Spectra of magnetic chain graphs

Transcription

1 Spectra of magnetic chain graphs Pavel Exner Doppler Institute for Mathematical Physics and Applied Mathematics Prague in collaboration with Stepan Manko and Daniel Vašata 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,

2 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,

3 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,

4 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,

5 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,

6 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,

7 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,

8 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,

9 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,

10 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,

11 The Hamiltonian The graph Γ is naturally parametrized by arc lengths of its edges, the corresponding state Hilbert space is L 2 (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,

12 The Hamiltonian The graph Γ is naturally parametrized by arc lengths of its edges, the corresponding state Hilbert space is L 2 (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, ψ j D 2 ψ j on each graph link, where D := i A. Its domain consists of all functions from the Sobolev space Hloc 2 (Γ) satisfying the δ-coupling conditions ψ i (0) = ψ j (0) =: ψ(0), i, j n, n Dψ i (0) = α ψ(0), i=1 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,

13 The Hamiltonian The graph Γ is naturally parametrized by arc lengths of its edges, the corresponding state Hilbert space is L 2 (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, ψ j D 2 ψ j on each graph link, where D := i A. Its domain consists of all functions from the Sobolev space Hloc 2 (Γ) satisfying the δ-coupling conditions ψ i (0) = ψ j (0) =: ψ(0), i, j n, n Dψ i (0) = α ψ(0), i=1 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,

14 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,

15 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 = {α j } j Z and A = {A j } 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,

16 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 = {α j } j Z and A = {A j } 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,

17 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 = {α j } j Z and A = {A j } 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 α j = which corresponds to Dirichlet decoupling of the chain in the particular vertex P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

18 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 = {α j } j Z and A = {A j } 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 α j = which corresponds to Dirichlet decoupling of the chain in the particular vertex Without loss of generality we may suppose that the circumference of 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,

19 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,

20 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 := k 2 0 in the form ψ L (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,

21 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 := k 2 0 in the form ψ L (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,

22 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 := k 2 0 in the form ψ L (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,

23 The fully periodic case, continued This yields quadratic equation for the phase factor e iθ, specifically sin kπ cos Aπ(e 2iθ 2ξ(k)e iθ + 1) = 0 with ξ(k) := 1 ( cos kπ + α ) cos Aπ 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,

24 The fully periodic case, continued This yields quadratic equation for the phase factor e iθ, specifically sin kπ cos Aπ(e 2iθ 2ξ(k)e iθ + 1) = 0 with ξ(k) := 1 ( cos kπ + α ) cos Aπ 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, otherwise k 2 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,

25 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 of infinitely degenerate ev s {k 2 R: ξ(k) = 0} and {k 2 R: k N}. On the other hand, if A 1 2 / Z, the spectrum of α consists of infinitely degenerate eigenvalues k 2 with k N, and absolutely continuous spectral bands. Each of these bands except the first one is contained in an interval (n 2, (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, otherwise it contains the point k 2 = 0. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

26 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 of infinitely degenerate ev s {k 2 R: ξ(k) = 0} and {k 2 R: k N}. On the other hand, if A 1 2 / Z, the spectrum of α consists of infinitely degenerate eigenvalues k 2 with k N, and absolutely continuous spectral bands. Each of these bands except the first one is contained in an interval (n 2, (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, otherwise it contains the point k 2 = 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 2 α P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

27 Determining the spectral bands 4 2 η γ > 0 γ = 0 γ ( 8/π, 0) γ < 8/π i 1 2 i z ir z R 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,

28 Determining the spectral bands 4 2 η γ > 0 γ = 0 γ ( 8/π, 0) γ < 8/π i 1 2 i z ir z R 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,

29 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,

30 Duality Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schrödinger (differential) equation in question to solutions of a suitable difference equation P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

31 Duality Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schrödinger (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,

32 Duality Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schrödinger (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 ( ) ψ(x, k) ( α,a k 2 ) = 0 ϕ(x, k) with the components referring to the upper and lower part of Γ, P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

33 Duality Let us pass to perturbations of the periodic system. We introduce first a useful trick relating solutions of the Schrödinger (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 ( ) ψ(x, k) ( α,a k 2 ) = 0 ϕ(x, k) with the components referring to the upper and lower part of Γ, on the other hand the difference one ψ j+1 (k) + ψ j 1 (k) = ξ j (k)ψ j (k), k K, where ψ j (k) := ψ(jπ, k). These two are intimately related. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

34 Duality, continued Theorem Let α j R, then any solution ψ(, k) ϕ(, k) with k 2 R and k K satisfies the difference equation, and conversely, the latter defines via ( ) [ ψ(x, k) = e ia(x jπ) ψ j (k) cos k(x jπ) ϕ(x, k) ] +(ψ j+1 (k)e ±iaπ sin k(x jπ) ψ j (k) cos kπ), x ( jπ, (j + 1)π ), sin kπ solutions to the former satisfying the δ-coupling conditions. In addition, the former belongs to L p (Γ) if and only if {ψ j (k)} j Z l p (Z), the claim being true for both p {2, }. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

35 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 α j = α + γ j, j M := {1,..., m}, α j = α, j Z \ M P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

36 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 α j = α + γ j, j M := {1,..., m}, α j = 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,

37 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 α j = α + γ j, j M := {1,..., m}, α j = 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, Φ j+1 (k) = N j (k)φ j (k), j Z,. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

38 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 α j = α + γ j, j M := {1,..., m}, α j = 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, Φ j+1 (k) = N j (k)φ j (k), j Z,. ( Outside the perturbation support the matrix N j (k) = 2ξ j (k) independent of j and we need that it has an eigenvalue less than one to ensure an exponential decay of the solution ) is P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

39 Coupling constant perturbations, continued To state the result, let us introduce P 0 (k) = 1, P 1 (k) = 2ξ 1 (k), P m (k) = 2ξ m (k)p m 1 (k) P m 2 (k), Q 0 (k) = 0, Q 1 (k) = 1, Q m (k) = 2ξ m (k)q m 1 (k) Q m 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,

40 Coupling constant perturbations, continued To state the result, let us introduce P 0 (k) = 1, P 1 (k) = 2ξ 1 (k), P m (k) = 2ξ m (k)p m 1 (k) P m 2 (k), Q 0 (k) = 0, Q 1 (k) = 1, Q m (k) = 2ξ m (k)q m 1 (k) Q m 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) k 2 R \ σ( α,a ) is an eigenvalue of α+γ,a iff for this k we have Q m 1 (k)λ(k) 2 (P m 1 (k) + Q m (k))λ(k) + P m (k) = 0 P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

41 Example: a single impurity In particular, if γ = {..., 0, γ 1, 0,...}, we have Proposition Let A / Z. The essential spectrum of α+γ,a coincides with that of α. If γ 1 < 0 there is precisely one simple impurity state in every odd gap, on the other hand, for γ 1 > 0 there is precisely one simple impurity state in every even gap. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

42 Example: a single impurity In particular, if γ = {..., 0, γ 1, 0,...}, we have Proposition Let A / Z. The essential spectrum of α+γ,a coincides with that of α. If γ 1 < 0 there is precisely one simple impurity state in every odd gap, on the other hand, for γ 1 > 0 there is precisely one simple impurity state in every even gap. The energy k 2 vs. γ 1 = 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,

43 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 γ j < 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 γ j > 0, then in the limit ε 0 there is exactly one simple impurity state in every even gap. The corresponding eigenvalue k 2 n,ε is given by the following asymptotic expansion k n,ε = k n + ( 1) 2n+1 K n ε 2 + O(ε 2 ), ε 0, n N. Here k1 2 < k2 2 <... are the non-dirichlet gap ends coming from the solutions to the equation ξ(k) = 1, and K n = sin k n π ( j M γ ) 2 j cos Aπ(32(k n π) 2 8αk n π cot k n π + 8α), n N. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

44 Distant perturbations Consider the case of two impurities at large distances: we change the coupling constants at two arbitrary but fixed points into α + γ 1 and α + γ 2 ; 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,

45 Distant perturbations Consider the case of two impurities at large distances: we change the coupling constants at two arbitrary but fixed points into α + γ 1 and α + γ 2 ; 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,

46 Distant perturbations Consider the case of two impurities at large distances: we change the coupling constants at two arbitrary but fixed points into α + γ 1 and α + γ 2 ; 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 γ 1 γ 2 < 0, then any for sufficiently large n, the operator α,a,n has precisely one simple impurity state in every gap of its essential spectrum. If γ 1 and γ 2 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 γ 1 = γ 2, 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,

47 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 = {A j } j Z, P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

48 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 = {A j } j Z, the same my be true for the ring (half-)perimeters, l = {l j } j Z, etc. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

49 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 = {A j } j Z, the same my be true for the ring (half-)perimeters, l = {l j } j Z, etc. What is important, the above duality holds again, with the difference relation being sin(kl j 1 ) cos(a j l j )ψ j+1 (k) + sin(kl j ) cos(a j 1 l j 1 )ψ j 1 (k) ( α ) = 2k sin(kl j 1) sin(kl j ) + sin k(l j 1 + l j ) ψ j (k), k K, where ψ j (k) := ψ(x j, k), P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

50 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 = {A j } j Z, the same my be true for the ring (half-)perimeters, l = {l j } j Z, etc. What is important, the above duality holds again, with the difference relation being sin(kl j 1 ) cos(a j l j )ψ j+1 (k) + sin(kl j ) cos(a j 1 l j 1 )ψ j 1 (k) ( α ) = 2k sin(kl j 1) sin(kl j ) + sin k(l j 1 + l j ) ψ j (k), k K, where ψ j (k) := ψ(x j, k), and the reconstruction formula becomes ( ) [ ψ(x, k) = e ia(x x j ) ψ j (k) cos k(x x j ) ϕ(x, k) +(ψ j+1 (k)e ±ial j ψ j (k) cos kl j ) sin k(x x j) sin kl j ], x ( x j, x j+1 ), P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

51 A remark: probability current The probability current on the jth circle in is easily found to be J ψ (k) = 2k { Re ψj (k)(re ψ j+1 (k) sin A j π + Im ψ j+1 (k) cos A j π) sin kπ Im ψ j (k)(re ψ j+1 (k) cos A j π Im ψ j+1 (k) sin A j π) } and a similar expression for the current J ϕ (k) on the lower part of the ring. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

52 A remark: probability current The probability current on the jth circle in is easily found to be J ψ (k) = 2k { Re ψj (k)(re ψ j+1 (k) sin A j π + Im ψ j+1 (k) cos A j π) sin kπ Im ψ j (k)(re ψ j+1 (k) cos A j π Im ψ j+1 (k) sin A j π) } and a similar expression for the current J ϕ (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,

53 A remark: probability current The probability current on the jth circle in is easily found to be J ψ (k) = 2k { Re ψj (k)(re ψ j+1 (k) sin A j π + Im ψ j+1 (k) cos A j π) sin kπ Im ψ j (k)(re ψ j+1 (k) cos A j π Im ψ j+1 (k) sin A j π) } and a similar expression for the current J ϕ (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,

54 A remark: probability current The probability current on the jth circle in is easily found to be J ψ (k) = 2k { Re ψj (k)(re ψ j+1 (k) sin A j π + Im ψ j+1 (k) cos A j π) sin kπ Im ψ j (k)(re ψ j+1 (k) cos A j π Im ψ j+1 (k) sin A j π) } and a similar expression for the current J ϕ (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,

55 A remark: probability current The probability current on the jth circle in is easily found to be J ψ (k) = 2k { Re ψj (k)(re ψ j+1 (k) sin A j π + Im ψ j+1 (k) cos A j π) sin kπ Im ψ j (k)(re ψ j+1 (k) cos A j π Im ψ j+1 (k) sin A j π) } and a similar expression for the current J ϕ (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 l 2 -sequences {ψ j (k)} j Z, the latter can be chosen real, whence the probability currents read as follows J ( ψ ϕ) (k) = ±2kψ j(k)ψ j+1 (k) sin A jπ sin kπ, and since the coefficients ψ j (k)ψ j+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,

56 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,

57 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 on neighboring rings, in particular, their spectral properties. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

58 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 on 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 A 1, A 2. For brevity, we denote the unperturbed operator A and the perturbed one A1,A 2, the coupling constant α and the half-perimeter are kept fixed. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

59 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 on 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 A 1, A 2. For brevity, we denote the unperturbed operator A and the perturbed one A1,A 2, the coupling constant α and the half-perimeter are kept fixed. Theorem (E-Manko 17) We have σ( A1,A 2 ) = σ( A ) = σ ess ( A ) unless the inequality (cos A 1 π) 2 + (cos A 2 π) 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,A 2 has precisely one simple eigenvalue in every gap of the essential spectrum. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

60 Magnetic perturbations, continued In particular, if the field is modified on a single ring, i.e. A 2 = A, we have a single simple eigenvalue in each gap provided cos A 1 π cos Aπ > 1, otherwise the spectrum does not change. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

61 Magnetic perturbations, continued In particular, if the field is modified on a single ring, i.e. A 2 = A, we have a single simple eigenvalue in each gap provided cos A 1 π cos Aπ > 1, otherwise 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,

62 Magnetic perturbations, continued In particular, if the field is modified on a single ring, i.e. A 2 = A, we have a single simple eigenvalue in each gap provided cos A 1 π cos Aπ > 1, otherwise 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,

63 Magnetic perturbations, continued In particular, if the field is modified on a single ring, i.e. A 2 = A, we have a single simple eigenvalue in each gap provided cos A 1 π cos Aπ > 1, otherwise 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,

64 Weak local perturbations Let now the parameters be of the form A + εa j and α + εα j 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,

65 Weak local perturbations Let now the parameters be of the form A + εa j and α + εα j 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 A j > 0. If n j=1 α j > 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 α j < 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 A j < 0. If n j=1 α j > 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 α j < 0, it has precisely one simple eigenvalue in every gap except possibly a finite number of even gaps of σ ess ( α,a ). P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

66 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,

67 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 α j = 0. The spectrum of ε coincides with that of α,a if the condition ( n ) sgn A j = sgn(cot Aπ) j=1 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,

68 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 α j = 0. The spectrum of ε coincides with that of α,a if the condition ( n ) sgn A j = sgn(cot Aπ) j=1 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,

69 The picture everybody knows representing the spectrum of the difference operator associated with the almost Mathieu equation u n+1 + u n 1 + 2λ cos(2π(ω + nα))u n = ɛu n for λ = 1, otherwise called Harper equation, as a function of α P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

70 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 only after Hofstadter made the structure visible P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

71 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 only 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,

72 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 only 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,

73 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 only 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 obstacles simulating the almost Mathieu relation [Kühl et al 98] P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

74 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 only 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 obstacles simulating the almost Mathieu relation [Kühl 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,

75 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,

76 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,

77 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, σ D = {k 2 : 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,

78 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, σ D = {k 2 : 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 \ σ D, the operator (H γ,a ) J is unitarily equivalent to the pre-image η ( 1)( (L A ) η(j) ), where LA is the operator on l 2 (Z) acting as (L A qϕ) j = 2 cos(a j π)ϕ j cos(a j 1 π)ϕ j 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,

79 Non-constant magnetic field, continued Corollary The spectrum of γ,a is bounded from below and can be decomposed into the discrete set σ D = {n 2 n N} of infinitely degenerate eigenvalues and the part σ LA determined by L A, σ( γ,a ) = σ p σ LA, where σ LA can be written as the union σ LA = with σ n = η ( 1)( σ(l A ) ) I n for n 0, I n = η ( 1)( [ 4, 4] ) ( n 2, (n + 1) 2) for n > 0, and I 0 = η ( 1)( [ 4, 4] ) (, 1 ). n=0 σ n P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

80 Non-constant magnetic field, continued Corollary The spectrum of γ,a is bounded from below and can be decomposed into the discrete set σ D = {n 2 n N} of infinitely degenerate eigenvalues and the part σ LA determined by L A, σ( γ,a ) = σ p σ LA, where σ LA can be written as the union σ LA = with σ n = η ( 1)( σ(l A ) ) I n for n 0, I n = η ( 1)( [ 4, 4] ) ( n 2, (n + 1) 2) for n > 0, and I 0 = η ( 1)( [ 4, 4] ) (, 1 ). When γ 0, the spectrum has always gaps between the σ n 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(l A ). n=0 σ n P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

81 Non-constant magnetic field, continued Corollary The spectrum of γ,a is bounded from below and can be decomposed into the discrete set σ D = {n 2 n N} of infinitely degenerate eigenvalues and the part σ LA determined by L A, σ( γ,a ) = σ p σ LA, where σ LA can be written as the union σ LA = with σ n = η ( 1)( σ(l A ) ) I n for n 0, I n = η ( 1)( [ 4, 4] ) ( n 2, (n + 1) 2) for n > 0, and I 0 = η ( 1)( [ 4, 4] ) (, 1 ). When γ 0, the spectrum has always gaps between the σ n 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(l A ). Remark: In general, the σ n s may very different from absolutely continuous spectral bands! P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19, n=0 σ n

82 A linear field growth Suppose now that A j = αj + θ holds for some α, θ R and every j Z. We denote the corresponding operator L A by L α,θ, i.e. (L α,θ ϕ) j = 2 cos ( π(αj + θ) ) ϕ j cos ( π(αj α + θ) ) ϕ j 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,

83 A linear field growth Suppose now that A j = αj + θ holds for some α, θ R and every j Z. We denote the corresponding operator L A by L α,θ, i.e. (L α,θ ϕ) j = 2 cos ( π(αj + θ) ) ϕ j cos ( π(αj α + θ) ) ϕ j 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 / Z for all j = 0,..., q 1, then L α,θ has purely ac spectrum that consists of q closed intervals possibly touching at the endpoints. In particular, σ(l α,θ ) = [ 4 cos(πθ), 4 cos(πθ) ] holds if q = 1. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

84 A linear field growth Suppose now that A j = αj + θ holds for some α, θ R and every j Z. We denote the corresponding operator L A by L α,θ, i.e. (L α,θ ϕ) j = 2 cos ( π(αj + θ) ) ϕ j cos ( π(αj α + θ) ) ϕ j 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 / Z for all j = 0,..., q 1, then L α,θ has purely ac spectrum that consists of q closed intervals possibly touching at the endpoints. In particular, σ(l α,θ ) = [ 4 cos(πθ), 4 cos(πθ) ] holds if q = 1. (b) If αj + θ Z for some j = 0,..., q 1, then the spectrum of L α,θ is of pure point type consisting of q distinct eigenvalues of infinite degeneracy. In particular, σ(l α,θ ) = {0} holds if q = 1. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

85 An irrational slope On the other hand, if α / Q the spectrum of L α,θ is closely related to that of the almost Mathieu operator H α,λ,θ in the critical situation, λ = 2, acting as ( Hα,θ,λ ϕ ) j = ϕ j+1 + ϕ j 1 + λ cos(2παj + θ)ϕ j for any ϕ l 2 (Z) and all j Z. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

86 An irrational slope On the other hand, if α / Q the spectrum of L α,θ is closely related to that of the almost Mathieu operator H α,λ,θ in the critical situation, λ = 2, acting as ( Hα,θ,λ ϕ ) j = ϕ j+1 + ϕ j 1 + λ cos(2παj + θ)ϕ j for any ϕ l 2 (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 H α,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,

87 An irrational slope On the other hand, if α / Q the spectrum of L α,θ is closely related to that of the almost Mathieu operator H α,λ,θ in the critical situation, λ = 2, acting as ( Hα,θ,λ ϕ ) j = ϕ j+1 + ϕ j 1 + λ cos(2παj + θ)ϕ j for any ϕ l 2 (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 H α,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 H α,θ,2 and L α,θ coincide P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

88 An irrational slope On the other hand, if α / Q the spectrum of L α,θ is closely related to that of the almost Mathieu operator H α,λ,θ in the critical situation, λ = 2, acting as ( Hα,θ,λ ϕ ) j = ϕ j+1 + ϕ j 1 + λ cos(2παj + θ)ϕ j for any ϕ l 2 (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 H α,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 H α,θ,2 and L α,θ coincide Combining all these results we can describe the spectrum of our original operator in case the magnetic field varies linearly along the chain P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

89 The linear-field spectrum Theorem (E-Vašata 16) Let A j = 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 ) = σ ac ( γ,a ) σ pp ( γ,a ) where σ ac ( γ,a ) = [0, ) and σ pp ( γ,a ) = {n 2 n N}. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,

90 The linear-field spectrum Theorem (E-Vašata 16) Let A j = 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 ) = σ ac ( γ,a ) σ pp ( γ,a ) where σ ac ( γ,a ) = [0, ) and σ pp ( γ,a ) = {n 2 n N}. (b) If α = p/q with p and q relatively prime, αj + θ / 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 {n 2 n N} and an ac part of the spectrum in each interval (, 1) and ( n 2, (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,

91 The linear-field spectrum Theorem (E-Vašata 16) Let A j = 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 ) = σ ac ( γ,a ) σ pp ( γ,a ) where σ ac ( γ,a ) = [0, ) and σ pp ( γ,a ) = {n 2 n N}. (b) If α = p/q with p and q relatively prime, αj + θ / 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 {n 2 n N} and an ac part of the spectrum in each interval (, 1) and ( n 2, (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 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 ( n 2, (n + 1) 2), n N there are exactly q distinct eigenvalues and the remaining eigenvalues form the set {n 2 n N}. All the eigenvalues are infinitely degenerate. P. Exner: Spectra of magnetic chain graphs OTIND 2016 Vienna December 19,