Spectra of a class of non-self-adjoint random Jacobi matrices Marko - - PowerPoint PPT Presentation

Spectra of a class of non-self-adjoint random Jacobi matrices

Marko Lindner, TU Chemnitz

2nd Najman Conference on Spectral Problems, Dubrovnik

15 May 2009

Marko Lindner Spectra of Random Jacobi Matrices

...with the help of...

This talk is based on joint work with Simon N. Chandler-Wilde, Reading, UK Ratchanikorn Chonchaiya, Reading, UK Bernd Silbermann, Chemnitz, Germany and supported by the Marie Curie Fellowships MEIF-CT-2005-009758 and PERG02-GA-2007-224761

• f the European Union.

Marko Lindner Spectra of Random Jacobi Matrices

The Spaces

This talk is about a class of bounded linear operators on a space of the form E = ℓp(ZN, X) with p ∈ [1, ∞], N ∈ N and X an arbitrary complex Banach space.

Marko Lindner Spectra of Random Jacobi Matrices

The Spaces

This talk is about a class of bounded linear operators on a space of the form E = ℓp(ZN, X) with p ∈ [1, ∞], N ∈ N and X an arbitrary complex Banach space. So u ∈ E iff u = (uk)k∈ZN where uk ∈ X for all k ∈ ZN and uE =

p

k∈ZN

ukp

X,

p < ∞, uE = sup

k∈ZN ukX,

p = ∞.

Marko Lindner Spectra of Random Jacobi Matrices

The Spaces

Simplest example E = ℓp = ℓp(Z, C), N = 1, X = C

Marko Lindner Spectra of Random Jacobi Matrices

The Spaces

Simplest example E = ℓp = ℓp(Z, C), N = 1, X = C Slightly more sophisticated example E = Lp(RN) ∼ = ℓp(ZN, Lp([0, 1]N)), X = Lp([0, 1]N) by identifying f ∈ Lp(RN) with (f |α+[0,1]N)α∈ZN

Marko Lindner Spectra of Random Jacobi Matrices

The Spaces

Simplest example E = ℓp = ℓp(Z, C), N = 1, X = C Slightly more sophisticated example E = Lp(RN) ∼ = ℓp(ZN, Lp([0, 1]N)), X = Lp([0, 1]N) by identifying f ∈ Lp(RN) with (f |α+[0,1]N)α∈ZN

Marko Lindner Spectra of Random Jacobi Matrices

The Operators

L(E) ... the space of all bounded & linear operators E → E, K(E) ... the space of all compact operators E → E. With every operator A ∈ L(E) we will associate a matrix     ... . . . ... · · · aij · · · ... . . . ...         . . . uj . . .     =     . . . bi . . .     with indices i, j ∈ ZN and operator entries aij : X → X. For simplicity, we will restrict ourselves to band matrices.

Marko Lindner Spectra of Random Jacobi Matrices

Operators: Notations

A ∈ L(E) is called a Fredholm operator if its null-space is finite-dimensional and its range has finite co-dimension. This holds iff A + K(E) is invertible in L(E)/K(E).

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Definition

To study the asymptotics of the matrix entries aij as (i, j) → ∞, we introduce so-called limit operators. Definition Take a sequence h(1), h(2), ... ∈ ZN with |h(n)| → ∞. If, for all i, j ∈ ZN, it holds that ai+h(n), j+h(n) ⇒ bij as n → ∞, then B with [B] = (bij) is called the limit operator of A with [A] = (aij) w.r.t. the sequence h = (h(1), h(2), ...), and we write Ah instead of B.

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example

Example: Let A be a discrete Schr¨

• dinger operator

         ... ... ... b−1 1 1 b0 1 1 b1 ... ... ...          with the following potential b = (..., β, β, β, β

• 4

, α, α, α

3

, β, β

• 2

, α

• 1

, β, β

• 2

, α, α, α

3

, β, β, β, β

• 4

, ...).

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example

b = (..., β, β, β, β

• 4

, α, α, α

3

, β, β

• 2

, α

• 1

, β, β

• 2

, α, α, α

3

, β, β, β, β

• 4

, ...). Then all limit operators of A are of the form

B B B B B B B @ ... ... ... β 1 1 β ... ... ... 1 C C C C C C C A , B B B B B B B @ ... ... ... α 1 1 α ... ... ... 1 C C C C C C C A ,

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example

b = (..., β, β, β, β

• 4

, α, α, α

3

, β, β

• 2

, α

• 1

, β, β

• 2

, α, α, α

3

, β, β, β, β

• 4

, ...). Then all limit operators of A are of the form

B B B B B B B @ ... ... ... β 1 1 β ... ... ... 1 C C C C C C C A , B B B B B B B @ ... ... ... α 1 1 α ... ... ... 1 C C C C C C C A , B B B B B B B @ ... ... ... β 1 1 α ... ... ... 1 C C C C C C C A , B B B B B B B @ ... ... ... α 1 1 β ... ... ... 1 C C C C C C C A

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example

b = (..., β, β, β, β

• 4

, α, α, α

3

, β, β

• 2

, α

• 1

, β, β

• 2

, α, α, α

3

, β, β, β, β

• 4

, ...). Then all limit operators of A are of the form

B B B B B B B @ ... ... ... β 1 1 β ... ... ... 1 C C C C C C C A , B B B B B B B @ ... ... ... α 1 1 α ... ... ... 1 C C C C C C C A , B B B B B B B @ ... ... ... β 1 1 α ... ... ... 1 C C C C C C C A , B B B B B B B @ ... ... ... α 1 1 β ... ... ... 1 C C C C C C C A

• r they are translates of the latter two matrices.

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators and Random Matrices

Example: Let again A be a discrete Schr¨

• dinger operator

         ... ... ... b−1 1 1 b0 1 1 b1 ... ... ...          but now with ..., b−1, b0, b1, ... independent samples from a random variable with values in a compact set Σ ⊂ L(X). Then, with probability 1, b = (..., b−1, b0, b1, ...) is a pseudoergodic sequence over Σ, by which we mean the following:

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators and Random Matrices

Then, with probability 1, b = (..., b−1, b0, b1, ...) is a pseudoergodic sequence over Σ, by which we mean the following: Definition A sequence b = (bk)k∈Z is called pseudoergodic over Σ ⊂ L(X) if, for all ε > 0 and all finite vectors c = (ci)i∈I with values ci ∈ Σ, there is a translate of b that matches c on I up to precision ε, i.e. ∃m ∈ Z : max

i∈I bi+m − ci < ε.

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators and Random Matrices

Then, with probability 1, b = (..., b−1, b0, b1, ...) is a pseudoergodic sequence over Σ, by which we mean the following: Definition A sequence b = (bk)k∈Z is called pseudoergodic over Σ ⊂ L(X) if, for all ε > 0 and all finite vectors c = (ci)i∈I with values ci ∈ Σ, there is a translate of b that matches c on I up to precision ε, i.e. ∃m ∈ Z : max

i∈I bi+m − ci < ε.

If the potential b is pseudoergodic then every discrete Schr¨

• dinger
• perator with a potential c = (..., c−1, c0, c1, ...) over Σ (including

A itself) is a limit operator of A – and vice versa.

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Theorem on Fredholmness

Let A ∈ L(E) be a band operator of the form A = invertible + locally compact with A = B∗. Then it is not hard to see that A Fredholm = ⇒ all limit operators of A are invertible.

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Theorem on Fredholmness

Let A ∈ L(E) be a band operator of the form A = invertible + locally compact with A = B∗. Then it is not hard to see that A Fredholm = ⇒ all limit operators of A are invertible. Under the additional condition that {aij : i, j ∈ ZN} is relatively compact in L(X), in which case we call A a rich operator, we also have the reverse implication: Theorem

Chandler-Wilde, ML 2007

The following are equivalent for all p ∈ [1, ∞]: A is Fredholm on ℓp(Z, X), all limit operators of A are invertible on ℓp(Z, X),

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Theorem on Fredholmness

Let A ∈ L(E) be a band operator of the form A = invertible + locally compact with A = B∗. Then it is not hard to see that A Fredholm = ⇒ all limit operators of A are invertible. Under the additional condition that {aij : i, j ∈ ZN} is relatively compact in L(X), in which case we call A a rich operator, we also have the reverse implication: Theorem

Chandler-Wilde, ML 2007

The following are equivalent for all p ∈ [1, ∞]: A is Fredholm on ℓp(Z, X), all limit operators of A are invertible on ℓp(Z, X), all limit operators of A are injective on ℓ∞(Z, X).

Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Theorem on Fredholmness

Theorem

Chandler-Wilde, ML 2007

The following are equivalent for all p ∈ [1, ∞]: A is Fredholm on ℓp(Z, X), all limit operators of A are invertible on ℓp(Z, X), all limit operators of A are injective on ℓ∞(Z, X). ...moreover, the Fredholm index of A does not depend on p. If we repeat the same argument with λI − A in place of A, we get: Spectral Formula

Chandler-Wilde, ML 2007

specp

ess(A) =

• h

spec p(Ah) =

• h

spec∞

point(Ah),

p ∈ [1, ∞]

Marko Lindner Spectra of Random Jacobi Matrices

Two Random Matrices from Models of Feinberg & Zee

Two Random Matrices (N = 1, X = C) Find the spectrum of          ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          and          ... ... ... τ−1 σ0 τ0 σ1 ... ... ...          , acting on E = ℓp(Z, C), where σi ∈ Σ and τi ∈ T are (not fully correlated) samples from two random variables with values in the compact sets Σ, T ⊂ C, respectively.

Marko Lindner Spectra of Random Jacobi Matrices

Two Random Matrices from Models of Feinberg & Zee

Two Random Matrices (N = 1, X = C) Find the spectrum of          ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          and          ... ... ... τ−1 σ0 τ0 σ1 ... ... ...          , acting on E = ℓp(Z, C), where σi ∈ Σ and τi ∈ T are (not fully correlated) samples from two random variables with values in the compact sets Σ, T ⊂ C, respectively. Both matrices are non-self-adjoint and even non-normal.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Finite Submatrices

In particular, the following approach goes wrong in the first case: Take n ∈ N and look at the finite submatrix An =       σ−n τ−n ... ... ... τn−1 σn      

• f the infinite matrix A.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Finite Submatrices

In particular, the following approach goes wrong in the first case: Take n ∈ N and look at the finite submatrix An =       σ−n τ−n ... ... ... τn−1 σn      

• f the infinite matrix A.

Clearly, spec An → Σ as n → ∞.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Finite Submatrices

In particular, the following approach goes wrong in the first case: Take n ∈ N and look at the finite submatrix An =       σ−n τ−n ... ... ... τn−1 σn      

• f the infinite matrix A.

Clearly, spec An → Σ = spec A as n → ∞.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Finite Submatrices

spec An → Σ = spec A as n → ∞. Note that this effect is not due to the random entries.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Finite Submatrices

spec An → Σ = spec A as n → ∞. Note that this effect is not due to the random entries. Indeed, already for singletons Σ = {σ} and T = {τ}, one has spec An = spec       σ τ ... ... ... τ σ      

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Finite Submatrices

spec An → Σ = spec A as n → ∞. Note that this effect is not due to the random entries. Indeed, already for singletons Σ = {σ} and T = {τ}, one has spec An = spec       σ τ ... ... ... τ σ       = {σ}

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Finite Submatrices

spec An → Σ = spec A as n → ∞. Note that this effect is not due to the random entries. Indeed, already for singletons Σ = {σ} and T = {τ}, one has spec An = spec       σ τ ... ... ... τ σ       = {σ} = σ+|τ|T = spec A no matter how big n is.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Limit Operators

Put A =                           ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          : σi ∈ Σ, τi ∈ T ∀i                  . If A ∈ A is random (as defined before) then, with probability 1, the diagonals of A are pseudoergodic, so that the set of limit

• perators of A equals A.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Limit Operators

Put A =                           ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          : σi ∈ Σ, τi ∈ T ∀i                  . If A ∈ A is random (as defined before) then, with probability 1, the diagonals of A are pseudoergodic, so that the set of limit

• perators of A equals A.

Theorem

Chandler-Wilde, ML 2007

If A ∈ A is random then, with probability 1, specessA =

• B∈A

spec B =

• B∈A

spec∞

pointB.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Limit Operators

Put A =                           ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          : σi ∈ Σ, τi ∈ T ∀i                  . If A ∈ A is random (as defined before) then, with probability 1, the diagonals of A are pseudoergodic, so that the set of limit

• perators of A equals A.

Theorem

Chandler-Wilde, ML 2007

If A ∈ A is random then, with probability 1, spec A = specessA =

• B∈A

spec B =

• B∈A

spec∞

pointB.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Limit Operators

Put A =                           ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          : σi ∈ Σ, τi ∈ T ∀i                  . If A ∈ A is random (as defined before) then, with probability 1, the diagonals of A are pseudoergodic, so that the set of limit

• perators of A equals A.

Theorem

Chandler-Wilde, ML 2007

If A ∈ A is random then, with probability 1, spec A =

• B∈A

spec∞

pointB.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i) = ⇒ x(i + 1) = λ − σi τi x(i)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i) = ⇒ x(i + 1) = λ − σi τi x(i)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i) = ⇒ x(i + 1) = λ − σi τi x(i)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i) = ⇒ x(i + 1) = λ − σi τi x(i)           σ+ τ+ σ+ ... ...          

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i) = ⇒ x(i + 1) = λ − σi τi x(i)           σ+ τ+ σ+ ... ...          

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i) = ⇒ x(i + 1) = λ − σi τi x(i)            ... ... σ− τ− σ− τ− σ+ τ+ σ+ ... ...           

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• B∈A

spec∞

pointB

So, given a λ ∈ C, we just have to look for a B ∈ A with ℓ∞-eigenvalue λ; that is, Bx = λx with x ∈ ℓ∞, i.e. σi x(i) + τi x(i + 1) = λx(i) = ⇒ x(i + 1) = λ − σi τi x(i) Theorem: Spectrum of Matrix #1

ML 2008

If A ∈ A is random then, with probability 1, spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD), where t := minτ∈T |τ| and T := maxτ∈T |τ|.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Spectrum

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Special Cases

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

1 If T = {1} then t = T = 1 and the result was known before

(Trefethen, Embree, Contedini 2001).

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Special Cases

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

1 If T = {1} then t = T = 1 and the result was known before

(Trefethen, Embree, Contedini 2001).

2 If Σ = {σ} and T = {τ} then A has constant diagonals and

• ur formula says spec A = (σ + |τ|D) \ (σ + |τ|D) = σ + |τ|T.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Special Cases

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

1 If T = {1} then t = T = 1 and the result was known before

(Trefethen, Embree, Contedini 2001).

2 If Σ = {σ} and T = {τ} then A has constant diagonals and

• ur formula says spec A = (σ + |τ|D) \ (σ + |τ|D) = σ + |τ|T.

3 If Σ = {σ} and T = {τ, τ ′} then letting |τ| → 0 demonstrates

what quantum physicists call “disk-annulus transition”.

Marko Lindner Spectra of Random Jacobi Matrices

Matrix #1: Special Cases

spec A =

• σ∈Σ

(σ + TD) \

• σ∈Σ

(σ + tD)

1 If T = {1} then t = T = 1 and the result was known before

(Trefethen, Embree, Contedini 2001).

2 If Σ = {σ} and T = {τ} then A has constant diagonals and

• ur formula says spec A = (σ + |τ|D) \ (σ + |τ|D) = σ + |τ|T.

3 If Σ = {σ} and T = {τ, τ ′} then letting |τ| → 0 demonstrates

what quantum physicists call “disk-annulus transition”.

4 If t = dist(T , 0) < diam Σ/2 then ∩σ∈Σ(σ + tD) = ∅ and

hence spec A = ∪σ∈Σ(σ + TD), showing that the upper bound in Gershgorin’s circle theorem is sharp in this case.

Marko Lindner Spectra of Random Jacobi Matrices

Remember...

...we were trying to find the spectrum of the two infinite matrices          ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          and          ... ... ... τ−1 σ0 τ0 σ1 ... ... ...          , where σi ∈ Σ and τi ∈ T are random samples from two compact sets Σ, T ⊂ C.

Marko Lindner Spectra of Random Jacobi Matrices

Remember...

...we were trying to find the spectrum of the two infinite matrices          ... ... σ−1 τ−1 σ0 τ0 σ1 ... ...          and          ... ... ... τ−1 σ0 τ0 σ1 ... ... ...          , where σi ∈ Σ and τi ∈ T are random samples from two compact sets Σ, T ⊂ C. Now we turn our attention to the 2nd matrix.

Marko Lindner Spectra of Random Jacobi Matrices

From a talk of Anthony Zee (MSRI Berkeley, 1999)

Marko Lindner Spectra of Random Jacobi Matrices

Notations: Pseudoergodic Bi-Infinite Matrix

Look at the bi-infinite matrix Ab =           

. . . . . . . . .

1 b−1 1 b0 1 b1

. . . . . . . . .

           , where b = (· · · , b−1, b0, b1, · · · ) ∈ {±1}Z is a pseudoergodic sequence

Marko Lindner Spectra of Random Jacobi Matrices

Notations: Pseudoergodic Bi-Infinite Matrix

Look at the bi-infinite matrix Ab =           

. . . . . . . . .

1 b−1 1 b0 1 b1

. . . . . . . . .

           , where b = (· · · , b−1, b0, b1, · · · ) ∈ {±1}Z is a pseudoergodic sequence; that means: every finite pattern of ±1’s can be found somewhere in the infinite sequence b.

Marko Lindner Spectra of Random Jacobi Matrices

Notations: Pseudoergodic Bi-Infinite Matrix

Look at the bi-infinite matrix Ab =           

. . . . . . . . .

1 b−1 1 b0 1 b1

. . . . . . . . .

           , where b = (· · · , b−1, b0, b1, · · · ) ∈ {±1}Z is a pseudoergodic sequence; that means: every finite pattern of ±1’s can be found somewhere in the infinite sequence b. We will look at Ab as an operator on ℓp(Z, C) with p ∈ [1, ∞].

Marko Lindner Spectra of Random Jacobi Matrices

Related Matrices: Semi-Infinite and Finite

We will also look at the semi-infinite and finite matrices Ab

+ =

      1 b1 1 b2

. . . . . . . . .

      and Ab

n =

        1 b1 1 b2

. . . . . . . . .

1 bn−1         , n ∈ N.

Marko Lindner Spectra of Random Jacobi Matrices

Related Matrices: Semi-Infinite and Finite

We will also look at the semi-infinite and finite matrices Ab

+ =

      1 b1 1 b2

. . . . . . . . .

      and Ab

n =

        1 b1 1 b2

. . . . . . . . .

1 bn−1         , n ∈ N. Questions: Spectra of Ab, Ab

+ and relations to Ab n as n → ∞.

Marko Lindner Spectra of Random Jacobi Matrices

Related Matrices: Semi-Infinite and Finite

We will also look at the semi-infinite and finite matrices Ab

+ =

      1 b1 1 b2

. . . . . . . . .

      and Ab

n =

        1 b1 1 b2

. . . . . . . . .

1 bn−1         , n ∈ N. Questions: Spectra of Ab, Ab

+ and relations to Ab n as n → ∞.

Pseudospectra, numerical ranges?

Marko Lindner Spectra of Random Jacobi Matrices

Tool 1: The Limit Operator Approach

If b is pseudoergodic then the set of limit operators of Ab equals {Ac : c ∈ {±1}Z}. As a consequence, we get: Spectral Formula

Chandler-Wilde, ML 2007

If b is pseudoergodic then spec Ab = specessAb =

• c∈{±1}Z

spec∞

pointAc.

In particular, spec Ab does not depend on the actual sequence b.

Marko Lindner Spectra of Random Jacobi Matrices

Tool 2: Similarity Transforms

Via a similarity transform with an appropriate ±1 diagonal matrix,

• ne can switch between two and one pseudoergodic diagonals:

         ... ... ... ±1 ±1 ±1 ±1 ... ... ...          ↔          ... ... ... 1 ±1 1 ±1 ... ... ...         

Marko Lindner Spectra of Random Jacobi Matrices

Tool 3: Reflection Ideas

...and similarly (repeated reflections) for finite vs. bi-infinite matrix.

Marko Lindner Spectra of Random Jacobi Matrices

Result 1: Symmetries

Here and in what follows, suppose that b = (· · · , b−1, b0, b1, · · · ) ∈ {±1}Z is pseudoergodic. Then we have the following results: Symmetry The spectrum of Ab is symmetric w.r.t. real and imaginary axis and w.r.t. the line where Re z = Im z.

Marko Lindner Spectra of Random Jacobi Matrices

Result 2: Semi-Infinite vs. Bi-Infinite Matrix

Pseudospectra: Semi-Infinite vs. Bi-Infinite Matrix It holds that specp

ε Ab + = specp ε Ab

for all p ∈ [1, ∞] and ε ≥ 0, including the case of spectra (ε = 0).

Marko Lindner Spectra of Random Jacobi Matrices

Result 3: Upper Bound on the Spectrum

Upper Bound by Square It holds that spec Ab ⊂ Num Ab = conv(2, 2i, −2, −2i), where Num Ab denotes the closure of the numerical range of Ab.

Marko Lindner Spectra of Random Jacobi Matrices

Result 4: Lower Bound on the Spectrum

Lower Bound by Disk The spectrum spec Ab contains the closed unit disk, spec Ab ⊃ D = {z ∈ C : |z| ≤ 1}.

Marko Lindner Spectra of Random Jacobi Matrices

Result 5: Finite vs. Infinite Matrix

Spectrum & Pseudospectrum: Finite vs. Infinite Matrix For all n ∈ N, p ∈ [1, ∞] and all ε ≥ 0, it holds that specp

ε Ab n ⊂

• c∈{±1}n−1

specp

ε Ac n ⊂ specp ε Ab.

Marko Lindner Spectra of Random Jacobi Matrices

Result 5: Finite vs. Infinite Matrix

Spectrum & Pseudospectrum: Finite vs. Infinite Matrix For all n ∈ N, p ∈ [1, ∞] and all ε ≥ 0, it holds that specp

ε Ab n ⊂

• c∈{±1}n−1

specp

ε Ac n ⊂ specp ε Ab.

In the case of spectra, ε = 0, one moreover has spec Ab

n ⊂

• c∈{±1}n−1

spec Ac

n ⊂ P2n+2 ⊂ spec Ab.

Marko Lindner Spectra of Random Jacobi Matrices

Result 5: Finite vs. Infinite Matrix

Spectrum & Pseudospectrum: Finite vs. Infinite Matrix For all n ∈ N, p ∈ [1, ∞] and all ε ≥ 0, it holds that specp

ε Ab n ⊂

• c∈{±1}n−1

specp

ε Ac n ⊂ specp ε Ab.

In the case of spectra, ε = 0, one moreover has spec Ab

n ⊂

• c∈{±1}n−1

spec Ac

n ⊂ P2n+2 ⊂ spec Ab.

In the above result, we have put Pm :=

• c∈{±1}Z, m−periodic

spec∞

pointAc ⊂

• c∈{±1}Z

spec∞

pointAc = spec Ab.

Marko Lindner Spectra of Random Jacobi Matrices

The Sets Pn: Spectra in the Periodic Case

Period 1

Marko Lindner Spectra of Random Jacobi Matrices

The Sets Pn: Spectra in the Periodic Case

Period 2 Periods 1, 2

Marko Lindner Spectra of Random Jacobi Matrices

The Sets Pn: Spectra in the Periodic Case

Period 3 Periods 1, ..., 3

Marko Lindner Spectra of Random Jacobi Matrices