By Niels Benedikter, Marcello Porta, Benjamin Schlein

ISBN-10: 3319248960

ISBN-13: 9783319248967

ISBN-10: 3319248987

ISBN-13: 9783319248981

These notes examine the time evolution of quantum platforms, and particularly the rigorous derivation of powerful equations approximating the many-body Schrödinger dynamics in yes bodily fascinating regimes. the focal point is totally on the derivation of time-dependent potent theories (non-equilibrium query) approximating many-body quantum dynamics. The ebook is split into seven sections, the 1st of which in brief stories the most houses of many-body quantum platforms and their time evolution. part 2 introduces the mean-field regime for bosonic platforms and explains how the many-body dynamics should be approximated during this restrict utilizing the Hartree equation. part three offers a style, in line with using coherent states, for carefully proving the convergence in the direction of the Hartree dynamics, whereas the fluctuations round the Hartree equation are thought of in part four. part five specializes in a dialogue of a extra refined regime, during which the many-body evolution will be approximated through the nonlinear Gross-Pitaevskii equation. part 6 addresses fermionic platforms (characterized via antisymmetric wave functions); the following, the fermionic mean-field regime is of course associated with a semiclassical regime, and it really is confirmed that the evolution of approximate Slater determinants may be approximated utilizing the nonlinear Hartree-Fock equation. In remaining, part 7 reexamines an identical fermionic mean-field regime, yet with a spotlight on combined quasi-free preliminary facts approximating thermal states at confident temperature.

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6) with the linear operator S : L 2 (R3 ) ⊕ L 2 (R3 ) → L 2 (R3 ) ⊕ L 2 (R3 ) defined by S( f, g) = ( f, −g), and where ·, · denotes the inner product in L 2 (R3 ) ⊕ L 2 (R3 ). 6). In other words, it is a linear map satisfying the two conditions ΘJ = J Θ and Θ ∗ SΘ = S. 7) Θ= V U¯ with two linear operators U, V : L 2 (R3 ) → L 2 (R3 ) satisfying U ∗ U − V ∗ V = 1 and U ∗ V¯ = V ∗ U¯ (the operators U¯ , V¯ are obtained taking the complex conjugate of the kernels of U , V ). A Bogoliubov transformation Θ : L 2 (R3 ) ⊕ L 2 (R3 ) → L 2 (R3 ) ⊕ L 2 (R3 ) is called implementable if there exists a unitary map Ξ : F → F defined on the bosonic Fock space F over L 2 (R3 ) such that Ξ ∗ A( f, g)Ξ = A(Θ( f, g)) for every f, g ∈ L 2 (R3 ).

In other words dx f N2 (x1 − x2 )(L 1 χ N )(x)ξ N (x) = = dx f N2 (x1 − x2 )χ N (x)(L 1 ξ N )(x) dx f 2 (N (x1 − x2 ))∇x1 χ N (x)∇x1 ξ N (x) and analogously for L 2 . 15) we conclude that ψ N , H N2 ψ N ≥ N (N − 1) dx f 2 (N (x1 − x2 ))L 1 φ N (x)L 2 φ N (x) = N (N − 1) dx f 2 (N (x1 − x2 ))∇x1 φ N (x)∇x1 L 2 φ N (x) = N (N − 1) dx f 2 (N (x1 − x2 )) ∇x2 ∇x1 φ N (x) +N (N − 1) 2 dx f 2 (N (x1 − x2 )) ∇x1 φ N (x)[∇x1 , L 2 ]φ N (x) 5 The Gross-Pitaevskii Regime = N (N − 1) + N (N − 1) 45 dx f 2 (N (x1 − x2 )) ∇x2 ∇x1 φ N (x) 2 dx f 2 (N (x1 − x2 )) ∇x1 φ N (x) ×∇ N (∇ f )(N (x1 − x2 )) ∇x2 φ N (x).

Here F+ denotes the bosonic Fock space over the orthogonal complement of span{ϕ}. We can think of Uϕ ψ N ∈ F+ as describing the fluctuations around the condensate. We assume that the initial data ψ N ∈ L 2s (R3N ) exhibits complete condensation in ϕ ∈ L 2 (R3 ) in the sense that the fluctuations φ(0) := Uϕ ψ N ∈ F+ can be bounded uniformly in N in the sense that for some C > 0 we have φ(0), dΓ (1 − Δ)φ(0) ≤ C independent of N . We let ψ N evolve with the Hamiltonian N HN = −Δx j + j=1 1 N N V (xi − x j ) .

### Effective Evolution Equations from Quantum Dynamics by Niels Benedikter, Marcello Porta, Benjamin Schlein

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