Spectra of magnetic chain graphs Pavel Exner Doppler Institute for - - PowerPoint PPT Presentation

Spectra of magnetic chain graphs

Pavel Exner

Doppler Institute for Mathematical Physics and Applied Mathematics Prague

in collaboration with Stepan Manko and Daniel Vaˇ sata A talk at the workshop Operator Theory and Indefinite Inner Product Spaces, Vienna, December 19, 2016

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 1 -

The talk outline

Setting the scene: magnetic chain graphs with a δ-coupling

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 2 -

The talk outline

Setting the scene: magnetic chain graphs with a δ-coupling The fully periodic case

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 2 -

The talk outline

Setting the scene: magnetic chain graphs with a δ-coupling The fully periodic case Coupling constant perturbations

◮ Duality with a difference operator ◮ Discrete spectrum due to local impurities ◮ Weak coupling ◮ Distant perturbations

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 2 -

The talk outline

Setting the scene: magnetic chain graphs with a δ-coupling The fully periodic case Coupling constant perturbations

◮ Duality with a difference operator ◮ Discrete spectrum due to local impurities ◮ Weak coupling ◮ Distant perturbations

Magnetic perturbations

◮ Duality with a difference operator ◮ Local changes of the magnetic field ◮ Weak perturbations of mixed type

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 2 -

The talk outline

Setting the scene: magnetic chain graphs with a δ-coupling The fully periodic case Coupling constant perturbations

◮ Duality with a difference operator ◮ Discrete spectrum due to local impurities ◮ Weak coupling ◮ Distant perturbations

Magnetic perturbations

◮ Duality with a difference operator ◮ Local changes of the magnetic field ◮ Weak perturbations of mixed type

Cantor spectra in chain graphs

◮ Can fractality occur in a ‘one-dimensional’ system? ◮ Duality with a difference operator, a stronger version ◮ Linear magnetic field

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 2 -

Quantum chain graphs

Quantum graphs are important both as models of nanostructures and as a source of interesting mathematical problems

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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Quantum chain graphs

Quantum graphs are important both as models of nanostructures and as a source of interesting mathematical problems In this talk we are going to consider a particular class, chain graphs, consisting of an array of rings coupled in the touching points

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 3 -

Quantum chain graphs

Quantum graphs are important both as models of nanostructures and as a source of interesting mathematical problems In this talk we are going to consider a particular class, chain graphs, consisting of an array of rings coupled in the touching points We consider a particular class of these quantum graphs such that the vertex coupling is of the δ type

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 3 -

Quantum chain graphs

Quantum graphs are important both as models of nanostructures and as a source of interesting mathematical problems In this talk we are going to consider a particular class, chain graphs, consisting of an array of rings coupled in the touching points We consider a particular class of these quantum graphs such that the vertex coupling is of the δ type the particle confined to the graph is charged and exposed to a magnetic field perpendicular to the graph plane

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 3 -

The Hamiltonian

The graph Γ is naturally parametrized by arc lengths of its edges, the corresponding state Hilbert space is L2(Γ). For simplicity we use units in which ℏ = 2m = e = 1, where e is the particle charge

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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The Hamiltonian

The graph Γ is naturally parametrized by arc lengths of its edges, the corresponding state Hilbert space is L2(Γ). For simplicity we use units in which ℏ = 2m = e = 1, where e is the particle charge Our Hamiltonian will be the magnetic Laplacian, ψj → −D2ψj on each graph link, where D := −i∇ − A. Its domain consists of all functions from the Sobolev space H2

loc(Γ) satisfying the δ-coupling conditions

ψi(0) = ψj(0) =: ψ(0) , i, j ∈ n ,

n

• i=1

Dψi(0) = α ψ(0) , where n = {1, 2, . . . , n} is the index set numbering the edges emanating from the vertex — in our case n = 4 — and α ∈ R is the coupling constant

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 4 -

The Hamiltonian

The graph Γ is naturally parametrized by arc lengths of its edges, the corresponding state Hilbert space is L2(Γ). For simplicity we use units in which ℏ = 2m = e = 1, where e is the particle charge Our Hamiltonian will be the magnetic Laplacian, ψj → −D2ψj on each graph link, where D := −i∇ − A. Its domain consists of all functions from the Sobolev space H2

loc(Γ) satisfying the δ-coupling conditions

ψi(0) = ψj(0) =: ψ(0) , i, j ∈ n ,

n

• i=1

Dψi(0) = α ψ(0) , where n = {1, 2, . . . , n} is the index set numbering the edges emanating from the vertex — in our case n = 4 — and α ∈ R is the coupling constant This is a particular case of the general conditions that make the operator self-adjoint [Kostrykin-Schrader’03]

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 4 -

Remarks

The detailed shape of the magnetic field is no important, what matters is the flux through each ring. Without loss of generality we may suppose that A is constant on each ring

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 5 -

Remarks

The detailed shape of the magnetic field is no important, what matters is the flux through each ring. Without loss of generality we may suppose that A is constant on each ring The field and the coupling constants may change from ring to ring. We denote the operator of interest as −∆α,A, where α = {αj}j∈Z and A = {Aj}j∈Z are sequences of real numbers; in any of them is constant we replace it simply by that number

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 5 -

Remarks

The detailed shape of the magnetic field is no important, what matters is the flux through each ring. Without loss of generality we may suppose that A is constant on each ring The field and the coupling constants may change from ring to ring. We denote the operator of interest as −∆α,A, where α = {αj}j∈Z and A = {Aj}j∈Z are sequences of real numbers; in any of them is constant we replace it simply by that number The simplest situation is the fully periodic case when both α and A are constant; we are going to consider various perturbations of this system

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 5 -

Remarks

The detailed shape of the magnetic field is no important, what matters is the flux through each ring. Without loss of generality we may suppose that A is constant on each ring The field and the coupling constants may change from ring to ring. We denote the operator of interest as −∆α,A, where α = {αj}j∈Z and A = {Aj}j∈Z are sequences of real numbers; in any of them is constant we replace it simply by that number The simplest situation is the fully periodic case when both α and A are constant; we are going to consider various perturbations of this system We exclude the case when some αj = ∞ which corresponds to Dirichlet decoupling of the chain in the particular vertex

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 5 -

Remarks

The detailed shape of the magnetic field is no important, what matters is the flux through each ring. Without loss of generality we may suppose that A is constant on each ring The field and the coupling constants may change from ring to ring. We denote the operator of interest as −∆α,A, where α = {αj}j∈Z and A = {Aj}j∈Z are sequences of real numbers; in any of them is constant we replace it simply by that number The simplest situation is the fully periodic case when both α and A are constant; we are going to consider various perturbations of this system We exclude the case when some αj = ∞ which corresponds to Dirichlet decoupling of the chain in the particular vertex Without loss of generality we may suppose that the circumference

• f each ring is 2π, later we may sometimes relax this condition and

consider rings of different sizes

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 5 -

The fully periodic case

In view of the periodicity of Γ and −∆α w.r.t. discrete shifts, we have Bloch-Floquet decomposition with the elementary cell

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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The fully periodic case

In view of the periodicity of Γ and −∆α w.r.t. discrete shifts, we have Bloch-Floquet decomposition with the elementary cell We write the wave function with energy E := k2 = 0 in the form ψL(x) = e−iAx(C +

L eikx + C − L e−ikx) ,

x ∈ [−π/2, 0] and similarly for the other three components; for E negative we put instead k = ıκ with κ > 0.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 6 -

The fully periodic case

In view of the periodicity of Γ and −∆α w.r.t. discrete shifts, we have Bloch-Floquet decomposition with the elementary cell We write the wave function with energy E := k2 = 0 in the form ψL(x) = e−iAx(C +

L eikx + C − L e−ikx) ,

x ∈ [−π/2, 0] and similarly for the other three components; for E negative we put instead k = ıκ with κ > 0. The wave function components have to be matched through (a) the δ-coupling and

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 6 -

The fully periodic case

In view of the periodicity of Γ and −∆α w.r.t. discrete shifts, we have Bloch-Floquet decomposition with the elementary cell We write the wave function with energy E := k2 = 0 in the form ψL(x) = e−iAx(C +

L eikx + C − L e−ikx) ,

x ∈ [−π/2, 0] and similarly for the other three components; for E negative we put instead k = ıκ with κ > 0. The wave function components have to be matched through (a) the δ-coupling and (b) Floquet-Bloch conditions

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 6 -

The fully periodic case, continued

This yields quadratic equation for the phase factor eiθ, specifically sin kπ cos Aπ(e2iθ − 2ξ(k)eiθ + 1) = 0 with ξ(k) := 1 cos Aπ

• cos kπ + α

4k sin kπ

• ,

which has real coefficients for any k ∈ R ∪ iR \ {0} and the discriminant equal to D = 4(ξ(k)2 − 1)

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 7 -

The fully periodic case, continued

This yields quadratic equation for the phase factor eiθ, specifically sin kπ cos Aπ(e2iθ − 2ξ(k)eiθ + 1) = 0 with ξ(k) := 1 cos Aπ

• cos kπ + α

4k sin kπ

• ,

which has real coefficients for any k ∈ R ∪ iR \ {0} and the discriminant equal to D = 4(ξ(k)2 − 1) The special cases A − 1

2 ∈ Z and k ∈ N have to be treated separately,

• therwise k2 ∈ σ(−∆α) holds if and only if the condition

|ξ(k)| ≤ 1 is satisfied. Together with the special case, we arrive thus at the description of the spectrum

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 7 -

The fully periodic case, continued

Theorem (E-Manko’15)

Let A / ∈ Z. If A − 1

2 ∈ Z, then the spectrum of −∆α consists of two series

• f infinitely degenerate ev’s {k2 ∈ R: ξ(k) = 0} and {k2 ∈ R: k ∈ N}.

On the other hand, if A − 1

2 /

∈ Z, the spectrum of −∆α consists of infinitely degenerate eigenvalues k2 with k ∈ N, and absolutely continuous spectral bands. Each of these bands except the first one is contained in an interval (n2, (n + 1)2) with n ∈ N. The first band is included in (0, 1) if α > 4(| cos Aπ| − 1)/π, or it is negative if α < −4(| cos Aπ| + 1)/π,

• therwise it contains the point k2 = 0.
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 8 -

The fully periodic case, continued

Theorem (E-Manko’15)

Let A / ∈ Z. If A − 1

2 ∈ Z, then the spectrum of −∆α consists of two series

• f infinitely degenerate ev’s {k2 ∈ R: ξ(k) = 0} and {k2 ∈ R: k ∈ N}.

On the other hand, if A − 1

2 /

∈ Z, the spectrum of −∆α consists of infinitely degenerate eigenvalues k2 with k ∈ N, and absolutely continuous spectral bands. Each of these bands except the first one is contained in an interval (n2, (n + 1)2) with n ∈ N. The first band is included in (0, 1) if α > 4(| cos Aπ| − 1)/π, or it is negative if α < −4(| cos Aπ| + 1)/π,

• therwise it contains the point k2 = 0.

Remark: The case A ∈ Z is by a simple gauge transformation equivalent to the non-magnetic case, A = 0. The spectrum is then easily obtained using the mirror symmetry: the antisymmetric component gives the Dirichlet eigenvalues, the symmetric one is the Kronig-Penney model with the coupling constant 1

2α

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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Determining the spectral bands

i

1 2i 1 2

1

3 2

2

5 2

3

7 2

−4 −2 2 4 η

γ > 0 γ = 0 γ ∈ (−8/π, 0) γ < −8/π

− → √z ∈ R+ ← − √z ∈ iR+

The picture refers to A = 0 with η(z) := 4ξ(√z) and γ = α

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 9 -

Determining the spectral bands

i

1 2i 1 2

1

3 2

2

5 2

3

7 2

−4 −2 2 4 η

γ > 0 γ = 0 γ ∈ (−8/π, 0) γ < −8/π

− → √z ∈ R+ ← − √z ∈ iR+

The picture refers to A = 0 with η(z) := 4ξ(√z) and γ = α For A − 1

2 /

∈ Z the situation is similar, just the width of the band varies, while for A − 1

2 ∈ Z it shrinks to a line

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 9 -

The first spectral band

The first spectral band of the operator −∆α,A vs. α at cos Aπ = 0.7.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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Duality

Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schr¨

• dinger (differential) equation in

question to solutions of a suitable difference equation

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 11 -

Duality

Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schr¨

• dinger (differential) equation in

question to solutions of a suitable difference equation The idea was put forward by physicists – Alexander and de Gennes – and later treated rigorously in [Cattaneo’97] [E’97], and [Pankrashkin’13]

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 11 -

Duality

Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schr¨

• dinger (differential) equation in

question to solutions of a suitable difference equation The idea was put forward by physicists – Alexander and de Gennes – and later treated rigorously in [Cattaneo’97] [E’97], and [Pankrashkin’13] We exclude possible Dirichlet eigenvalues from our considerations assuming k ∈ K := {z : Im z ≥ 0 ∧ z / ∈ Z}. On the one hand, we have the differential equation (−∆α,A − k2)

• ψ(x, k)

ϕ(x, k)

• = 0

with the components referring to the upper and lower part of Γ,

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 11 -

Duality

Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schr¨

• dinger (differential) equation in

question to solutions of a suitable difference equation The idea was put forward by physicists – Alexander and de Gennes – and later treated rigorously in [Cattaneo’97] [E’97], and [Pankrashkin’13] We exclude possible Dirichlet eigenvalues from our considerations assuming k ∈ K := {z : Im z ≥ 0 ∧ z / ∈ Z}. On the one hand, we have the differential equation (−∆α,A − k2)

• ψ(x, k)

ϕ(x, k)

• = 0

with the components referring to the upper and lower part of Γ, on the

• ther hand the difference one

ψj+1(k) + ψj−1(k) = ξj(k)ψj(k) , k ∈ K , where ψj(k) := ψ(jπ, k). These two are intimately related.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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Duality, continued

Theorem

Let αj ∈ R, then any solution

  ψ(·, k) ϕ(·, k)   with k2 ∈ R and k ∈ K satisfies

the difference equation, and conversely, the latter defines via

• ψ(x, k)

ϕ(x, k)

• = e∓iA(x−jπ)
• ψj(k) cos k(x − jπ)

+(ψj+1(k)e±iAπ − ψj(k) cos kπ)sin k(x − jπ) sin kπ

• , x ∈
• jπ, (j + 1)π
• ,

solutions to the former satisfying the δ-coupling conditions. In addition, the former belongs to Lp(Γ) if and only if {ψj(k)}j∈Z ∈ ℓp(Z), the claim being true for both p ∈ {2, ∞}.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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Coupling constant perturbations

Consider now the situation when the coupling is modified at a finite subset of vertices, the constant sequence α being modified to αj = α + γj , j ∈ M := {1, . . . , m} , αj = α , j ∈ Z \ M

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 13 -

Coupling constant perturbations

Consider now the situation when the coupling is modified at a finite subset of vertices, the constant sequence α being modified to αj = α + γj , j ∈ M := {1, . . . , m} , αj = α , j ∈ Z \ M It follows from general principles that such an operator −∆α,A can have at most m eigenvalues in each gap of the unperturbed one

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 13 -

Coupling constant perturbations

Consider now the situation when the coupling is modified at a finite subset of vertices, the constant sequence α being modified to αj = α + γj , j ∈ M := {1, . . . , m} , αj = α , j ∈ Z \ M It follows from general principles that such an operator −∆α,A can have at most m eigenvalues in each gap of the unperturbed one To find these eigenvalues, we have investigate behavior of the matrices relating solutions on the neighboring rings, Φj+1(k) = Nj(k)Φj(k) , j ∈ Z , .

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 13 -

Coupling constant perturbations

Consider now the situation when the coupling is modified at a finite subset of vertices, the constant sequence α being modified to αj = α + γj , j ∈ M := {1, . . . , m} , αj = α , j ∈ Z \ M It follows from general principles that such an operator −∆α,A can have at most m eigenvalues in each gap of the unperturbed one To find these eigenvalues, we have investigate behavior of the matrices relating solutions on the neighboring rings, Φj+1(k) = Nj(k)Φj(k) , j ∈ Z , . Outside the perturbation support the matrix Nj(k) =

• 2ξj(k)

−1 1

• is

independent of j and we need that it has an eigenvalue less than one to ensure an exponential decay of the solution

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 13 -

Coupling constant perturbations, continued

To state the result, let us introduce P0(k) = 1 , P1(k) = 2ξ1(k) , Pm(k) = 2ξm(k)Pm−1(k) − Pm−2(k) , Q0(k) = 0 , Q1(k) = 1 , Qm(k) = 2ξm(k)Qm−1(k) − Qm−2(k) , and furthermore, λ(k) := ξ(k) − sgn(ξ(k))

• ξ(k)2 − 1 .
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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Coupling constant perturbations, continued

To state the result, let us introduce P0(k) = 1 , P1(k) = 2ξ1(k) , Pm(k) = 2ξm(k)Pm−1(k) − Pm−2(k) , Q0(k) = 0 , Q1(k) = 1 , Qm(k) = 2ξm(k)Qm−1(k) − Qm−2(k) , and furthermore, λ(k) := ξ(k) − sgn(ξ(k))

• ξ(k)2 − 1 .

Inspecting the conditions of the solution decay, we arrive at

Theorem (E-Manko’15)

k2 ∈ R \ σ(−∆α,A) is an eigenvalue of −∆α+γ,A iff for this k we have Qm−1(k)λ(k)2 − (Pm−1(k) + Qm(k))λ(k) + Pm(k) = 0

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

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Example: a single impurity

In particular, if γ = {. . . , 0, γ1, 0, . . .}, we have

Proposition

Let A / ∈ Z. The essential spectrum of −∆α+γ,A coincides with that of −∆α. If γ1 < 0 there is precisely one simple impurity state in every odd gap, on the other hand, for γ1 > 0 there is precisely one simple impurity state in every even gap.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 15 -

Example: a single impurity

In particular, if γ = {. . . , 0, γ1, 0, . . .}, we have

Proposition

Let A / ∈ Z. The essential spectrum of −∆α+γ,A coincides with that of −∆α. If γ1 < 0 there is precisely one simple impurity state in every odd gap, on the other hand, for γ1 > 0 there is precisely one simple impurity state in every even gap. The energy k2 vs. γ1 = f (k) for cos Aπ = 0.6 and the coupling strength (i) α = 1, (ii) α = −1, (iii) α = −3

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 15 -

Weak coupling

Theorem (E-Manko’15)

Suppose that A / ∈ Z. For any ε ∈ (0, 1) the essential spectrum of −∆α+εγ,A coincides with that of −∆α,A. Let

j∈M γj < 0, then in the

limit ε → 0, the operator −∆α+εγ,A has exactly one simple impurity state in every odd gap. If

j∈M γj > 0, then in the limit ε → 0 there is exactly

• ne simple impurity state in every even gap. The corresponding eigenvalue

k2

n,ε is given by the following asymptotic expansion

kn,ε = kn + (−1)2n+1Knε2 + O(ε2) , ε → 0 , n ∈ N . Here k2

1 < k2 2 < . . . are the ‘non-Dirichlet’ gap ends coming from the

solutions to the equation |ξ(k)| = 1, and Kn = sin knπ

j∈M γj

2 cos Aπ(32(knπ)2 − 8αknπ cot knπ + 8α) , n ∈ N .

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 16 -

Distant perturbations

Consider the case of two impurities at large distances: we change the coupling constants at two arbitrary but fixed points into α + γ1 and α + γ2; we suppose that there are exactly n graph vertices between the chosen two.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 17 -

Distant perturbations

Consider the case of two impurities at large distances: we change the coupling constants at two arbitrary but fixed points into α + γ1 and α + γ2; we suppose that there are exactly n graph vertices between the chosen two. For brevity, let −∆α,A,n denote the Hamiltonian; we are interested in its spectral properties of for large n.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 17 -

Distant perturbations

Consider the case of two impurities at large distances: we change the coupling constants at two arbitrary but fixed points into α + γ1 and α + γ2; we suppose that there are exactly n graph vertices between the chosen two. For brevity, let −∆α,A,n denote the Hamiltonian; we are interested in its spectral properties of for large n.

Theorem (E-Manko’15)

Let A / ∈ Z. For any n ∈ N the essential spectrum of −∆α,A,n coincides with that of −∆α,A. If γ1γ2 < 0, then any for sufficiently large n, the

• perator −∆α,A,n has precisely one simple impurity state in every gap of

its essential spectrum. If γ1 and γ2 are both positive (negative), then for sufficiently large n, −∆α,A,n has two simple impurity states in every even (respectively, odd) gap of its essential spectrum and no impurity state in every odd (respectively, even) one (provided we start counting from the first gap). If γ1 = γ2, the impurity states in every even or odd gap are exponentially close to each other with respect to n.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 17 -

More general systems: the duality

We may consider more general chain graphs in several respects. We have already mentioned that, the magnetic field can vary, A = {Aj}j∈Z,

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 18 -

More general systems: the duality

We may consider more general chain graphs in several respects. We have already mentioned that, the magnetic field can vary, A = {Aj}j∈Z, the same my be true for the ring (half-)perimeters, ℓ = {ℓj}j∈Z, etc.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 18 -

More general systems: the duality

We may consider more general chain graphs in several respects. We have already mentioned that, the magnetic field can vary, A = {Aj}j∈Z, the same my be true for the ring (half-)perimeters, ℓ = {ℓj}j∈Z, etc. What is important, the above duality holds again, with the difference relation being sin(kℓj−1) cos(Ajℓj)ψj+1(k) + sin(kℓj) cos(Aj−1ℓj−1)ψj−1(k) = α 2k sin(kℓj−1) sin(kℓj) + sin k(ℓj−1 + ℓj)

• ψj(k) ,

k ∈ K , where ψj(k) := ψ(xj, k),

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 18 -

More general systems: the duality

We may consider more general chain graphs in several respects. We have already mentioned that, the magnetic field can vary, A = {Aj}j∈Z, the same my be true for the ring (half-)perimeters, ℓ = {ℓj}j∈Z, etc. What is important, the above duality holds again, with the difference relation being sin(kℓj−1) cos(Ajℓj)ψj+1(k) + sin(kℓj) cos(Aj−1ℓj−1)ψj−1(k) = α 2k sin(kℓj−1) sin(kℓj) + sin k(ℓj−1 + ℓj)

• ψj(k) ,

k ∈ K , where ψj(k) := ψ(xj, k), and the reconstruction formula becomes

• ψ(x, k)

ϕ(x, k)

• = e∓iA(x−xj)
• ψj(k) cos k(x − xj)

+(ψj+1(k)e±iAℓj − ψj(k) cos kℓj)sin k(x − xj) sin kℓj

• , x ∈
• xj, xj+1
• ,
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 18 -

A remark: probability current

The probability current on the jth circle in is easily found to be Jψ(k) = 2k sin kπ

• Re ψj(k)(Re ψj+1(k) sin Ajπ + Im ψj+1(k) cos Ajπ)

−Im ψj(k)(Re ψj+1(k) cos Ajπ − Im ψj+1(k) sin Ajπ)

• and a similar expression for the current Jϕ(k) on the lower part of the ring.
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 19 -

A remark: probability current

The probability current on the jth circle in is easily found to be Jψ(k) = 2k sin kπ

• Re ψj(k)(Re ψj+1(k) sin Ajπ + Im ψj+1(k) cos Ajπ)

−Im ψj(k)(Re ψj+1(k) cos Ajπ − Im ψj+1(k) sin Ajπ)

• and a similar expression for the current Jϕ(k) on the lower part of the ring.

Note that the expression makes sense as long as we consider k ∈ K.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 19 -

A remark: probability current

The probability current on the jth circle in is easily found to be Jψ(k) = 2k sin kπ

• Re ψj(k)(Re ψj+1(k) sin Ajπ + Im ψj+1(k) cos Ajπ)

−Im ψj(k)(Re ψj+1(k) cos Ajπ − Im ψj+1(k) sin Ajπ)

• and a similar expression for the current Jϕ(k) on the lower part of the ring.

Note that the expression makes sense as long as we consider k ∈ K. In the the non-magnetic case, A ∈ Z, currents on upper and lower edges are the same, as one expects from the symmetry of the δ coupling.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 19 -

A remark: probability current

The probability current on the jth circle in is easily found to be Jψ(k) = 2k sin kπ

• Re ψj(k)(Re ψj+1(k) sin Ajπ + Im ψj+1(k) cos Ajπ)

−Im ψj(k)(Re ψj+1(k) cos Ajπ − Im ψj+1(k) sin Ajπ)

• and a similar expression for the current Jϕ(k) on the lower part of the ring.

Note that the expression makes sense as long as we consider k ∈ K. In the the non-magnetic case, A ∈ Z, currents on upper and lower edges are the same, as one expects from the symmetry of the δ coupling. If δ is replaced by an asymmetric coupling, interesting ‘switching’ patterns between the upper and lower parts may occur [Cheon-Poghosyan’15]

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 19 -

A remark: probability current

The probability current on the jth circle in is easily found to be Jψ(k) = 2k sin kπ

• Re ψj(k)(Re ψj+1(k) sin Ajπ + Im ψj+1(k) cos Ajπ)

−Im ψj(k)(Re ψj+1(k) cos Ajπ − Im ψj+1(k) sin Ajπ)

• and a similar expression for the current Jϕ(k) on the lower part of the ring.

Note that the expression makes sense as long as we consider k ∈ K. In the the non-magnetic case, A ∈ Z, currents on upper and lower edges are the same, as one expects from the symmetry of the δ coupling. If δ is replaced by an asymmetric coupling, interesting ‘switching’ patterns between the upper and lower parts may occur [Cheon-Poghosyan’15] For those k that produce ℓ2-sequences {ψj(k)}j∈Z, the latter can be chosen real, whence the probability currents read as follows J(ψ

ϕ)(k) = ±2kψj(k)ψj+1(k)sin Ajπ

sin kπ , and since the coefficients ψj(k)ψj+1(k) decay for a fixed k as |j| → ∞, the probability current is ‘circling’ around such localized solutions.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 19 -

Magnetic perturbations

Using the above duality, we can treat local changes of the magnetic field.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 20 -

Magnetic perturbations

Using the above duality, we can treat local changes of the magnetic

• field. The core is again analysis of the ‘transfer’ matrices relating solutions
• n neighboring rings, in particular, their spectral properties.
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 20 -

Magnetic perturbations

Using the above duality, we can treat local changes of the magnetic

• field. The core is again analysis of the ‘transfer’ matrices relating solutions
• n neighboring rings, in particular, their spectral properties.

As an example, consider the basic magnetic field given be vector potential A is modified on two adjacent rings to A1, A2. For brevity, we denote the unperturbed operator −∆A and the perturbed one −∆A1,A2, the coupling constant α and the half-perimeter are kept fixed.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 20 -

Magnetic perturbations

Using the above duality, we can treat local changes of the magnetic

• field. The core is again analysis of the ‘transfer’ matrices relating solutions
• n neighboring rings, in particular, their spectral properties.

As an example, consider the basic magnetic field given be vector potential A is modified on two adjacent rings to A1, A2. For brevity, we denote the unperturbed operator −∆A and the perturbed one −∆A1,A2, the coupling constant α and the half-perimeter are kept fixed.

Theorem (E-Manko’17)

We have σ(−∆A1,A2) = σ(−∆A) = σess(−∆A) unless the inequality (cos A1π)2 + (cos A2π)2 2(cos Aπ)2 > 1

• holds. If, on the other hand, this is the case, the essential spectra of

the two operators are the same and −∆A1,A2 has precisely one simple eigenvalue in every gap of the essential spectrum.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 20 -

Magnetic perturbations, continued

In particular, if the field is modified on a single ring, i.e. A2 = A, we have a single simple eigenvalue in each gap provided | cos A1π| | cos Aπ| > 1 ,

• therwise the spectrum does not change.
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 21 -

Magnetic perturbations, continued

In particular, if the field is modified on a single ring, i.e. A2 = A, we have a single simple eigenvalue in each gap provided | cos A1π| | cos Aπ| > 1 ,

• therwise the spectrum does not change.

Note that the additional eigenvalues appear for perturbations which can be regarded as being ‘closer to the non-magnetic case’

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 21 -

Magnetic perturbations, continued

In particular, if the field is modified on a single ring, i.e. A2 = A, we have a single simple eigenvalue in each gap provided | cos A1π| | cos Aπ| > 1 ,

• therwise the spectrum does not change.

Note that the additional eigenvalues appear for perturbations which can be regarded as being ‘closer to the non-magnetic case’ Note also that very spectral gap of the unperturbed system lies between a spectral band and an eigenvalue of infinite multiplicity. When we change the magnetic field, the eigenvalue may emerge from the spectral band and return it. On the other hand it never emerges from the degenerate band.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 21 -

Magnetic perturbations, continued

In particular, if the field is modified on a single ring, i.e. A2 = A, we have a single simple eigenvalue in each gap provided | cos A1π| | cos Aπ| > 1 ,

• therwise the spectrum does not change.

Note that the additional eigenvalues appear for perturbations which can be regarded as being ‘closer to the non-magnetic case’ Note also that very spectral gap of the unperturbed system lies between a spectral band and an eigenvalue of infinite multiplicity. When we change the magnetic field, the eigenvalue may emerge from the spectral band and return it. On the other hand it never emerges from the degenerate band. One can also treat more complicated perturbations, including combined modifications of the geometry and magnetic field – we refer to the paper [E-Manko’17] for discussion of other examples.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 21 -

Weak local perturbations

Let now the parameters be of the form A + εAj and α + εαj and ask about the spectrum in the asymptotic regime ε → 0, assuming the perturbation is restricted to ring indices j ∈ {1, . . . , n}. For brevity, we denote the perturbed operator −∆ε

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 22 -

Weak local perturbations

Let now the parameters be of the form A + εAj and α + εαj and ask about the spectrum in the asymptotic regime ε → 0, assuming the perturbation is restricted to ring indices j ∈ {1, . . . , n}. For brevity, we denote the perturbed operator −∆ε

Theorem (E-Manko’17)

Let cot Aπ n

j=1 Aj > 0. If n j=1 αj > 0, the operator −∆ε has no

eigenvalues as ε → 0+ except in a finite number of even gaps, at most one per gap. Similarly, for n

j=1 αj < 0 and −∆ε has no eigenvalues except in

a finite number of odd gaps, at most one per gap. On the other hand, let cot Aπ n

j=1 Aj < 0. If n j=1 αj > 0, the operator

−∆ε has precisely one simple eigenvalue as ε → 0 in every gap except possibly a finite number of odd gaps. If n

j=1 αj < 0, it has precisely one

simple eigenvalue in every gap except possibly a finite number of even gaps of σess(−∆α,A).

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 22 -

Zero total flux change

The case left out in the above theorem concerns the situation when the perturbation does not change the total magnetic flux. Then we have

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 23 -

Zero total flux change

The case left out in the above theorem concerns the situation when the perturbation does not change the total magnetic flux. Then we have

Theorem (E-Manko’17)

Let n

j=1 αj = 0. The spectrum of −∆ε coincides with that of −∆α,A if

the condition sgn

• n
• j=1

Aj

• = −sgn(cot Aπ)

does not hold. If it does, the essential spectrum is preserved and −∆ε has additionally precisely one simple eigenvalue in every gap of it.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 23 -

Zero total flux change

The case left out in the above theorem concerns the situation when the perturbation does not change the total magnetic flux. Then we have

Theorem (E-Manko’17)

Let n

j=1 αj = 0. The spectrum of −∆ε coincides with that of −∆α,A if

the condition sgn

• n
• j=1

Aj

• = −sgn(cot Aπ)

does not hold. If it does, the essential spectrum is preserved and −∆ε has additionally precisely one simple eigenvalue in every gap of it. Remark: One prove various results also for nonlocal perturbations, for instance a weak-coupling version of the Saxon-Hutner conjecture in the present context, we refer to [E-Manko’17] for more information

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 23 -

The picture everybody knows

representing the spectrum of the difference operator associated with the almost Mathieu equation un+1 + un−1 + 2λ cos(2π(ω + nα))un = ǫun for λ = 1, otherwise called Harper equation, as a function of α

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 24 -

Are such things in nature?

Fractal nature of spectra for electron on a lattice in a homogeneous magnetic field was conjectured by [Azbel’64] but it caught the imagination

• nly after Hofstadter made the structure visible
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 25 -

Are such things in nature?

Fractal nature of spectra for electron on a lattice in a homogeneous magnetic field was conjectured by [Azbel’64] but it caught the imagination

• nly after Hofstadter made the structure visible

It triggered a long and fruitful mathematical quest culminating by the proof of the Ten Martini Conjecture by Avila and Jitomirskaya in 2009, that is that the spectrum for an irrational field is a Cantor set

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 25 -

Are such things in nature?

Fractal nature of spectra for electron on a lattice in a homogeneous magnetic field was conjectured by [Azbel’64] but it caught the imagination

• nly after Hofstadter made the structure visible

It triggered a long and fruitful mathematical quest culminating by the proof of the Ten Martini Conjecture by Avila and Jitomirskaya in 2009, that is that the spectrum for an irrational field is a Cantor set On the physical side, the effect remained theoretical for a long time and thought of in terms of the mentioned setting, with lattice and and a homogeneous field providing the needed two length scales, generically incommensurable, from the lattice spacing and the cyclotron radius

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 25 -

Are such things in nature?

Fractal nature of spectra for electron on a lattice in a homogeneous magnetic field was conjectured by [Azbel’64] but it caught the imagination

• nly after Hofstadter made the structure visible

It triggered a long and fruitful mathematical quest culminating by the proof of the Ten Martini Conjecture by Avila and Jitomirskaya in 2009, that is that the spectrum for an irrational field is a Cantor set On the physical side, the effect remained theoretical for a long time and thought of in terms of the mentioned setting, with lattice and and a homogeneous field providing the needed two length scales, generically incommensurable, from the lattice spacing and the cyclotron radius The first experimental demonstration of such a spectral character was done instead in a microwave waveguide system with suitably placed

• bstacles simulating the almost Mathieu relation [K¨

uhl et al’98]

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 25 -

Are such things in nature?

Fractal nature of spectra for electron on a lattice in a homogeneous magnetic field was conjectured by [Azbel’64] but it caught the imagination

• nly after Hofstadter made the structure visible

It triggered a long and fruitful mathematical quest culminating by the proof of the Ten Martini Conjecture by Avila and Jitomirskaya in 2009, that is that the spectrum for an irrational field is a Cantor set On the physical side, the effect remained theoretical for a long time and thought of in terms of the mentioned setting, with lattice and and a homogeneous field providing the needed two length scales, generically incommensurable, from the lattice spacing and the cyclotron radius The first experimental demonstration of such a spectral character was done instead in a microwave waveguide system with suitably placed

• bstacles simulating the almost Mathieu relation [K¨

uhl et al’98] Only recently an experimental realization of the original concept was achieved using a graphene lattice [Dean et al’13], [Ponomarenko’13]

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 25 -

Globally non-constant magnetic field

Our goal is now to investigate whether a similar effect can be seen in a ‘one-dimensional’ system. The coupling constant will be from now on denoted γ! To his aim we again employ duality

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 26 -

Globally non-constant magnetic field

Our goal is now to investigate whether a similar effect can be seen in a ‘one-dimensional’ system. The coupling constant will be from now on denoted γ! To his aim we again employ duality However, the above version dealing with weak solutions is not sufficient, we need a stronger one proved in [Pankrashkin’13] using boundary triples

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 26 -

Globally non-constant magnetic field

Our goal is now to investigate whether a similar effect can be seen in a ‘one-dimensional’ system. The coupling constant will be from now on denoted γ! To his aim we again employ duality However, the above version dealing with weak solutions is not sufficient, we need a stronger one proved in [Pankrashkin’13] using boundary triples We exclude the Dirichlet eigenvalues, σD = {k2 : k ∈ N}, and introduce s(x; z) = sin(x√z)

√z

for z = 0, x for z = 0, and c(x; z) = cos(x√z)

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 26 -

Globally non-constant magnetic field

Our goal is now to investigate whether a similar effect can be seen in a ‘one-dimensional’ system. The coupling constant will be from now on denoted γ! To his aim we again employ duality However, the above version dealing with weak solutions is not sufficient, we need a stronger one proved in [Pankrashkin’13] using boundary triples We exclude the Dirichlet eigenvalues, σD = {k2 : k ∈ N}, and introduce s(x; z) = sin(x√z)

√z

for z = 0, x for z = 0, and c(x; z) = cos(x√z)

Theorem (after Pankrashkin’13)

For any interval J ⊂ R \ σD, the operator (Hγ,A)J is unitarily equivalent to the pre-image η(−1) (LA)η(J)

• , where LA is the operator on ℓ2(Z)

acting as (LAqϕ)j = 2 cos(Ajπ)ϕj+1 + 2 cos(Aj−1π)ϕj−1 and η(z) := γs(π; z) + 2c(π; z) + 2s′(π; z)

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 26 -

Non-constant magnetic field, continued

Corollary

The spectrum of −∆γ,A is bounded from below and can be decomposed into the discrete set σD = {n2| n ∈ N} of infinitely degenerate eigenvalues and the part σLA determined by LA, σ(−∆γ,A) = σp ∪ σLA, where σLA can be written as the union σLA =

∞

• n=0

σn with σn = η(−1) σ(LA)

• ∩ In for n ≥ 0, In = η(−1)

[−4, 4]

• ∩
• n2, (n + 1)2

for n > 0, and I0 = η(−1) [−4, 4]

• ∩
• − ∞, 1
• .
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 27 -

Non-constant magnetic field, continued

Corollary

The spectrum of −∆γ,A is bounded from below and can be decomposed into the discrete set σD = {n2| n ∈ N} of infinitely degenerate eigenvalues and the part σLA determined by LA, σ(−∆γ,A) = σp ∪ σLA, where σLA can be written as the union σLA =

∞

• n=0

σn with σn = η(−1) σ(LA)

• ∩ In for n ≥ 0, In = η(−1)

[−4, 4]

• ∩
• n2, (n + 1)2

for n > 0, and I0 = η(−1) [−4, 4]

• ∩
• − ∞, 1
• .

When γ = 0, the spectrum has always gaps between the σn’s. For γ > 0, the spectrum is positive. For γ < −8π, the spectrum has a negative part and does not contain zero. Finally, 0 ∈ σ(−∆γ,A) holdsif and only if γπ + 4 ∈ σ(LA).

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 27 -

Non-constant magnetic field, continued

Corollary

The spectrum of −∆γ,A is bounded from below and can be decomposed into the discrete set σD = {n2| n ∈ N} of infinitely degenerate eigenvalues and the part σLA determined by LA, σ(−∆γ,A) = σp ∪ σLA, where σLA can be written as the union σLA =

∞

• n=0

σn with σn = η(−1) σ(LA)

• ∩ In for n ≥ 0, In = η(−1)

[−4, 4]

• ∩
• n2, (n + 1)2

for n > 0, and I0 = η(−1) [−4, 4]

• ∩
• − ∞, 1
• .

When γ = 0, the spectrum has always gaps between the σn’s. For γ > 0, the spectrum is positive. For γ < −8π, the spectrum has a negative part and does not contain zero. Finally, 0 ∈ σ(−∆γ,A) holdsif and only if γπ + 4 ∈ σ(LA). Remark: In general, the σn’s may very different from absolutely continuous spectral bands!

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 27 -

A linear field growth

Suppose now that Aj = αj + θ holds for some α, θ ∈ R and every j ∈ Z. We denote the corresponding operator LA by Lα,θ, i.e. (Lα,θϕ)j = 2 cos

• π(αj + θ)
• ϕj+1 + 2 cos
• π(αj − α + θ)
• ϕj−1

for all j ∈ Z. The rational case, α = p/q, is easily dealt with.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 28 -

A linear field growth

Suppose now that Aj = αj + θ holds for some α, θ ∈ R and every j ∈ Z. We denote the corresponding operator LA by Lα,θ, i.e. (Lα,θϕ)j = 2 cos

• π(αj + θ)
• ϕj+1 + 2 cos
• π(αj − α + θ)
• ϕj−1

for all j ∈ Z. The rational case, α = p/q, is easily dealt with.

Proposition

Assume that α = p/q, where p and q are relatively prime. Then (a) If αj + θ + 1

2 /

∈ Z for all j = 0, . . . , q − 1, then Lα,θ has purely ac spectrum that consists of q closed intervals possibly touching at the

• endpoints. In particular, σ(Lα,θ) =
• − 4| cos(πθ)|, 4| cos(πθ)|
• holds if

q = 1.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 28 -

A linear field growth

Suppose now that Aj = αj + θ holds for some α, θ ∈ R and every j ∈ Z. We denote the corresponding operator LA by Lα,θ, i.e. (Lα,θϕ)j = 2 cos

• π(αj + θ)
• ϕj+1 + 2 cos
• π(αj − α + θ)
• ϕj−1

for all j ∈ Z. The rational case, α = p/q, is easily dealt with.

Proposition

Assume that α = p/q, where p and q are relatively prime. Then (a) If αj + θ + 1

2 /

∈ Z for all j = 0, . . . , q − 1, then Lα,θ has purely ac spectrum that consists of q closed intervals possibly touching at the

• endpoints. In particular, σ(Lα,θ) =
• − 4| cos(πθ)|, 4| cos(πθ)|
• holds if

q = 1. (b) If αj + θ + 1

2 ∈ Z for some j = 0, . . . , q − 1, then the spectrum of

Lα,θ is of pure point type consisting of q distinct eigenvalues of infinite

• degeneracy. In particular, σ(Lα,θ) = {0} holds if q = 1.
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 28 -

An irrational slope

On the other hand, if α / ∈ Q the spectrum of Lα,θ is closely related to that of the almost Mathieu operator Hα,λ,θ in the critical situation, λ = 2, acting as

• Hα,θ,λϕ
• j = ϕj+1 + ϕj−1 + λ cos(2παj + θ)ϕj

for any ϕ ∈ ℓ2(Z) and all j ∈ Z.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 29 -

An irrational slope

On the other hand, if α / ∈ Q the spectrum of Lα,θ is closely related to that of the almost Mathieu operator Hα,λ,θ in the critical situation, λ = 2, acting as

• Hα,θ,λϕ
• j = ϕj+1 + ϕj−1 + λ cos(2παj + θ)ϕj

for any ϕ ∈ ℓ2(Z) and all j ∈ Z. From the mentioned deep results of Avila, Jitomirskaya, and Krikorian we know that for any α / ∈ Q, the spectrum of Hα,2,θ does not depend on θ and it is a Cantor set of Lebesgue measure zero

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 29 -

An irrational slope

On the other hand, if α / ∈ Q the spectrum of Lα,θ is closely related to that of the almost Mathieu operator Hα,λ,θ in the critical situation, λ = 2, acting as

• Hα,θ,λϕ
• j = ϕj+1 + ϕj−1 + λ cos(2παj + θ)ϕj

for any ϕ ∈ ℓ2(Z) and all j ∈ Z. From the mentioned deep results of Avila, Jitomirskaya, and Krikorian we know that for any α / ∈ Q, the spectrum of Hα,2,θ does not depend on θ and it is a Cantor set of Lebesgue measure zero In the same way as in [Shubin’94] one can demonstrate an unitary equivalence which means, in particular, that the spectra of Hα,θ,2 and Lα,θ coincide

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 29 -

An irrational slope

On the other hand, if α / ∈ Q the spectrum of Lα,θ is closely related to that of the almost Mathieu operator Hα,λ,θ in the critical situation, λ = 2, acting as

• Hα,θ,λϕ
• j = ϕj+1 + ϕj−1 + λ cos(2παj + θ)ϕj

for any ϕ ∈ ℓ2(Z) and all j ∈ Z. From the mentioned deep results of Avila, Jitomirskaya, and Krikorian we know that for any α / ∈ Q, the spectrum of Hα,2,θ does not depend on θ and it is a Cantor set of Lebesgue measure zero In the same way as in [Shubin’94] one can demonstrate an unitary equivalence which means, in particular, that the spectra of Hα,θ,2 and Lα,θ coincide Combining all these results we can describe the spectrum of our original

• perator in case the magnetic field varies linearly along the chain
• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 29 -

The linear-field spectrum

Theorem (E-Vaˇ sata’16)

Let Aj = αj + θ for some α, θ ∈ R and every j ∈ Z. Then for the spectrum σ(−∆γ,A) the following holds: (a) If α, θ ∈ Z and γ = 0, then σ(−∆γ,A) = σac(−∆γ,A) ∪ σpp(−∆γ,A) where σac(−∆γ,A) = [0, ∞) and σpp(−∆γ,A) = {n2| n ∈ N}.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 30 -

The linear-field spectrum

Theorem (E-Vaˇ sata’16)

Let Aj = αj + θ for some α, θ ∈ R and every j ∈ Z. Then for the spectrum σ(−∆γ,A) the following holds: (a) If α, θ ∈ Z and γ = 0, then σ(−∆γ,A) = σac(−∆γ,A) ∪ σpp(−∆γ,A) where σac(−∆γ,A) = [0, ∞) and σpp(−∆γ,A) = {n2| n ∈ N}. (b) If α = p/q with p and q relatively prime, αj + θ + 1

2 /

∈ Z for all j = 0, . . . , q − 1 and assumptions of (a) do not hold, then −∆γ,A has infinitely degenerate ev’s at the points of {n2| n ∈ N} and an ac part

• f the spectrum in each interval (−∞, 1) and
• n2, (n + 1)2

, n ∈ N consisting of q closed intervals possibly touching at the endpoints.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 30 -

The linear-field spectrum

Theorem (E-Vaˇ sata’16)

Let Aj = αj + θ for some α, θ ∈ R and every j ∈ Z. Then for the spectrum σ(−∆γ,A) the following holds: (a) If α, θ ∈ Z and γ = 0, then σ(−∆γ,A) = σac(−∆γ,A) ∪ σpp(−∆γ,A) where σac(−∆γ,A) = [0, ∞) and σpp(−∆γ,A) = {n2| n ∈ N}. (b) If α = p/q with p and q relatively prime, αj + θ + 1

2 /

∈ Z for all j = 0, . . . , q − 1 and assumptions of (a) do not hold, then −∆γ,A has infinitely degenerate ev’s at the points of {n2| n ∈ N} and an ac part

• f the spectrum in each interval (−∞, 1) and
• n2, (n + 1)2

, n ∈ N consisting of q closed intervals possibly touching at the endpoints. (c) If α = p/q, where p and q are relatively prime, and αj + θ + 1

2 ∈ Z for

some j = 0, . . . , q − 1, then the spectrum −∆γ,A is of pure pure type and such that in each interval (−∞, 1) and

• n2, (n + 1)2

, n ∈ N there are exactly q distinct eigenvalues and the remaining eigenvalues form the set {n2| n ∈ N}. All the eigenvalues are infinitely degenerate.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 30 -

The linear-field spectrum, continued

Theorem (E-Vaˇ sata’16, cont’d)

(d) If α / ∈ Q, then σ(−∆γ,A) does not depend on θ and it is a disjoint union of the isolated-point family {n2| n ∈ N} and Cantor sets, one inside each interval (−∞, 1) and

• n2, (n + 1)2

, n ∈ N. Moreover, the overall Lebesgue measure of σ(−∆γ,A) is zero.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 31 -

The linear-field spectrum, continued

Theorem (E-Vaˇ sata’16, cont’d)

(d) If α / ∈ Q, then σ(−∆γ,A) does not depend on θ and it is a disjoint union of the isolated-point family {n2| n ∈ N} and Cantor sets, one inside each interval (−∞, 1) and

• n2, (n + 1)2

, n ∈ N. Moreover, the overall Lebesgue measure of σ(−∆γ,A) is zero. Using a fresh result of [Last-Shamis’16] we can also show

Proposition

Let Aj = αj + θ for some α, θ ∈ R and every j ∈ Z. There exist a dense Gδ set of the slopes α for which, and all θ, the Haussdorff dimension dimH σ(−∆γ,A) = 0

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 31 -

The linear-field spectrum, continued

Theorem (E-Vaˇ sata’16, cont’d)

(d) If α / ∈ Q, then σ(−∆γ,A) does not depend on θ and it is a disjoint union of the isolated-point family {n2| n ∈ N} and Cantor sets, one inside each interval (−∞, 1) and

• n2, (n + 1)2

, n ∈ N. Moreover, the overall Lebesgue measure of σ(−∆γ,A) is zero. Using a fresh result of [Last-Shamis’16] we can also show

Proposition

Let Aj = αj + θ for some α, θ ∈ R and every j ∈ Z. There exist a dense Gδ set of the slopes α for which, and all θ, the Haussdorff dimension dimH σ(−∆γ,A) = 0 Remark: If you regard a linear field unphysical, you may either view it as an idealization or to replace it a quasiperiodic function with the same slope leading to the same result.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 31 -

The talk was based on

[EM15] P.E., Stepan Manko: Spectra of magnetic chain graphs: coupling constant perturbations, J. Phys. A: Math. Theor. 48 (2015), 125302 (20pp) [EM17] P.E., Stepan Manko: Spectral properties of magnetic chain graphs, Ann.

• H. Poincar´

e (2017), to appear [EV16] P.E., Daniel Vaˇ sata: Cantor spectra of magnetic chain graphs, arXiv:1611.04559

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 32 -

The talk was based on

[EM15] P.E., Stepan Manko: Spectra of magnetic chain graphs: coupling constant perturbations, J. Phys. A: Math. Theor. 48 (2015), 125302 (20pp) [EM17] P.E., Stepan Manko: Spectral properties of magnetic chain graphs, Ann.

• H. Poincar´

e (2017), to appear [EV16] P.E., Daniel Vaˇ sata: Cantor spectra of magnetic chain graphs, arXiv:1611.04559

as well as the other papers mentioned in the course of the presentation.

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 32 -

It remains to say

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 33 -

It remains to say

Thank you for your attention!

• P. Exner: Spectra of magnetic chain graphs

OTIND 2016 Vienna December 19, 2016

• 33 -