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 - 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 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 L 2 (Γ). For simplicity we use units in which ℏ = 2 m = e = 1, where e is the particle charge 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 L 2 (Γ). For simplicity we use units in which ℏ = 2 m = 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 H 2 loc (Γ) satisfying the δ -coupling conditions n � ψ i (0) = ψ j (0) =: ψ (0) , i , j ∈ n , D ψ 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, 2016 - 4 -
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 = 2 m = 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 H 2 loc (Γ) satisfying the δ -coupling conditions n � ψ i (0) = ψ j (0) =: ψ (0) , i , j ∈ n , D ψ 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, 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 = { 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, 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 = { 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, 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 = { 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, 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 = { 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 2 π , 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 - 6 -
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