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)