Random Schrödinger Operators
Transcript of Random Schrödinger Operators
Random Schrodinger Operators
Francisco Hoecker-EscutiLaGa, Universite Paris 13
Technische Universitat Chemnitz
Maths Physics Young Researchers MeetingApril 11th-12th , 2013
Institut Henri Poincare, Paris, France
Solids have enormous number of particles (1023 ∼ ∞).
Given a particle in an initial state ψ0 ∈ H, a Hilbert space, thestate of the particle at time t is given by
ψ(·) : t 7→ ψ(t) ∈ H
satisfying
i
~∂
∂tψ = Hψ (1)
where
H = − p2
2m+ V .
We will choose the units so that
~2
2m= 1
There exists a one-parameter group of unitary operators
t 7→ e−itH
so thatϕ(t) = ϕ0e−itH .
Example of a periodic Schrodinger operator :
H = −4+ V on L2(Rd)
where for any x ∈ Rd and γ ∈ Zd
V (x + γ) = V (γ).
We could also write :
V (x) =∑γ∈Zd
ω · u(x − γ)
where u is a single site potential.
Spatially homogeneous solids.
Alloy type Schrodinger operators :
Hω := −4+ Vω
whereVω :=
∑γ
ωγu(x − γ)
and ωγ are random variables. This is known as the (continuous)Anderson Model.
Alloy type Schrodinger operators :
Hω := −4+ Vω
whereVω :=
∑γ
ωγu(x − γ)
and ωγ are random variables.
This is known as the (continuous)Anderson Model.
Alloy type Schrodinger operators :
Hω := −4+ Vω
whereVω :=
∑γ
ωγu(x − γ)
and ωγ are random variables. This is known as the (continuous)Anderson Model.
Amorphous Schrodinger operators :
I Random displacement :
Vω :=∑γ
u(x − γ − ωγ)
I Poisson model :
Vω :=
∫Rd
u(x − y)dµγ(y)
I Gaussian model : Vω is a Gaussian field
The Anderson model.
Discrete model (tight-binding or semi-classical approximation) :
Hω := −4+ Vω
defined on `2(Zd) where (Vωu)n = ωnun foru = (un)n∈Zd ∈ `2(Zd).
Measurable family of operators.
Let (Hω)ω∈Ω a family of bounded operators on H, a separableHilbert space.
Definition(Hω)ω∈Ω measurable if and only if ω 7→ 〈ψ,Hωϕ〉 measurable.
I The discrete Anderson model on `2(Zd) is measurable.
Let (Hω)ω∈Ω a family of self-adjoint operators.
Definition(Hω)ω∈Ω measurable if f (Hω) bounded for every Borelian boundedfunction f .
Proposition
I (Hω)ω∈Ω measurable iff (Hω + z)−1 for some z ∈ C .
I (Hω)ω∈Ω measurable iff e itHω measurable for all t ∈ R.
Application.
Proposition
If Vω is stochastic process measurable in x and ω such thatHω = H + Vω is essentially self-adjoint on C∞0 (Rd), then Hω ismeasurable.
All the models presented in this talk all measurable.
Ergodic operators.
DefinitionA measure-preserving group of transformations (τγ)γ∈Γ is said tobe ergodic if for X measurable
∀γ ∈ Γ, X τγ = X a.s.⇒ X = constant a.s.
DefinitionA self-adjoint and measurable family (Hω)ω∈Ω is ergodic if thereexists an ergodic group (τγ)γ∈Γ of automorphisms of Ω and afamily (Uγ)γ∈Γ of unitary operators on H such that
Hτγω = UγHωU∗γ .
Proposition
If (Hω)ω∈Ω then (f (Hω))ω∈Ω for any bounded measurable f .
Consequences
LemmaIf a (Πω)ω is a family of ergodic projectors then rank(Πω) is aconstant almost surely.
Theorem (Pastur)
If (Hω)ω∈Ω is ergodic then there exists a closed set Σ such thatΣ = σ(Hω) almost surely.
TheoremIf (Hω)ω∈Ω is ergodic then the discrete spectrum is constant.
Almost sure spectrum
Discrete model :
TheoremFor P-almost all ω we have σ(Hω) = [0, 4d ] + supp ω0.
Continuous model :
TheoremIf Vω ≥ 0 then Σ = [0,+∞) a.s.
Almost sure spectrum
Discrete model :
TheoremFor P-almost all ω we have σ(Hω) = [0, 4d ] + supp ω0.
Continuous model :
TheoremIf Vω ≥ 0 then Σ = [0,+∞) a.s.
Almost sure spectrum
Discrete model :
TheoremFor P-almost all ω we have σ(Hω) = [0, 4d ] + supp ω0.
Continuous model :
TheoremIf Vω ≥ 0 then Σ = [0,+∞) a.s.
Spectral types.
Lebesgue decomposition of measures :
I pure point measure
I absolutely continuous measure
I singular continuous measure
Spectral types. Note that, if A is a Borel set,
A 7→ 〈ϕ, χA(H)ϕ〉
is a well defined positive measure.
I Hpp := ϕ : 〈ϕ, µ(A)ϕ〉 is pure point.I Hac := ϕ : 〈ϕ, µ(A)ϕ〉 is absolutely continuous.I Hsc := ϕ : 〈ϕ, µ(A)ϕ〉 is singular continuous.
They are orthogonal and H = Hpp ⊕Hac ⊕Hsc . Furthermore, theoperator H maps each of these spaces into itself. We can thusrestrict H to each of these spaces. This gives a decomposition :
σ(H) = σpp(H) ∪ σac(H) ∪ σsc(H)
Spectral types. Note that, if A is a Borel set,
A 7→ 〈ϕ, χA(H)ϕ〉
is a well defined positive measure.
I Hpp := ϕ : 〈ϕ, µ(A)ϕ〉 is pure point.
I Hac := ϕ : 〈ϕ, µ(A)ϕ〉 is absolutely continuous.I Hsc := ϕ : 〈ϕ, µ(A)ϕ〉 is singular continuous.
They are orthogonal and H = Hpp ⊕Hac ⊕Hsc . Furthermore, theoperator H maps each of these spaces into itself. We can thusrestrict H to each of these spaces. This gives a decomposition :
σ(H) = σpp(H) ∪ σac(H) ∪ σsc(H)
Spectral types. Note that, if A is a Borel set,
A 7→ 〈ϕ, χA(H)ϕ〉
is a well defined positive measure.
I Hpp := ϕ : 〈ϕ, µ(A)ϕ〉 is pure point.I Hac := ϕ : 〈ϕ, µ(A)ϕ〉 is absolutely continuous.
I Hsc := ϕ : 〈ϕ, µ(A)ϕ〉 is singular continuous.They are orthogonal and H = Hpp ⊕Hac ⊕Hsc . Furthermore, theoperator H maps each of these spaces into itself. We can thusrestrict H to each of these spaces. This gives a decomposition :
σ(H) = σpp(H) ∪ σac(H) ∪ σsc(H)
Spectral types. Note that, if A is a Borel set,
A 7→ 〈ϕ, χA(H)ϕ〉
is a well defined positive measure.
I Hpp := ϕ : 〈ϕ, µ(A)ϕ〉 is pure point.I Hac := ϕ : 〈ϕ, µ(A)ϕ〉 is absolutely continuous.I Hsc := ϕ : 〈ϕ, µ(A)ϕ〉 is singular continuous.
They are orthogonal and H = Hpp ⊕Hac ⊕Hsc . Furthermore, theoperator H maps each of these spaces into itself. We can thusrestrict H to each of these spaces. This gives a decomposition :
σ(H) = σpp(H) ∪ σac(H) ∪ σsc(H)
Spectral types. Note that, if A is a Borel set,
A 7→ 〈ϕ, χA(H)ϕ〉
is a well defined positive measure.
I Hpp := ϕ : 〈ϕ, µ(A)ϕ〉 is pure point.I Hac := ϕ : 〈ϕ, µ(A)ϕ〉 is absolutely continuous.I Hsc := ϕ : 〈ϕ, µ(A)ϕ〉 is singular continuous.
They are orthogonal and H = Hpp ⊕Hac ⊕Hsc . Furthermore, theoperator H maps each of these spaces into itself. We can thusrestrict H to each of these spaces. This gives a decomposition :
σ(H) = σpp(H) ∪ σac(H) ∪ σsc(H)
Theorem (Kunz–Souillard, Kirsch–Martinelli)
If (Hω)ω∈Ω is ergodic then there exist closed sets Σpp,Σac ,Σsc ofR such that
Σpp = σpp(Hω),Σac = σac(Hω),Σsc = σsc(Hω)
almost surely.
Ruelle-Amrein-Georgescu-Enss theorem.
Pure point :
TheoremLet H be a self-adjoint operator on `2(Zd), take ψ ∈ Hpp and letΛL be a cube in Zd centered at the origin with sidelength 2L + 1.Then
limL→∞
supt≥0
∑x∈ΛL
∣∣∣e−itHψ(x)∣∣∣2 = ‖ψ‖2
and
limL→∞
supt≥0
∑x 6∈ΛL
∣∣∣e−itHψ(x)∣∣∣2 = 0.
Ruelle-Amrein-Georgescu-Enss theorem.
Absolutely continuous :
TheoremLet H be a self-adjoint operator on `2(Zd), take ψ ∈ Hac and letΛL be a cube in Zd centered at the origin with sidelength 2L + 1.Then
limL→∞
supt≥0
∑x∈ΛL
∣∣∣e−itHψ(x)∣∣∣2 = 0
and
limL→∞
supt≥0
∑x 6∈ΛL
∣∣∣e−itHψ(x)∣∣∣2 = ‖ψ‖2 .
Known results.
I In dimension 1, Σ = Σpp and Σac = Σsc = ∅.
I If the disorder is large, Σ = Σpp and Σac = Σsc = ∅.
I In dimension d ≥ 2, Σ ∩ (0, ε) = Σpp ∩ (0, ε) andΣac ∩ (0, ε) = Σsc ∩ (0, ε) = ∅.
Phase diagram
Consider the discrete Anderson model on `2(Zd)
Hω := −4+ λVω
with symmetric, zero mean random variables ωγ .
Anderson localization
I Exponential localisation : in the interval of energy I thespectrum of Hω, is pure point almost surely and theeigenfunctions are exponentially localised.
I Dynamical localisation : if we denote by PI (Hω) the spectralprojector of Hω on the interval I , then there exists a > 0 suchthat
E[
supt∈R
∣∣∣⟨δn, e itHωPI (Hω)δm⟩∣∣∣] ≤ 1
ae−γ(E)|m−n|.
I Absence of level repulsion : in the interval I , the localspacing of the eigenvalues of Hω exhibits Poisson statistics.
Photon localisation in photonic crystals
The integrated density of states.
Let Λ be a cube of sidelenth L on Rd or Zd and −4N,DL the
Laplacian with Neumann and Dirichlet boundary conditions. DefineHN,Dω,L := −4N,D
L + Vω.
TheoremThere exist NN,D positives, non-decreasing and right-continuoussuch that for any energy E for which NN,D we have
NN,D = limL→+∞
1
Ld#eigenvalues of HD,N
ω,L ≤ E
Density of states.
Theorem (Pastur–Shubin formula)
Let ϕ ∈ C∞0 (Rd). We have
I For the discrete Anderson model on `2(Zd)∫Rϕ(E )dN(E ) = E 〈δ0, ϕ(Hω)δ0〉 ;
I For the continuous Anderson model on L2(Rd)∫Rϕ(E )dN(E ) = E
(tr[χ[0,1]d , ϕ(Hω)χ[0,1]d
]).
Wegner estimate
Let Hω be the discrete or the continuous Anderson model. Let ussuppose that the single site potential u is positive and that therandom variables ωγ are regular (i.e. they admit a compactlysupported bounded density).
TheoremLet Λ be a cube Rd or Zd of sidelength L and K ⊂ R compact,there exists CK > 0 such that if J ⊂ K , then
E(tr[χJ(HD,N
ω,L )])≤ CK |J||Λ|
Lifschitz tails
Let H0 = −4 on Rd and Vω a random potential. We assume
Vω ≥ 0.
Let(H0 + Vω)Λ
the restriction to a cube Λ with Neumann boundary conditions.This restriction has a compact resolvent. This means that
I σ ((H0 + Vω)Λ) is a discrete subset of R with no finiteacumulation point.
Therefore the spectral counting function
n (E , (H0 + Vω)Λ) := tr[χ[0,E ](H0 + Vω)Λ
]is finite.
In absence of disorder, i.e. Vω = 0, we have
n (E , (H0 + Vω)Λ) = CdEd/2 (|Λ|+ o(|Λ|)) , E ≥ 0.
and thus
limΛRd
1
|Λ|n (E , (H0)Λ) =: N0(E ) = CdEd/2.
Lifschitz tails
Let us assume now that Vω ≥ 0 and (ωγ)γ are any non-trivialrandom variables.
TheoremLet E− = inf Σ. Then
limE→E−E≥E−
log | log N(E )|log(E − E−)
≤ −d/2
This can be proven in many cases.
Proof
Under very mild ergodicity assumptions on Vω, the limit
N(E ) := limΛRd
1
|Λ|n (E , (H0 + Vω)Λ) .
exists. It is independent of the choice of ω (outside some set ofmeasure zero) and equals
N(E ) := infΛ
1
|Λ|E [n (E , (H0 + Vω)Λ)] .
Proof
First step : Reduction to a probability estimation.
N(E ) = infΛ
1
|Λ|E [n (E , (H0 + Vω)Λ)]
≤ 1
|Λ|
∫n (E , (H0 + Vω)Λ)χΩ(E)(ω)dP(ω)
≤ CEd/2PΩ(E )
withΩ := ω : inf σ((H0 + Vω)Λ) ≤ E.
Second step : note that if λ ∈ [0, 1]
(H0 + Vω)Λ ≥ (H0 + λVω)Λ
as operators. This implies that
n (E , (H0 + Vω)Λ) ≤ n (E , (H0 + λVω)Λ) .
By taking λ small we can use perturbation theory.
Let us consider HΛ(ω, λ) = H0 + λVω defined on L2(Λ) withNeumann boundary conditions. The first eigenvalue behaves like
E1(HΛ(ω)) = E1(ω, λ) ∼ E1(H0) + λE ′1(ω, 0)
for small λ, withE ′1(ω, 0) = 〈Vωϕ0, ϕ0〉 .
Here, ϕ0 is the normalized ground state of H0. How large can wetake λ ?
LemmaThere exists a constant C such that for 0 ≤ λ ≤ Cl−2 we have
|E1(ω, λ)− λE ′1(ω, 0)| ≤ 1
Cl2λ2.
The constants are independent of ω.
Assume thatE1(ω) ≤ Cl−2.
Then we haveλE ′1(0) ≤ Cl2λ2 + Cl−2.
This suggests to take λ ∼ l−2. This gives
E ′1(0) = 〈Vωϕ0, ϕ0〉 ≤ Cl−2
In probabilistic terms :
P(E1(ω) ≤ Cl−2
)≤ P
(〈Vωϕ0, ϕ0〉 ≤ cl−2.
).
But
〈Vωϕ0, ϕ0〉 =
∫Rd
u(x)dx · 1
ld
∑|γ|≤l
ωγ
Third step : large deviation inequality.
Lemma (Chernoff’s inequality)
P
(∣∣∣∣∣ 1
N
N∑n=1
ωn − E(ω0)
∣∣∣∣∣ ≥ t
). e−ct
2N
Application :
P(E1(ω) ≤ Cl−2
)≤ e−c(E(ω0)−l−2)ld = e−c
′ld
End of the proof : from the first step,
N(E ) ≤ CEd/2PE1(ω) ≤ E.
Now choose l E−1/2 to get,
N(E ) ≤ Cl−d · PE1(ω) ≤ Cl−2
≤ l−d · e−c ′ld
≤ ce−c′ld = ce−c
′E−d/2
We state this result as
Theorem
lim supE0
logN(E )
E−d/2≤ −c
Merci Jimena !Merci a tous !
(applause)