6. Model System
6.2. Framework for Many-Electron Systems
6.2.3. Schrödinger Operator and Time Evolution
The interaction between the electrons enters the model via a two body potentialΦ:Zd →R. We make the following assumptions onΦ:
Assumption
(S) Φis symmetric, i.e.Φ(x)= Φ(−x) for allx∈Zd.
(C) Φhas finite support, i.e. there is anR≥0 such thatΦ(x)=0 for|x| ≥R.
The interaction term in the Schrödinger operator is constructed in the following way. Using the notation introduced as first construction in Appendix B.1.2, fork,lthe operatorΦ(XN,k−XN,l) models the interaction between thek-th and thel-th electron. Since any two electrons interact in this way and since we do not want to consider interactions of higher order than two-body interactions, the total term in theN-electron Schödinger operator implementing the electron- electron interaction is the multiplication operator
WN,−:= 1 2 N X k,l=1, k,l Φ(XN,k−XN,l) (6.71)
6.2. Framework for Many-Electron Systems 61
for allN≥2 andWN,−:=0 forN ∈ {0,1}. The interaction term is a well-defined multiplication operator onhN,−, i.e. WN,−(hN,−) ⊂ hN,−), for the fact that it is symmetric in the components of the position operator. For Λ ⊂ Zd and N ∈ N0 the random Schrödinger operator of the
N-electron system that is restricted toΛis the bounded, linear and self-adjoint operator
HΛ(E,ω,,µ)N,−(t) :=dΓN,− H( E,µ)
Λ,ω (t)+WN,−ΓN,−(χΛ), (6.72) where ω ∈ Ω and t ∈ R are arbitrary andχΛ is the multiplication operator induced by the characteristic function ofΛ.
Due to (Yos80, BGKS05) by the same methods as presented in 6.1.7 the random Schrödinger operator in (6.72) possesses a unitary propagator, i.e. for any Λ ⊂ Zd, ω ∈ Ω andN ∈ N0
there exists a mappingUΛ(E,ω,,µ)N,− :R2→U(hN,−), (t,r) 7→ U( E,µ)
Λ,ω,N,−(t,r) such that for arbitrary
t,r,q∈Rthe following relations are satisfied
U(ΛE,ω,,µ)N,−(t,t)=idN,−, UΛ(E,ω,,µ)N,−(t,r)UΛ(E,ω,,µ)N,−(r,q)=UΛ(E,ω,,µ)N,−(t,q), i∂tU( E,µ) Λ,ω,N,−(t,r)=H (E,µ) Λ,ω,N,−(t)U (E,µ) Λ,ω,N,−(t,r), −i∂rU(ΛE,ω,,µ)N,−(t,r)=UΛ(E,ω,,µ)N,−(t,r)HΛ(E,ω,,µ)N,−(r).
So far, we described systems of a fixed number of electronsN∈N0. We are interested mostly in the special caseΛ =Zdfor which a finite particle number corresponds to a vanishing mean electron density. However, the formalism we are trying to achieve should describe systems of finite electron density, so we transfer to Fock space. Onh−for anyΛ⊂ Rd,ω ∈Ωandt ∈R the energy of the many-electron system is described by the random Schrödinger operator
H(ΛE,ω,,µ)−(t) :=M
N∈N0
HΛ(E,ω,,µ)N,−(t). (6.73)
For the case of vanishing electric field, we just writeHΛ(µ,ω,) −. Because of the existence of unitary propagators for arbitrary finite particle number N ∈ N0, there also is a unitary propagator
UΛ(E,ω,,µ)− : R2 → U(h−), (t,r) 7→ U( E,µ)
Λ,ω,−(t,r) for the random Schrödinger operator on Fock space (6.73) given by
UΛ(E,ω,,µ)−(t,r)= M N∈N0
U(ΛE,ω,,µ)N,−(t,r). (6.74)
For the reason that the random Schrödinger operator in general is an unbounded operator, the propagator is only strongly differentiable, i.e. for anyt,r,q∈Randψ−∈D HΛ(µ,ω,) −
U(ΛE,ω,,µ)−(t,t)=id−, (6.75) U(ΛE,ω,,µ)−(t,r)UΛ(E,ω,,µ)−(r,q)=UΛ(E,ω,,µ)−(t,q), (6.76) i∂tUΛ(E,ω,,µ)−(t,r)ψ−=HΛ(E,ω,,µ)−(t)UΛ(E,ω,,µ)−(t,r)ψ−, (6.77) −i∂rU( E,µ) Λ,ω,−(t,r)ψ−=U (E,µ) Λ,ω,−(t,r)H (E,µ) Λ,ω,−(r)ψ−. (6.78) Moreover, for vanishing electric field the unitary propagator depends only on time differences. Accordingly, for anyt∈Rwe set
UΛ(µ,ω,) −(t) :=U(0Λ,ω,,µ)−(t,0)=e−itH (µ)
62 6. Model System
Next, we translate the above constructions to an algebraic setting. The Schrödinger operator defines a derivation onBc,−via
HΛ(E,ω,,µ)t,−(B−) :=iH( E,µ)
Λ,ω,−(t),B−
(6.80) for anyΛ⊂Zd,ω∈Ω,t∈RandB−∈Bc,−. For vanishing electric field this is independent of
t∈R. Then, we suppress these arguments such thatHΛ(E,ω,,µ)t,−=γt(,E−)◦ HΛ(µ,ω,) −◦ γ(t,E−)−1
for any
t∈R.
Similarly, for anyΛ ⊂ Zd,ω ∈ Ω, t,r ∈ Rand B− ∈ B− the unitary propagator defined in Equation (6.74) induces automorphismsτ(ΛE,ω,,µ)t,r,−:B−→B−via
τ(E,µ) Λ,ω,t,r,−(B−) :=U (E,µ) Λ,ω,−(t,r)B−U (E,µ) Λ,ω,−(r,t). (6.81) For fixed B− ∈ DHΛ(µ,ω,) −Equation (6.81) is differentiable with respect to r and because of Equations (6.77) and (6.78) one has
∂rτ(ΛE,ω,,µ)t,r,−(B−)=τΛ(E,ω,,µ)t,r,− HΛ(E,ω,,µ)r,−(B−). (6.82) For the special case of vanishing electric field, the unitary propagator defines a strongly con- tinuous one parameter group of automorphisms{τ(Λµ,ω,) t,−:t∈R}onB−via
τ(µ) Λ,ω,t,−(B−) :=U (µ) Λ,ω,−(t)B−U (µ) Λ,ω,−(t) ∗= e−itH (µ) Λ,ω,−B −eitH (µ) Λ,ω,− =e−tH (µ) Λ,ω,−(B −). (6.83) For fixedω ∈ ΩandB− ∈Bc,−the mappingR →B−, t 7→τ(Λµ,ω,) t,−(B−) is differentiable with respect to the variabletand one obtains the differential equation
∂tτ(Λµ,ω,) t,−(B−)=τΛ(µ,ω,) − HΛ(µ,ω,) −(B−). (6.84) For Λ = Zd we suppress the label Λ in the definitions above. Because in this case the Schrödinger operator satisfies a covariant transformation law, we get that for fixedt ∈ Rthe mappingHt(,E−,µ):Ω→Der(Bc,−), ω7→→ Hω,(Et,µ,−)is a covariant derivation, i.e.
ϕa,− H( E,µ)
ω,t,−(B−)=H( E,µ)
φa(ω),t,−(ϕa,−(B−)) (6.85) for all a ∈ Zd, t ∈ R, B− ∈ Bc,− and almost allω ∈ Ω. Moreover, for any fixedt,r ∈ Rthe mappingτ(t,Er,,µ−):Ω→Aut(B−), ω7→→τ(ω,Et,µ,r),−is a covariant automorphism, i.e. for alla∈Zd,
t,r∈R,B−∈Bc,−and almost allω∈Ωwe have
ϕa,− τ(ω,Et,µ,r,)−(B−)=τ(φE,µ)
a(ω),t,r,−(ϕa,−(B−)). (6.86)