PDF Properties of Spectrum for Anderson type random operator

(1)

Properties of Spectrum for Anderson type random operator

Anish Mallick

ICTS-TIFR Bengaluru

November 14, 2018

Properties of Spectrum for Anderson type random operator 1 / 25

(2)

1 Introduction

2 Multiplicity bounds of spectrum

3 Anderson type operators on Cayley like graphs

4 Local Eigenvalue Statistics and regularity of DOS

5 Future directions

6 References

Properties of Spectrum for Anderson type random operator 2 / 25

(3)

Anderson type operator

The model was introduced to study spin wave propagation over doped semi-conductors.

Anderson developed it to show that random medium causes the eigenfunctions to decay exponentially. This phenomenon is now known as Anderson localization.

The continuous version of the Anderson Hamiltonian is given by H^ω =−

d

X

i=1

∂²

∂x²_i + X

n∈Z^d

ω_nχ_n+[0,1)d

onL²(R^d).

There are basically three type of questions

Spectrum of the operator (including behavior of density of states).

Local structure of spectrum (behavior of gaps between eigenvalue).

Multiplicity of spectrum.

Properties of Spectrum for Anderson type random operator Introduction 3 / 25

(4)

Anderson type operator

Anderson tight binding model/Random Schr¨odinger operator

For ergodic case, the spectrum is almost surely constant and there are regularity results for density of states.

Recently there is some results on local spectral statistics in region of Anderson localization, which in turn provides answer for multiplicity.

Anderson type operator

There are many intermediate models like dimer/polymer models which falls between Anderson tight binding model and random Schr¨odinger operator.

For the ergodic case, many results involving density of state can be extended from theory of Anderson tight binding model to these cases.

The questions of spectral statistics and multiplicity are being studied.

Properties of Spectrum for Anderson type random operator Introduction 4 / 25

(5)

Multiplicity bounds of singular spectrum for certain Random operators

Properties of Spectrum for Anderson type random operator Multiplicity bounds of spectrum 5 / 25

(6)

Multiplicity bounds of spectrum

Family of operator

The family of random operator A^ω is of the form A^ω=A+X

n

ω_nC_n,

where Ais essentially self adjoint operator and {C_n}_n are finite rank non-negative operators. Here {ω_n}_nare independent real random variables with absolutely continuous distribution.

Example

On`²(Z)consider the operator ∆ +V^ω, where∆is Laplacian and (V^ωu)(n) =ωbⁿ

Ncu(n) ∀n∈Z.

This example can be generalized toZ^d case.

(7)

Multiplicity bounds of spectrum

Main results of [AM, D. R. Dolai, arXiv:1709.01774]¹

Under the assumption that A^ω is almost surely essentially self adjoint operator.

The multiplicity of singular spectrum is bounded above by maximum multiplicity of eigenvalues of √

Cn(A^ω−z)⁻¹√

Cn for almost all (ω, z).

As an expression, it is given by sup

n

ess-sup

z∈C\R

M ult^ω_n(z), where

M ult^ω_n(z) :=Maximum multiplicity of roots of polynomial det(p

Cn(A^ω−z)⁻¹p

Cn−xI)in x.

It can be shown that M ult^ω_n(z) is constant for almost all ω.

1Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian

(8)

Multiplicity bounds of spectrum

Main results of [AM, M. Krishna, arXiv:1803.06895]² WhenCn are finite rank projection satisfyingP

nCn=I and {ω_n}_n are independent real random variables following an absolutely

continuous distribution with full support.

Let J be an interval in the region where Simon-Wolff criteria is satisfied, such that the multiplicity of eigenvalues of A^ω in J is bounded below byK almost surely.

Then, the multiplicity of spectrum, in the region where Simon-Wolff criteria is satisfied, is bounded below byK.

Simon-Wolff criteria

Is the region where only pure-point spectrum exists

\

n

E∈R: lim

↓0

(A^ω−E−ι)⁻¹Cn

<∞ a.s.

2Global multiplicity bounds and Spectral Statistics Random Operators

(9)

Example

On`²(Z× {1,· · · ,5}) consider the operator H^ω = ˜∆ +X

n

ωnPn, where

( ˜∆u)(x, n) =_n

3

(u(x+ 1, n) +u(x−1, n)),

P0 P1 P2 P3 P4 P5

and ωn are i.i.d real random variables with absolutely continuous distribution. The projection is given by (Pnu)(x, m) =

u(x, m) x=n,

0 o.w .

Observe that H^ω restricted to `²(Z× {1}),`²(Z× {2})and `²(Z× {3}) are unitarily equivalent. Similarly `²(Z× {4}) and`²(Z× {5}) are unitarily equivalent.

First result will imply that the maximum multiplicity is bounded by three.

And second result will imply that lower bound of multiplicity is two.

(10)

Technical details (Assuming C

_n

are projections)

SettingHn^ω ={f(A^ω)φ:f ∈Cc(R), φ∈CnH} we have (A^ω,Hn^ω)∼= (M_Id, L²(R,Σ^ω_n,C^rank(Cⁿ⁾)), whereΣ^ω_n is matrix valued measure which satisfies

R ₁

x−zΣ^ω(dx) =Cn(A^ω−z)⁻¹Cn.

WritingdΣ^ω_n(E) =W_n^ω(E)dσ^ω_n(E) whereσ_n^ω =tr(Σ^ω_n), Poltoratskii’s theorem gives

W_n^ω(E) = lim↓0 1

tr(Cn(A^ω−E−ι)⁻¹Cn)C_n(A^ω−E−ι)⁻¹C_n, for almost all E w.r.tσ^ω_n,sing.

For E in support ofσ_n,sing^ω , we havetr(Cn(A^ω−E−ι)⁻¹Cn)→ ∞ as goes to zero.

IfHn^ω∩Hm^ω={0}, then σ_n,sing^ω andσ^ω_m,sing are singular.

(11)

Technical details (Assuming C

_n

are projections)

To establish multiplicity of W_n^ω(E) for almost all E, it is enough to study the case A_λ =A+λC_n. This is done as:

Let G_λ(z) =C_n(A_λ−z)⁻¹C_n, then using resolvent equation G_λ(z) =G₀(z)(I+λG₀(z))⁻¹ = (G₀(z)⁻¹+λI)⁻¹ A consequence of Poltoratskii’s theorem is

supp(σ_λ,sing) ={E : det(I+λG₀(E+ι0)) = 0}

So multiplicity of G₀(E+ι0)for almost allE provides the required answer.

Viewingdet(I +λG₀(z))as polynomial of λ, we need to find the multiplicity roots as function of z. This can be re-stated as a polynomial being non-zero.

(12)

Anderson type operators on Cayley like graphs

Properties of Spectrum for Anderson type random operator Anderson type operators on

Cayley like graphs 12 / 25

(13)

Anderson type operators on Cayley like graphs

Definition [AM, P. A. Narayanan arXiv:1808.05820]⁴

Given a finitely generated group Gwith generatorsg1,· · · , gn and a sequence of vertices v±1, v±2,· · · , v±n from a finite undirected graph H= (V,E), define the infinite graph H_G = (V_G,E_G) by

V_G=V ×G={(v, g) :v∈ V, g∈G}, The edge set E_G) is union of

{{(v, g),(w, g)}:{v, w} ∈ E, g ∈G}

and

{{(v_−i, g),(vi, ggi)}:g∈G,1≤i≤n}.

4On multiplicity of spectrum for Anderson type operators with higher rank perturbations

(14)

Anderson type operators on Cayley like graphs

On`²(V_G) let ∆denote the adjacency operator. Define the Anderson operator asH^ω = ∆ +X

g∈G

ω_gP_g,whereP_g is the projection onto

`²(V × {g}) and{ω_g}_g are i.i.d random variables.

Viewing ωg as projection from (Ω,B,P) = (R^G,⊗_GB_R,⊗_Gµ) to R, for any h∈Gdefine θ_h: Ω→Ωby(θ_hω)g =ω_gh. Then observe that

H^θ^h^ω =U_hH^ωU_h^∗,

where for any automorphism φof H_G define the unitary operatorUφ on

`²(V_G) by

(U_φu)((v, g)) =u(φ(v, g)), so the operator H^ω is ergodic.

(15)

Anderson type operators on Cayley like graphs

special Unitaries preserving H^ω

Aut_And(H_G) ={φ∈Aut(H_G) :H^ω=U_φH^ωU_φ^∗ a.s.}

Theorem

The group homomorphism Θ : Y

g∈G

Aut(H|{v_±i}ⁿ_i=1)→Aut_And(H_G)

defined byΘ((φg)g)((v, h)) = (φ_h(v), h), for (v, h)∈ V_G, is ismorphism.

Here

Aut(H|{v±i}ⁿ_i=1) :={φ∈Aut(H) :φ(v±i) =v±i 1≤i≤n}.

So ifAut(H|{v_±i}ⁿ_i=1) is trivial, then AutAnd(H_G) is also trivial.

(16)

Anderson type operators on Cayley like graphs

Observe that the zero eigenvalue for Laplacian over Hhas non-trivial multiplicity.

But by construction of the graph

Aut(H|{x₁, x2}) ={1}. xH₂

x₁

Another important observation is that, there is multiple eigenvectors corresponding to zero eigenvalue of Laplacian which is zero at x₁ and x₂. If we construct H_Z2 for the groupZ² and define the Anderson operator described as above, then the random operator H^ω has non-trivial multiplicity. By construction of the graph, Aut_And(H

Z²) is trivial, so automorphisms of underlying space is not contributing to any multiplicity of the operator.

(17)

Local Eigenvalue Statistics and regularity of DOS

Properties of Spectrum for Anderson type random operator Local Eigenvalue Statistics and

regularity of DOS 17 / 25

(18)

Local Eigenvalue Statistics and regularity of DOS

Model in [AM, D.R. Dolai, arXiv:1506.07132]³ On the Hilbert space `²(Z^d) consider the operator

(H^ωu)(n) = X

kn−mk1=1

(u(n)−u(m)) + (1 +knk^α₂)ωnu(n) n∈Z^d, where {ω_n}_n are i.i.d random variables following uniform distribution in [0,1].

It was shown by Gordon-Jakˇsi´c-Mochanov-Simon that σ_ess(H^ω) = [a_k,∞).

For E∈(a_j, aj−1) for j≤k the limit N_j(E) = lim

L→∞

1

L^d−jα#{E_n∈σ(H_L^ω) :E_n< E}

exists and is non-zero almost surely, whereH_L^ω is restriction ofH^ω in {−L,· · · , L}^d.

3Spectral statistics of random Schr¨odinger operator with growing potential

(19)

Smoothness property of N

₁

Result [AM, D.R. Dolai, arXiv:1506.07132]

For E >2dwe haveN₁(E) =C_d,α(E−2d).

For Λ_L,m_L:={−L,· · ·, L}^d\ {−m_L,· · · , m_L}^d define H_L,m^ω

L. The function GL(z) = _L¹dE

h T r

(H_L,m^ω

L−z)⁻¹ i

has analytic continuation for any compact subset of {z:Rez >2d²}for large enoughL.

L m_L

Random walk expansion δ0,(H^ω−z)⁻¹δ0

=

∞

X

k=0

X

γ∈Γ^k_L(0,0) k

Y

i=0

1

(1 +kγ_ik^α₂)ω_γ_i+ 2d−z where Γ^k_L(0,0)is set of path starting and ending at0 of lengthk.

(20)

Local Statistics for unfolded eigenvalues

Result [AM, D.R. Dolai, arXiv:1506.07132]

The point process Λ^ω,t_L (f) =T r(f(L^d−jα(Nj(H_L^ω)−t))) for f ∈Cc(R) converges to Poisson point process.

The set ΛL,m_L can be divided into {Λ_l_L(p)}_p, such that for most eigenvalues of H_L,m^ω

L there is a unique p where some eigenvalueH_Λ^ω

lL(p) is exponentially close.

So we can re-define the point process with re- spect to smaller boxes.

Λ^ω,t_L,p(f) =T r(f(L^d−jα(N_j(H_Λ^ω

lL(p))−t))).

These point processes are independent.

Rest of the work is to show that the point process P

pΛ^ω,t_L,p converges to Poisson point process.

(21)

Current problems and future directions

Properties of Spectrum for Anderson type random operator Future directions 21 / 25

(22)

Current Problems (in progress)

Regularity properties of density of state

Joint work with Prof. Manjunath: To show that the density of states measure for random Toeplitz/Hankel matrix is absolutely continuous. In case of Toeplitz matrix, the density of states measure is C^∞.

Joint work with D.R. Dolai and Prof. M. Krishna: To show that the density of state measure isC^∞ in the region of dynamical localization, random Schr¨odinger operator on L²(R^d).

The fundamental difference between above work is, there is no well defined understanding of behavior of eigenvectors for the case of Toeplitz/Hankel matrix, but for the other work we are explicitly focusing on the region where the eigenvectors are exponentially localized.

(23)

Future direction

Consider the sequence of random matrices {A^ω_α,L}_L∈_N given by

A^ω_α,L= ∆L+_L¹αV^ω, where∆L is Laplacian andV^ω is multiplication by i.i.d random variables, defined on C^L.

Observe that A^ω_α,L converges strongly to discrete Laplacian on`²(N) for α >0. ForE₀ ∈(−2,2), consider the point process

Λ^ω_α,E₀_,L(f) =T r(f(L(A^ω_α,L−E0))) for f ∈Cc(R).

For α > ¹₂, it can be shown thatΛ^ω_α,E

0,L converges to clock process, in sense of distribution, as L→ ∞.

The main goal is to show that for α < ¹₂, the point process Λ^ω_α,E

0,L

converges to Poisson point process.

This would imply that local eigenvalue statistics is not a good indicator of the nature of spectrum locally.

(24)

References

AM, D. R. Dolai,Schr¨odinger operators with decaying randomness - Pure point spectrum,arXiv:1808:05822.

AM, P. A. Narayanan,On multiplicity of spectrum for Anderson type operators with higher rank perturbations,arXiv:1808:05820.

AM, M. Krishna, Global multiplicity bounds and Spectral Statistics Random Operators,arXiv:1803:06895.

AM, D. R. Dolai,Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian,arXiv:1709:01774.

AM, D. R. Dolai,Spectral statistics of random Schr¨odinger operator with growing potential,arXiv:1506.07132.

Properties of Spectrum for Anderson type random operator References 24 / 25

(25)

Properties of Spectrum for Anderson type random operator References 25 / 25