ; A L 2 loc ( R 3 ). Theorem (H A,V [MC, Falconi, Olivieri 17]) Under the above assumptions, a self-adjoint microscopic Pauli-Fierz Hamiltonian H A,V on L 2 (R 3N ) Γ s (H) and a family (Ψ ε ) ε (0,1) of quantum states of the field such that, up to subsequences, Ψε H A,V dγ(ω) s res Ψε Γ s(h) H A,V. ε 0 There is a freedom in the choice of the form factor λ and states Ψ ε. In order to recover unbounded potentials the field energy typically diverges, i.e., c ε + as ε 0. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

47 ❷ Main Results Convergence: Generic Potentials Let H A,V = N j=1( i j A(x j ) ) 2 + V (x1,..., x N ) be a magnetic Schrödinger operator with V L 1 loc ( R 3N ; R +) + K ( R 3N ) ; A L 2 loc ( R 3 ). Theorem (H A,V [MC, Falconi, Olivieri 17]) Under the above assumptions, a self-adjoint microscopic Pauli-Fierz Hamiltonian H A,V on L 2 (R 3N ) Γ s (H) and a family (Ψ ε ) ε (0,1) of quantum states of the field such that, up to subsequences, Ψε H A,V dγ(ω) s res Ψε Γ s(h) H A,V. ε 0 There is a freedom in the choice of the form factor λ and states Ψ ε. In order to recover unbounded potentials the field energy typically diverges, i.e., c ε + as ε 0. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

48 ❷ Main Results Convergence: Generic Potentials Let H A,V = N j=1( i j A(x j ) ) 2 + V (x1,..., x N ) be a magnetic Schrödinger operator with V L 1 loc ( R 3N ; R +) + K ( R 3N ) ; A L 2 loc ( R 3 ). Theorem (H A,V [MC, Falconi, Olivieri 17]) Under the above assumptions, a self-adjoint microscopic Pauli-Fierz Hamiltonian H A,V on L 2 (R 3N ) Γ s (H) and a family (Ψ ε ) ε (0,1) of quantum states of the field such that, up to subsequences, Ψε H A,V dγ(ω) s res Ψε Γ s(h) H A,V. ε 0 There is a freedom in the choice of the form factor λ and states Ψ ε. In order to recover unbounded potentials the field energy typically diverges, i.e., c ε + as ε 0. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

49 ❷ Main Results A Key Example Example Consider a single particle moving in a uniform magnetic field, i.e., A = 1 2 x = 1 2 ( y, x, 0) with V 0. Suppose the particle form factor is given by λ(x; k) = 1 e 1 2 k 2 e ik x ; k For any ξ ε (k) C0 (R3 ), with ξ ε 1, pick the coherent states { i 2 Ψ ε = exp dk k e ( 1 2 k 2 a 8ε γ(k) ( e γ ) ) } ξ ε (k) + h.c. Ω, R 3 γ=1 where = ( y, x, 0) and Ω is the vacuum. Then Ψ ε H A,V dγ(ω) Ψ ε Γ s(h) s res H A,0 = ( i 1 ε 0 2 x ) 2. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

50 ❷ Main Results A Key Example Example Consider a single particle moving in a uniform magnetic field, i.e., A = 1 2 x = 1 2 ( y, x, 0) with V 0. Suppose the particle form factor is given by λ(x; k) = 1 e 1 2 k 2 e ik x ; k For any ξ ε (k) C0 (R3 ), with ξ ε 1, pick the coherent states { i 2 Ψ ε = exp dk k e ( 1 2 k 2 a 8ε γ(k) ( e γ ) ) } ξ ε (k) + h.c. Ω, R 3 γ=1 where = ( y, x, 0) and Ω is the vacuum. Then Ψ ε H A,V dγ(ω) Ψ ε Γ s(h) s res H A,0 = ( i 1 ε 0 2 x ) 2. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

51 ❷ Main Results A Key Example Example Consider a single particle moving in a uniform magnetic field, i.e., A = 1 2 x = 1 2 ( y, x, 0) with V 0. Suppose the particle form factor is given by λ(x; k) = 1 e 1 2 k 2 e ik x ; k For any ξ ε (k) C0 (R3 ), with ξ ε 1, pick the coherent states { i 2 Ψ ε = exp dk k e ( 1 2 k 2 a 8ε γ(k) ( e γ ) ) } ξ ε (k) + h.c. Ω, R 3 γ=1 where = ( y, x, 0) and Ω is the vacuum. Then Ψ ε H A,V dγ(ω) Ψ ε Γ s(h) s res H A,0 = ( i 1 ε 0 2 x ) 2. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

52 ❷ Main Results A Key Example Example Consider a single particle moving in a uniform magnetic field, i.e., A = 1 2 x = 1 2 ( y, x, 0) with V 0. Suppose the particle form factor is given by λ(x; k) = 1 e 1 2 k 2 e ik x ; k For any ξ ε (k) C0 (R3 ), with ξ ε 1, pick the coherent states { i 2 Ψ ε = exp dk k e ( 1 2 k 2 a 8ε γ(k) ( e γ ) ) } ξ ε (k) + h.c. Ω, R 3 γ=1 where = ( y, x, 0) and Ω is the vacuum. Then Ψ ε H A,V dγ(ω) Ψ ε Γ s(h) s res H A,0 = ( i 1 ε 0 2 x ) 2. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

53 ❷ Main Results Quasi-classical Limit of the Energy H eff (µ) = { ( i j A µ ) 2 + Wµ B µ σ j } + U Once one knows that the effective Schrödinger operator converges in the quasi-classical limit ε 0, it is natural to ask the following question: does the ground state energy of the whole system inf σ(h) converge as ε 0? and to what? Suppose that inf σ(h) does converge, then its limit should be inf [inf σ (H eff(µ)) + c(µ)], c(µ) = dµ(z) ωz 2 µ M (H H ω) H ω where lim c ε = c(µ), but is this quantity bounded from below in first ε 0 place? M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

54 ❷ Main Results Quasi-classical Limit of the Energy H eff (µ) = { ( i j A µ ) 2 + Wµ B µ σ j } + U Once one knows that the effective Schrödinger operator converges in the quasi-classical limit ε 0, it is natural to ask the following question: does the ground state energy of the whole system inf σ(h) converge as ε 0? and to what? Suppose that inf σ(h) does converge, then its limit should be inf [inf σ (H eff(µ)) + c(µ)], c(µ) = dµ(z) ωz 2 µ M (H H ω) H ω where lim c ε = c(µ), but is this quantity bounded from below in first ε 0 place? M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

55 ❷ Main Results Ground State Energy One can consider a smaller minimization domain for the measure µ M (L 2 (R d )), i.e., states with finite field energy: { M 2,ω := M 2 (H ω ) = µ M (H ω ) dµ(z) ωz } 2 H < + H ω The bottom of the spectrum inf σ(h) is finite uniformly in ε. In view of the Theorem below, one also obtains the boundedness of the r.h.s. (which was not obvious a priori). Theorem (G.s. energy [MC, Falconi, Olivieri 17]) Under the same assumptions as before, ( ) lim inf σ(h) = inf inf σ (H eff (µ)) + c(µ). ε 0 µ M 2,ω M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

56 ❷ Main Results Ground State Energy One can consider a smaller minimization domain for the measure µ M (L 2 (R d )), i.e., states with finite field energy: { M 2,ω := M 2 (H ω ) = µ M (H ω ) dµ(z) ωz } 2 H < + H ω The bottom of the spectrum inf σ(h) is finite uniformly in ε. In view of the Theorem below, one also obtains the boundedness of the r.h.s. (which was not obvious a priori). Theorem (G.s. energy [MC, Falconi, Olivieri 17]) Under the same assumptions as before, ( ) lim inf σ(h) = inf inf σ (H eff (µ)) + c(µ). ε 0 µ M 2,ω M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

57 ❷ Main Results Ground State Energy One can consider a smaller minimization domain for the measure µ M (L 2 (R d )), i.e., states with finite field energy: { M 2,ω := M 2 (H ω ) = µ M (H ω ) dµ(z) ωz } 2 H < + H ω The bottom of the spectrum inf σ(h) is finite uniformly in ε. In view of the Theorem below, one also obtains the boundedness of the r.h.s. (which was not obvious a priori). Theorem (G.s. energy [MC, Falconi, Olivieri 17]) Under the same assumptions as before, ( ) lim inf σ(h) = inf inf σ (H eff (µ)) + c(µ). ε 0 µ M 2,ω M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

58 ❷ Main Results Ground State Energy: Remarks The role of c(µ) is to guarantee boundedness from below. The convergence of the ground state is much more difficult to obtain, since in first place it is not clear whether a measure µ minimizing inf σ(h eff (µ)) + c(µ) in M 2,ω actually exists. The result applies also to the Nelson model and to the polaron. The Nelson massless model (as the massless Pauli-Fierz model) is a little more complicate because ω is not bounded below by c 1 = the limit measure µ is just a cylindrical measure on H, but a true measure on H ω. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

59 ❷ Main Results Ground State Energy: Remarks The role of c(µ) is to guarantee boundedness from below. The convergence of the ground state is much more difficult to obtain, since in first place it is not clear whether a measure µ minimizing inf σ(h eff (µ)) + c(µ) in M 2,ω actually exists. The result applies also to the Nelson model and to the polaron. The Nelson massless model (as the massless Pauli-Fierz model) is a little more complicate because ω is not bounded below by c 1 = the limit measure µ is just a cylindrical measure on H, but a true measure on H ω. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

60 ❷ Main Results Ground State Energy: Remarks The role of c(µ) is to guarantee boundedness from below. The convergence of the ground state is much more difficult to obtain, since in first place it is not clear whether a measure µ minimizing inf σ(h eff (µ)) + c(µ) in M 2,ω actually exists. The result applies also to the Nelson model and to the polaron. The Nelson massless model (as the massless Pauli-Fierz model) is a little more complicate because ω is not bounded below by c 1 = the limit measure µ is just a cylindrical measure on H, but a true measure on H ω. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

61 ❸ Perspectives Dynamics ψ Ξ ε e iht Ψ ε (t) Tr ψε (t). ψ Ξ ε Tr ψ ψ, ψε (t) ε 0 ε 0 ψ(t). Key questions 1 Does exist a particle Hamiltonian H eff such that ψ e ih eff t ψ(t), and, if yes, what is H eff? 2 What happens to the field? It becomes classical but what is its classical dynamics? The answers to both questions depend on time and energy scaling in ε. Similar questions investigated for the strong coupling of Fröhlich polaron when Ξ ε is a coherent state [Lieb, Thomas 97; Frank, Schlein 14; Frank, Gang 17; Griesemer 17]. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

62 ❸ Perspectives Dynamics ψ Ξ ε e iht Ψ ε (t) Tr ψε (t). ψ Ξ ε Tr ψ ψ, ψε (t) ε 0 ε 0 ψ(t). Key questions 1 Does exist a particle Hamiltonian H eff such that ψ e ih eff t ψ(t), and, if yes, what is H eff? 2 What happens to the field? It becomes classical but what is its classical dynamics? The answers to both questions depend on time and energy scaling in ε. Similar questions investigated for the strong coupling of Fröhlich polaron when Ξ ε is a coherent state [Lieb, Thomas 97; Frank, Schlein 14; Frank, Gang 17; Griesemer 17]. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

63 ❸ Perspectives Dynamics ψ Ξ ε e iht Ψ ε (t) Tr ψε (t). ψ Ξ ε Tr ψ ψ, ψε (t) ε 0 ε 0 ψ(t). Key questions 1 Does exist a particle Hamiltonian H eff such that ψ e ih eff t ψ(t), and, if yes, what is H eff? 2 What happens to the field? It becomes classical but what is its classical dynamics? The answers to both questions depend on time and energy scaling in ε. Similar questions investigated for the strong coupling of Fröhlich polaron when Ξ ε is a coherent state [Lieb, Thomas 97; Frank, Schlein 14; Frank, Gang 17; Griesemer 17]. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

64 ❸ Perspectives Dynamics (in progress) 1 Frozen field [CFO2]: The state of the field Ξ ε converges to a measure µ; The effective evolution is generated by H eff (µ). 2 Ξ ε is a (suitable) coherent state but the field evolves freely [CCFO]: The effective model is a time-dependent point interaction, i.e., formally H eff (t) = + α(t)δ(x). 3 Generic initial state Ξ ε or non-product state Ψ ε, non-frozed field [CFO2]: The dynamics of the field is free to leading order; The effective dynamics of the particles is highly non-trivial and takes the form of a sort of transport equation; The effective dynamics is typically not a group but norm preserving. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

65 ❸ Perspectives Dynamics (in progress) 1 Frozen field [CFO2]: The state of the field Ξ ε converges to a measure µ; The effective evolution is generated by H eff (µ). 2 Ξ ε is a (suitable) coherent state but the field evolves freely [CCFO]: The effective model is a time-dependent point interaction, i.e., formally H eff (t) = + α(t)δ(x). 3 Generic initial state Ξ ε or non-product state Ψ ε, non-frozed field [CFO2]: The dynamics of the field is free to leading order; The effective dynamics of the particles is highly non-trivial and takes the form of a sort of transport equation; The effective dynamics is typically not a group but norm preserving. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

66 ❸ Perspectives Dynamics (in progress) 1 Frozen field [CFO2]: The state of the field Ξ ε converges to a measure µ; The effective evolution is generated by H eff (µ). 2 Ξ ε is a (suitable) coherent state but the field evolves freely [CCFO]: The effective model is a time-dependent point interaction, i.e., formally H eff (t) = + α(t)δ(x). 3 Generic initial state Ξ ε or non-product state Ψ ε, non-frozed field [CFO2]: The dynamics of the field is free to leading order; The effective dynamics of the particles is highly non-trivial and takes the form of a sort of transport equation; The effective dynamics is typically not a group but norm preserving. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

67 ❸ Perspectives GRAN SASSO QUANTUM GSSI FROM MANY PARTICLE SYSTEMS TO QUANTUM FLUIDS 28 NOVEMBER 1 DECEMBER, 2018 LECTURERS Rémi Carles* (CNRS & Université de Rennes 1) Nicolas Rougerie (CNRS & Université Grenoble-Alpes) Juan J. L. Velázquez (University of Bonn) SPEAKERS Carlo Barenghi (Newcastle University) Iacopo Carusotto (INO-CNR BEC Center & University of Trento) Marco Falconi (University of Tübingen) Evelyne Miot-Desecures (CNRS & Université Grenoble-Alpes) Marcin Napiórkowski (University of Warsaw) Vedran Sohinger (University of Warwick) M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

68 ❸ Perspectives Traps? In all the three cases considered the effective potentials are not trapping, i.e., V µ (x) remains bounded as x. Can one generate trapping potentials in the quasi-classical limit? If one drops the assumptions on the expectation value of dγ(1) and dγ(ω), the semiclassical scheme still applies [Falconi 16] but the classical measure µ is in general only a cylindrical measure. Example Let f ε L 2 (R d ) be s.t. f ε ε 0 f 0 D (R d ) \ L 2 (R d ) in distributional sense, then Ξ(f ε ) ε 0 δ f0 = δ(z f 0 ) M (D ), δ f0 (L 2 (R d )) = 0. Hence δ f0 / M (L 2 (R d )) and in fact δ f0 M cyl (L 2 (R d )). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

69 ❸ Perspectives Traps? In all the three cases considered the effective potentials are not trapping, i.e., V µ (x) remains bounded as x. Can one generate trapping potentials in the quasi-classical limit? If one drops the assumptions on the expectation value of dγ(1) and dγ(ω), the semiclassical scheme still applies [Falconi 16] but the classical measure µ is in general only a cylindrical measure. Example Let f ε L 2 (R d ) be s.t. f ε ε 0 f 0 D (R d ) \ L 2 (R d ) in distributional sense, then Ξ(f ε ) ε 0 δ f0 = δ(z f 0 ) M (D ), δ f0 (L 2 (R d )) = 0. Hence δ f0 / M (L 2 (R d )) and in fact δ f0 M cyl (L 2 (R d )). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

70 ❸ Perspectives Traps? In all the three cases considered the effective potentials are not trapping, i.e., V µ (x) remains bounded as x. Can one generate trapping potentials in the quasi-classical limit? If one drops the assumptions on the expectation value of dγ(1) and dγ(ω), the semiclassical scheme still applies [Falconi 16] but the classical measure µ is in general only a cylindrical measure. Example Let f ε L 2 (R d ) be s.t. f ε ε 0 f 0 D (R d ) \ L 2 (R d ) in distributional sense, then Ξ(f ε ) ε 0 δ f0 = δ(z f 0 ) M (D ), δ f0 (L 2 (R d )) = 0. Hence δ f0 / M (L 2 (R d )) and in fact δ f0 M cyl (L 2 (R d )). M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

71 ❸ Perspectives Trapping Potentials Theorem (Traps [MC, Falconi 17]) In case ❷A assume that λ admits a multiplicative inverse and λ, ωλ L 2 (R d ). Then for any V L 2 loc (Rd ; R + ) there exists f V,ε such that Ξ(f V,ε ) H Ξ(f V,ε ) s res H eff = ε 0 N ( j + V (x j )) + U(x 1,..., x N ), j=1 and H eff is essentially self-adjoint on C 0 (Rd ). The typical example V (x) = x 2 is included. One can pick f V,ε (k) = 1 2(2π) d/2 λ (k) F (ϕ ε V ) (k); A price to pay to get a trapping potential is that the field energy must be very large and c ε ε 0 +. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

72 ❸ Perspectives Trapping Potentials Theorem (Traps [MC, Falconi 17]) In case ❷A assume that λ admits a multiplicative inverse and λ, ωλ L 2 (R d ). Then for any V L 2 loc (Rd ; R + ) there exists f V,ε such that Ξ(f V,ε ) H Ξ(f V,ε ) s res H eff = ε 0 N ( j + V (x j )) + U(x 1,..., x N ), j=1 and H eff is essentially self-adjoint on C 0 (Rd ). The typical example V (x) = x 2 is included. One can pick f V,ε (k) = 1 2(2π) d/2 λ (k) F (ϕ ε V ) (k); A price to pay to get a trapping potential is that the field energy must be very large and c ε ε 0 +. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22

73 ❸ Perspectives Trapping Potentials Theorem (Traps [MC, Falconi 17]) In case ❷A assume that λ admits a multiplicative inverse and λ, ωλ L 2 (R d ). Then for any V L 2 loc (Rd ; R + ) there exists f V,ε such that Ξ(f V,ε ) H Ξ(f V,ε ) s res H eff = ε 0 N ( j + V (x j )) + U(x 1,..., x N ), j=1 and H eff is essentially self-adjoint on C 0 (Rd ). The typical example V (x) = x 2 is included. One can pick f V,ε (k) = 1 2(2π) d/2 λ (k) F (ϕ ε V ) (k); A price to pay to get a trapping potential is that the field energy must be very large and c ε ε 0 +. M. Correggi (Roma 1) Quasi-Classical Limit CRM 14/9/ / 22