Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit. Michele Correggi

Transcription

1 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 with M. Falconi (Zurich) M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

2 Outline 1 Introduction: Physical motivations: cold atoms in magneto-optical traps or optical lattices; Quantum particles + field models (Nelson model & polaron): quasi-classical limit. 2 Main results [CF]: Derivation of effective potentials in the quasi-classical limit; Ground state energy convergence. 3 Perspectives: Pauli-Fierz model: derivation of magnetic Schrödinger operators [CFO]. Main References [CF] M.C., M. Falconi, Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit, preprint arxiv: [math-ph] (2017); [CFO] M.C., M. Falconi, M. Olivieri, in preparation. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

3 Outline 1 Introduction: Physical motivations: cold atoms in magneto-optical traps or optical lattices; Quantum particles + field models (Nelson model & polaron): quasi-classical limit. 2 Main results [CF]: Derivation of effective potentials in the quasi-classical limit; Ground state energy convergence. 3 Perspectives: Pauli-Fierz model: derivation of magnetic Schrödinger operators [CFO]. Main References [CF] M.C., M. Falconi, Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit, preprint arxiv: [math-ph] (2017); [CFO] M.C., M. Falconi, M. Olivieri, in preparation. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

4 Outline 1 Introduction: Physical motivations: cold atoms in magneto-optical traps or optical lattices; Quantum particles + field models (Nelson model & polaron): quasi-classical limit. 2 Main results [CF]: Derivation of effective potentials in the quasi-classical limit; Ground state energy convergence. 3 Perspectives: Pauli-Fierz model: derivation of magnetic Schrödinger operators [CFO]. Main References [CF] M.C., M. Falconi, Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit, preprint arxiv: [math-ph] (2017); [CFO] M.C., M. Falconi, M. Olivieri, in preparation. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

5 ❶ Introduction Physical Motivations 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 LAGA 05/07/ / 16

6 ❶ Introduction Physical Motivations 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 LAGA 05/07/ / 16

7 ❶ Introduction Physical Motivations 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 LAGA 05/07/ / 16

8 ❶ Introduction Physical Motivations Optical lattices Another typical setting for cold atom experiments is give by optical lattices; Optical lattice 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 LAGA 05/07/ / 16

9 ❶ Introduction Physical Motivations Optical lattices Another typical setting for cold atom experiments is give by optical lattices; Optical lattice 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 LAGA 05/07/ / 16

10 ❶ Introduction Physical Motivations Optical lattices Another typical setting for cold atom experiments is give by optical lattices; Optical lattice 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 LAGA 05/07/ / 16

11 ❶ Introduction Particle-Field Models Model We want to describe N non-relativistic quantum particles (cold atoms) interacting with a bosonic quantum field (lasers or magnetic fields). Space of states: H = L 2 (R Nd ) Γ sym (h): L 2 (R dn ) describes the particle degrees of freedom; h is the one-excitation space for the field (e.g., h = l 2 (Z d ) or L 2 (R d )). The full Hamiltonian is N H = ( j + A(x j )) + U(x 1,..., x N ) + dγ(ω) j=1 k n a na n dγ(ω) = n Z d dk ω(k)a (k)a(k) R d (discrete modes); (Nelson model or polaron). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

12 ❶ Introduction Particle-Field Models Model We want to describe N non-relativistic quantum particles (cold atoms) interacting with a bosonic quantum field (lasers or magnetic fields). Space of states: H = L 2 (R Nd ) Γ sym (h): L 2 (R dn ) describes the particle degrees of freedom; h is the one-excitation space for the field (e.g., h = l 2 (Z d ) or L 2 (R d )). The full Hamiltonian is N H = ( j + A(x j )) + U(x 1,..., x N ) + dγ(ω) j=1 k n a na n dγ(ω) = n Z d dk ω(k)a (k)a(k) R d (discrete modes); (Nelson model or polaron). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

13 ❶ Introduction Particle-Field Models Model We want to describe N non-relativistic quantum particles (cold atoms) interacting with a bosonic quantum field (lasers or magnetic fields). Space of states: H = L 2 (R Nd ) Γ sym (h): L 2 (R dn ) describes the particle degrees of freedom; h is the one-excitation space for the field (e.g., h = l 2 (Z d ) or L 2 (R d )). The full Hamiltonian is N H = ( j + A(x j )) + U(x 1,..., x N ) + dγ(ω) j=1 k n a na n dγ(ω) = n Z d dk ω(k)a (k)a(k) R d (discrete modes); (Nelson model or polaron). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

14 ❶ Introduction Interactions H = ( j + A(x j )) + U(x 1,..., x N ) + dγ(ω) Particle interaction The potential U(x 1,..., x N ) takes into account: the interaction between the particles (e.g., Coulomb potential); some external trapping potential not due to the field. Particle-field interaction The particle-field interaction A(x) is linear as in the Nelson model, i.e., ( λ n a ne ikn x + λ n a n e ikn x) (discrete modes); A(x) = n Z d ( ) dk λ(k)a (k)e ik x + h.c. R d (Nelson/polaron). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

15 ❶ Introduction Interactions H = ( j + A(x j )) + U(x 1,..., x N ) + dγ(ω) Particle interaction The potential U(x 1,..., x N ) takes into account: the interaction between the particles (e.g., Coulomb potential); some external trapping potential not due to the field. Particle-field interaction The particle-field interaction A(x) is linear as in the Nelson model, i.e., ( λ n a ne ikn x + λ n a n e ikn x) (discrete modes); A(x) = n Z d ( ) dk λ(k)a (k)e ik x + h.c. R d (Nelson/polaron). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

16 ❶ Introduction Quasi-classical Limit In physics effective models the fields are replaced by macroscopic non-relativistic potentials. This approximation is justified if the number of field excitations is very large, i.e., N = a a, which implies that 1 = [a, a] N and therefore the quantum nature of the fields 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. The quasi-classical limit is not a classical limit 0, but rather an effective regime, e.g., the strong coupling regime for the polaron; The quantum nature of the particle is preserved! M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

17 ❶ Introduction Quasi-classical Limit In physics effective models the fields are replaced by macroscopic non-relativistic potentials. This approximation is justified if the number of field excitations is very large, i.e., N = a a, which implies that 1 = [a, a] N and therefore the quantum nature of the fields 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. The quasi-classical limit is not a classical limit 0, but rather an effective regime, e.g., the strong coupling regime for the polaron; The quantum nature of the particle is preserved! M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

18 ❶ Introduction Quasi-classical Limit In physics effective models the fields are replaced by macroscopic non-relativistic potentials. This approximation is justified if the number of field excitations is very large, i.e., N = a a, which implies that 1 = [a, a] N and therefore the quantum nature of the fields 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. The quasi-classical limit is not a classical limit 0, but rather an effective regime, e.g., the strong coupling regime for the polaron; The quantum nature of the particle is preserved! M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

19 ❶ Introduction Quasi-classical Limit In physics effective models the fields are replaced by macroscopic non-relativistic potentials. This approximation is justified if the number of field excitations is very large, i.e., N = a a, which implies that 1 = [a, a] N and therefore the quantum nature of the fields 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. The quasi-classical limit is not a classical limit 0, but rather an effective regime, e.g., the strong coupling regime for the polaron; The quantum nature of the particle is preserved! M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

20 ❶ Introduction Effective Models In order to identify the particle effective model we have to trace out the field degrees of freedom. 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, we define H ε := Ψ ε H Ψ ε Γsym(h) c ε, c ε := Ψ ε dγ(ω) Ψ ε Γsym(h), as the operator associated to the quadratic form ψ Ψ ε H ψ Ψ ε H, i.e., setting V ε,ψε (x) := Ψ ε A(x) Ψ ε Γsym(h), H ε = N ( j + V ε,ψε (x j )) + U(x 1,..., x N ). j=1 M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

21 ❶ Introduction Effective Models In order to identify the particle effective model we have to trace out the field degrees of freedom. 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, we define H ε := Ψ ε H Ψ ε Γsym(h) c ε, c ε := Ψ ε dγ(ω) Ψ ε Γsym(h), as the operator associated to the quadratic form ψ Ψ ε H ψ Ψ ε H, i.e., setting V ε,ψε (x) := Ψ ε A(x) Ψ ε Γsym(h), H ε = N ( j + V ε,ψε (x j )) + U(x 1,..., x N ). j=1 M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

22 ❶ Introduction Assumptions The main questions are thus: does H ε converge as ε 0? in which sense? and to what? H is self-adjoint on D( + U) D(dΓ(ω)). Assumptions H = ( j + A(x j )) + U(x 1,..., x N ) +dγ(ω) }{{} L 2 (R d ;R + )+K (R dn ) 1 (discrete modes) ω n = k n, A(x) = ( λ n a ne ikn x + h.c. ) with {λ n } n Z d l 2 (Z d ) (UV cut-off); ( ) 2 A(x) = dk λ(k)a (k)e ik x + h.c. with A B (Nelson) ω(k) = k 2 + m 2, m 0, and λ L 2 (R d ) (UV cut-off); (polaron) ω(k) = 1 and λ(k) = k 1 d 2. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

23 ❶ Introduction Assumptions The main questions are thus: does H ε converge as ε 0? in which sense? and to what? H is self-adjoint on D( + U) D(dΓ(ω)). Assumptions H = ( j + A(x j )) + U(x 1,..., x N ) +dγ(ω) }{{} L 2 (R d ;R + )+K (R dn ) 1 (discrete modes) ω n = k n, A(x) = ( λ n a ne ikn x + h.c. ) with {λ n } n Z d l 2 (Z d ) (UV cut-off); ( ) 2 A(x) = dk λ(k)a (k)e ik x + h.c. with A B (Nelson) ω(k) = k 2 + m 2, m 0, and λ L 2 (R d ) (UV cut-off); (polaron) ω(k) = 1 and λ(k) = k 1 d 2. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

24 ❶ Introduction Assumptions The main questions are thus: does H ε converge as ε 0? in which sense? and to what? H is self-adjoint on D( + U) D(dΓ(ω)). Assumptions H = ( j + A(x j )) + U(x 1,..., x N ) +dγ(ω) }{{} L 2 (R d ;R + )+K (R dn ) 1 (discrete modes) ω n = k n, A(x) = ( λ n a ne ikn x + h.c. ) with {λ n } n Z d l 2 (Z d ) (UV cut-off); ( ) 2 A(x) = dk λ(k)a (k)e ik x + h.c. with A B (Nelson) ω(k) = k 2 + m 2, m 0, and λ L 2 (R d ) (UV cut-off); (polaron) ω(k) = 1 and λ(k) = k 1 d 2. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

25 ❶ Introduction Assumptions The main questions are thus: does H ε converge as ε 0? in which sense? and to what? H is self-adjoint on D( + U) D(dΓ(ω)). Assumptions H = ( j + A(x j )) + U(x 1,..., x N ) +dγ(ω) }{{} L 2 (R d ;R + )+K (R dn ) 1 (discrete modes) ω n = k n, A(x) = ( λ n a ne ikn x + h.c. ) with {λ n } n Z d l 2 (Z d ) (UV cut-off); ( ) 2 A(x) = dk λ(k)a (k)e ik x + h.c. with A B (Nelson) ω(k) = k 2 + m 2, m 0, and λ L 2 (R d ) (UV cut-off); (polaron) ω(k) = 1 and λ(k) = k 1 d 2. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

26 ❶ Introduction Semiclassical Scheme 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 dµ, µ M (h). ε 0 h The limit measure µ might depend on the subsequence and the convergence is meant in the sense that any observable on the right has to be replaced by the classical counterpart. Theorem (Classical limit [Ammari, Nier 08]) If dγ(1) δ Ψ ε C, for some δ 1/2, and dγ(ω) Ψε C, then a subsequence {ε k } k N and µ M (h) s.t. for any g h dγ(ω) ε dµ(z) ωz 2 k h, (a + a)(g) 2R dµ(z) z g ε h k h h M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

27 ❶ Introduction Semiclassical Scheme 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 dµ, µ M (h). ε 0 h The limit measure µ might depend on the subsequence and the convergence is meant in the sense that any observable on the right has to be replaced by the classical counterpart. Theorem (Classical limit [Ammari, Nier 08]) If dγ(1) δ Ψ ε C, for some δ 1/2, and dγ(ω) Ψε C, then a subsequence {ε k } k N and µ M (h) s.t. for any g h dγ(ω) ε dµ(z) ωz 2 k h, (a + a)(g) 2R dµ(z) z g ε h k h h M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

28 ❶ Introduction Semiclassical Scheme 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 dµ, µ M (h). ε 0 h The limit measure µ might depend on the subsequence and the convergence is meant in the sense that any observable on the right has to be replaced by the classical counterpart. Theorem (Classical limit [Ammari, Nier 08]) If dγ(1) δ Ψ ε C, for some δ 1/2, and dγ(ω) Ψε C, then a subsequence {ε k } k N and µ M (h) s.t. for any g h dγ(ω) ε dµ(z) ωz 2 k h, (a + a)(g) 2R dµ(z) z g ε h k h h M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

29 ❷ Main Results Convergence of the Hamiltonian Theorem (Quasi-classical limit [MC, Falconi 17]) Under assumptions ❶ (resp. ❷A or ❷B) on the field state Ψ ε, H ε res H eff (µ) = ε 0 N ( j + V µ (x j )) + U(x 1,..., x N ), j=1 dµ(z) {λ n e ikn x} z ❶ l 2 (Z d ) n Zd l 2 (Z d ) V µ (x) = 2R (2π) d/2 dµ(z) F 1( zλ ) (x) ❷A L 2 (R d ) ( ) (2π) d/2 dµ(z) F 1 k 1 d 2 z (x) ❷B L 2 (R d ) where µ is the classical measure associated to Ψ ε and H eff is self-adjoint on the same domain of self-adjointness of H ε. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

30 ❷ Main Results Convergence of the Hamiltonian Theorem (Quasi-classical limit [MC, Falconi 17]) Under assumptions ❶ (resp. ❷A or ❷B) on the field state Ψ ε, H ε res H eff (µ) = ε 0 N ( j + V µ (x j )) + U(x 1,..., x N ), j=1 dµ(z) {λ n e ikn x} z ❶ l 2 (Z d ) n Zd l 2 (Z d ) V µ (x) = 2R (2π) d/2 dµ(z) F 1( zλ ) (x) ❷A L 2 (R d ) ( ) (2π) d/2 dµ(z) F 1 k 1 d 2 z (x) ❷B L 2 (R d ) where µ is the classical measure associated to Ψ ε and H eff is self-adjoint on the same domain of self-adjointness of H ε. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

31 ❷ Main Results Effective Potentials (❶ discrete modes): one can reproduce optical lattice potentials, i.e., µ M (l 2 (Z d )) and λ l 2 (Z d ) such that V µ (x) = n Z d (a n cos(k n x) + b n sin(k n x)). (❷A Nelson model): for any µ M (L 2 (R d )) and λ L 2 (R d ), the potential is continuous and vanishes at. (❷B polaron): for any µ M (L 2 (R d )), the potential might not be bounded but still it is Kato infinitesimally small w.r.t.. Example Let W L 1 (R d ) be real s. t. λ 1 Ŵ L 2 (R d ), then the( potential W (x) ) can be generated by a coherent state of the field, i.e., Ξ Ŵ, Ξ(f) = e a (f/ε) a(f/ε) Ω, 1 2(2π) d/2 λ Ξ(f) ε 0 δ(z f). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

32 ❷ Main Results Effective Potentials (❶ discrete modes): one can reproduce optical lattice potentials, i.e., µ M (l 2 (Z d )) and λ l 2 (Z d ) such that V µ (x) = n Z d (a n cos(k n x) + b n sin(k n x)). (❷A Nelson model): for any µ M (L 2 (R d )) and λ L 2 (R d ), the potential is continuous and vanishes at. (❷B polaron): for any µ M (L 2 (R d )), the potential might not be bounded but still it is Kato infinitesimally small w.r.t.. Example Let W L 1 (R d ) be real s. t. λ 1 Ŵ L 2 (R d ), then the( potential W (x) ) can be generated by a coherent state of the field, i.e., Ξ Ŵ, Ξ(f) = e a (f/ε) a(f/ε) Ω, 1 2(2π) d/2 λ Ξ(f) ε 0 δ(z f). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

33 ❷ Main Results Effective Potentials (❶ discrete modes): one can reproduce optical lattice potentials, i.e., µ M (l 2 (Z d )) and λ l 2 (Z d ) such that V µ (x) = n Z d (a n cos(k n x) + b n sin(k n x)). (❷A Nelson model): for any µ M (L 2 (R d )) and λ L 2 (R d ), the potential is continuous and vanishes at. (❷B polaron): for any µ M (L 2 (R d )), the potential might not be bounded but still it is Kato infinitesimally small w.r.t.. Example Let W L 1 (R d ) be real s. t. λ 1 Ŵ L 2 (R d ), then the( potential W (x) ) can be generated by a coherent state of the field, i.e., Ξ Ŵ, Ξ(f) = e a (f/ε) a(f/ε) Ω, 1 2(2π) d/2 λ Ξ(f) ε 0 δ(z f). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

34 ❷ Main Results Effective Potentials (❶ discrete modes): one can reproduce optical lattice potentials, i.e., µ M (l 2 (Z d )) and λ l 2 (Z d ) such that V µ (x) = n Z d (a n cos(k n x) + b n sin(k n x)). (❷A Nelson model): for any µ M (L 2 (R d )) and λ L 2 (R d ), the potential is continuous and vanishes at. (❷B polaron): for any µ M (L 2 (R d )), the potential might not be bounded but still it is Kato infinitesimally small w.r.t.. Example Let W L 1 (R d ) be real s. t. λ 1 Ŵ L 2 (R d ), then the( potential W (x) ) can be generated by a coherent state of the field, i.e., Ξ Ŵ, Ξ(f) = e a (f/ε) a(f/ε) Ω, 1 2(2π) d/2 λ Ξ(f) ε 0 δ(z f). M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

35 ❷ Main Results 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 LAGA 05/07/ / 16

36 ❷ Main Results 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 LAGA 05/07/ / 16

37 ❷ Main Results 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 LAGA 05/07/ / 16

38 ❷ Main Results 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 W L 2 loc (Rd ; R + ) there exists f W,ε such that Ξ(f W,ε ) H Ξ(f W,ε ) s res H eff = ε 0 N ( j + W (x j ))+U(x 1,..., x N ), j=1 and H eff is essentially self-adjoint on C 0 (Rd ). The typical example W (x) = x 2 is included. One can pick f W,ε (k) = 1 2(2π) d/2 λ (k) F (ϕ ε W ) (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 LAGA 05/07/ / 16

39 ❷ Main Results 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 W L 2 loc (Rd ; R + ) there exists f W,ε such that Ξ(f W,ε ) H Ξ(f W,ε ) s res H eff = ε 0 N ( j + W (x j ))+U(x 1,..., x N ), j=1 and H eff is essentially self-adjoint on C 0 (Rd ). The typical example W (x) = x 2 is included. One can pick f W,ε (k) = 1 2(2π) d/2 λ (k) F (ϕ ε W ) (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 LAGA 05/07/ / 16

40 ❷ Main Results 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 W L 2 loc (Rd ; R + ) there exists f W,ε such that Ξ(f W,ε ) H Ξ(f W,ε ) s res H eff = ε 0 N ( j + W (x j ))+U(x 1,..., x N ), j=1 and H eff is essentially self-adjoint on C 0 (Rd ). The typical example W (x) = x 2 is included. One can pick f W,ε (k) = 1 2(2π) d/2 λ (k) F (ϕ ε W ) (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 LAGA 05/07/ / 16

41 ❷ Main Results Limit of the Energy 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) R d L 2 (R d ) with lim ε 0 c ε = c(µ), but is this quantity bounded from below? In cases ❷A and ❷B our assumptions guarantee that H is bounded from below uniformly in ε. For this to be true, the field energy c ε has to be retained and in fact it is crucial to guarantee the boundedness from below. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

42 ❷ Main Results Limit of the Energy 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) R d L 2 (R d ) with lim ε 0 c ε = c(µ), but is this quantity bounded from below? In cases ❷A and ❷B our assumptions guarantee that H is bounded from below uniformly in ε. For this to be true, the field energy c ε has to be retained and in fact it is crucial to guarantee the boundedness from below. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

43 ❷ Main Results Limit of the Energy 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) R d L 2 (R d ) with lim ε 0 c ε = c(µ), but is this quantity bounded from below? In cases ❷A and ❷B our assumptions guarantee that H is bounded from below uniformly in ε. For this to be true, the field energy c ε has to be retained and in fact it is crucial to guarantee the boundedness from below. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

44 ❷ Main Results Ground State Energy We consider the cases ❷A (massive Nelson model) and ❷B (polaron). The minimization domain for the meaure µ M (L 2 (R d )) is given by states with finite field energy, i.e., { M ω := µ µ ( L 2 ω(r d ) ) = 1, µ } L 2 ω (R d ) is Borel, c(µ) < L 2 ω := {f L 2 (R d ) } ωf L 2 (R d ) <. Theorem (Ground state energy [MC, Falconi 17]) Under the assumptions ❷A (with ω > c > 0) or ❷B, lim inf σ(h) = inf [inf σ(h eff (µ)) + c(µ)]. ε 0 µ M ω M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

45 ❷ Main Results Ground State Energy We consider the cases ❷A (massive Nelson model) and ❷B (polaron). The minimization domain for the meaure µ M (L 2 (R d )) is given by states with finite field energy, i.e., { M ω := µ µ ( L 2 ω(r d ) ) = 1, µ } L 2 ω (R d ) is Borel, c(µ) < L 2 ω := {f L 2 (R d ) } ωf L 2 (R d ) <. Theorem (Ground state energy [MC, Falconi 17]) Under the assumptions ❷A (with ω > c > 0) or ❷B, lim inf σ(h) = inf [inf σ(h eff (µ)) + c(µ)]. ε 0 µ M ω M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

46 ❷ Main Results Ground State Energy We consider the cases ❷A (massive Nelson model) and ❷B (polaron). The minimization domain for the meaure µ M (L 2 (R d )) is given by states with finite field energy, i.e., { M ω := µ µ ( L 2 ω(r d ) ) = 1, µ } L 2 ω (R d ) is Borel, c(µ) < L 2 ω := {f L 2 (R d ) } ωf L 2 (R d ) <. Theorem (Ground state energy [MC, Falconi 17]) Under the assumptions ❷A (with ω > c > 0) or ❷B, lim inf σ(h) = inf [inf σ(h eff (µ)) + c(µ)]. ε 0 µ M ω M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

47 ❷ Main Results Ground State: Remarks Since inf σ(h) is uniformly bounded from below in ε, one deduces that inf µ Mω [inf σ(h eff (µ)) + c(µ)] is a bounded from below quantity as well: inf σ(h eff (µ)) is also bounded from below but c(µ) is essential for the boundedness of the whole expression. The result can be extended to the massless Nelson model but, since in this case ω is not bounded below by c 1, then the classical measure is in general a cylindrical measure on L 2 (R d ) and thus one has to use a different approach [Falconi 17]. The convergence of the ground state is much more difficult to obtain, since in first place it is not clear whether a measure minimizing inf σ(h eff (µ)) + c(µ) in M ω actually exists. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

48 ❷ Main Results Ground State: Remarks Since inf σ(h) is uniformly bounded from below in ε, one deduces that inf µ Mω [inf σ(h eff (µ)) + c(µ)] is a bounded from below quantity as well: inf σ(h eff (µ)) is also bounded from below but c(µ) is essential for the boundedness of the whole expression. The result can be extended to the massless Nelson model but, since in this case ω is not bounded below by c 1, then the classical measure is in general a cylindrical measure on L 2 (R d ) and thus one has to use a different approach [Falconi 17]. The convergence of the ground state is much more difficult to obtain, since in first place it is not clear whether a measure minimizing inf σ(h eff (µ)) + c(µ) in M ω actually exists. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

49 ❷ Main Results Ground State: Remarks Since inf σ(h) is uniformly bounded from below in ε, one deduces that inf µ Mω [inf σ(h eff (µ)) + c(µ)] is a bounded from below quantity as well: inf σ(h eff (µ)) is also bounded from below but c(µ) is essential for the boundedness of the whole expression. The result can be extended to the massless Nelson model but, since in this case ω is not bounded below by c 1, then the classical measure is in general a cylindrical measure on L 2 (R d ) and thus one has to use a different approach [Falconi 17]. The convergence of the ground state is much more difficult to obtain, since in first place it is not clear whether a measure minimizing inf σ(h eff (µ)) + c(µ) in M ω actually exists. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

50 ❸ Perspectives Work in Progress 1 Quasi-classical limit of the Pauli-Fierz model [MC, Falconi, Olivieri 17]: The minimal coupling generates in the limit an effective electric + magnetic potential; The convergence of the ground state energy requires the machinery of cylindrical measures. 2 Dynamics in the quasi-classical limit: Does the partial trace of e iht converge as ε 0 to a unitary group? for what initial states? and in case is the generator of such a group H eff (µ)? For the polaron such a question has already been studied in [Frank, Schlein 14, Frank, Gang 15, Griesemer 16] for initial states given by coherent states. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

51 ❸ Perspectives Work in Progress 1 Quasi-classical limit of the Pauli-Fierz model [MC, Falconi, Olivieri 17]: The minimal coupling generates in the limit an effective electric + magnetic potential; The convergence of the ground state energy requires the machinery of cylindrical measures. 2 Dynamics in the quasi-classical limit: Does the partial trace of e iht converge as ε 0 to a unitary group? for what initial states? and in case is the generator of such a group H eff (µ)? For the polaron such a question has already been studied in [Frank, Schlein 14, Frank, Gang 15, Griesemer 16] for initial states given by coherent states. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

52 ❸ Perspectives Work in Progress 1 Quasi-classical limit of the Pauli-Fierz model [MC, Falconi, Olivieri 17]: The minimal coupling generates in the limit an effective electric + magnetic potential; The convergence of the ground state energy requires the machinery of cylindrical measures. 2 Dynamics in the quasi-classical limit: Does the partial trace of e iht converge as ε 0 to a unitary group? for what initial states? and in case is the generator of such a group H eff (µ)? For the polaron such a question has already been studied in [Frank, Schlein 14, Frank, Gang 15, Griesemer 16] for initial states given by coherent states. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

53 ❸ Perspectives Work in Progress 1 Quasi-classical limit of the Pauli-Fierz model [MC, Falconi, Olivieri 17]: The minimal coupling generates in the limit an effective electric + magnetic potential; The convergence of the ground state energy requires the machinery of cylindrical measures. 2 Dynamics in the quasi-classical limit: Does the partial trace of e iht converge as ε 0 to a unitary group? for what initial states? and in case is the generator of such a group H eff (µ)? For the polaron such a question has already been studied in [Frank, Schlein 14, Frank, Gang 15, Griesemer 16] for initial states given by coherent states. M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16

54 ❸ Perspectives Thank you for the attention! School & Workshop Mathematical Challenges in Quantum Mechanics 2 (MCQM2) Rome, February 2018 courses by R. Frank, B. Schlein, S. Teufel talks by G. Berkolaiko, N. Datta, A. Giuliani, M. Griesemer, M. Zirnbauer support for young participants + contributed talks M. Correggi (Roma 1) Quasi-Classical Limit LAGA 05/07/ / 16