Analysis of the dispersion equation for the Schrödinger operator on periodic metric graphs

Oleinik, V. L.; Павлов, Б. С.; Sibirev, N. V.

doi:10.1088/0959-7174/14/2/006

Cited by 11 publications

(14 citation statements)

References 16 publications

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“…This theory also holds in the quantum graph case, (see, e.g., [14,15,30,35,49] u(x + p 1 e 1 + p 2 e 2 ) = e ip·θ u(x) = e i(p 1 θ 1 +p 2 θ 2 ) u(x) (3.1) for any vector p = (p 1 , p 2 ) ∈ Z 2 and any x ∈ G. Due to the conditions (3.1), functions u are uniquely determined by their restrictions to the fundamental domain W . Then conditions (2.4) and (3.1) reduce to…”

Section: Spectra Of Graphene Operatorsmentioning

confidence: 84%

On the Spectra of Carbon Nano-Structures

Kuchment

Post

2007

Commun. Math. Phys.

158

218

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Section: Spectra Of Graphene Operatorsmentioning

confidence: 84%

On the Spectra of Carbon Nano-Structures

Kuchment

Post

2007

Commun. Math. Phys.

158

218

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“…In the particular case of graphs, this theory is also described, for instance, in [16], [34]- [40], [49].…”

Section: The Main Graph Operatorsmentioning

confidence: 99%

“…The Hilbert space where the operator acts is Here, f ′ e (v) denotes the outgoing derivative of f at v along the edge e. It is well known (e.g., [16,40,43,49]) and easy to check that Floquet theory applies to the quantum graph case. In particular, the spectrum σ(H) coincides with the union over the Brillouin zone B of the spectra of Floquet Hamiltonians H(k) .16), and with the additional cyclic (Bloch, Floquet) condition (2.3):…”

Section: Quantum Graph Casementioning

confidence: 99%

On occurrence of spectral edges for periodic operators inside the Brillouin zone

Harrison

Kuchment

Sobolev

et al. 2007

J. Phys. A: Math. Theor.

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Abstract. The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schrödinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of "corner" high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community.In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiply-periodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a "generic" case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds.

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“…When studying scattering on one-dimensional quantum networks we obtain the scattered waves of the full Hamiltonian on the network by matching solutions of the boundary problems on the interior part of the network with exponentials, or, more generally, with Jost functions in the semi-infinite wires [29], [15], [37], [83], [84] and [64].…”

Section: Spectral Properties Of L λ L λmentioning

confidence: 99%

“…In the one-dimensional case, each splitting in finite number of points is a Glazman splitting. Periodic one-dimensional networks based on quantum graphs are studied with the aid of the standard Dirichlet-to-Neumann map in [64].…”

Section: Spectral Properties Of L λ L λmentioning

confidence: 99%

Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix‐functions

Mikhailova

Павлов

Prokhorov

2007

Mathematische Nachrichten

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We consider the spectral problem for the Schrödinger operator L on a quantum network, represented as a union of a finite number of quantum wells and finite or semi-infinite straight quantum wires connecting them to each other and to infinity. The absolutely continuous spectrum of the Schrödinger operator, with real piecewisecontinuous potential on the wells and constant potential on the wires, has a band structure defined by the geometry and the potentials on the semi-infinite wires. We split the Schrödinger operator, depending on the Fermi level Λ, into an orthogonal sum of two operators L → lΛ ⊕ LΛ, which have the complementary branches of the absolutely continuous spectra coincident, counting multiplicity, respectively with the unionsof absolutely continuous branches σ = [λ σ , ∞) of open and closed channels in the semi-infinite wires, with thresholds λσ < Λ, λσ > Λ, respectively. This enables us to reduce the problem of evaluation of the scattering matrix on the network to the solution of a finite linear system, involving the Dirichlet-to-Neumann map of the intermediate Hamiltonian LΛ. We calculate the intermediate Dirichlet-to-Neumann map approximately, for thin networks, based on the classical DN-map of the relevant Schrödinger operator on the compact part of the network. To this end we develop a modified analytic perturbation procedure for meromorphic operatorfunctions with small one-dimensional residues.

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Analysis of the dispersion equation for the Schrödinger operator on periodic metric graphs

Cited by 11 publications

References 16 publications

On the Spectra of Carbon Nano-Structures

On the Spectra of Carbon Nano-Structures

On occurrence of spectral edges for periodic operators inside the Brillouin zone

Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix‐functions

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