The Spectrum of Magnetic Schrödinger Operators on a Graph with Periodic Structure

Higuchi, Yusuke; Shirai, Tomoyuki

doi:10.1006/jfan.1999.3478

Cited by 25 publications

(36 citation statements)

References 12 publications

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“…The group G acts on the graph by the left translations g : v → gv. The properties of the Laplacian depend on G and can be rather exotic, even the band spectrum can be guaranteed only under additional assumptions like the positivity of the Kadison constant [20,29,36]. Nevertheless, theorem 6 holds and shows the location of gaps also in this case.…”

Section: Remarkmentioning

confidence: 98%

Spectra of Schrödinger Operators on Equilateral Quantum Graphs

Pankrashkin

2006

Lett Math Phys

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We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum graph is the preimage of the combinatorial spectrum under a certain entire function. Using this correspondence we show that that the number of gaps in the spectrum of the Schrödinger operators admits an estimate from below in terms of the Hill operator independently of the graph structure.

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Section: Remarkmentioning

confidence: 98%

Spectra of Schrödinger Operators on Equilateral Quantum Graphs

Pankrashkin

2006

Lett Math Phys

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“…2) Note that the decomposition of the discrete magnetic Schrödinger operators on periodic graphs into the constant fiber direct integral (2.2) (without an exact form of fiber operators) was discussed by Higuchi and Shirai [HS99a]. The precise form of the fiber Laplacian ∆ α (ϑ) defined by (2.4) is important to study spectral properties of the magnetic Laplacians and Schrödinger operators acting on periodic graphs (see the proof of Theorems 2.3 -2.5).…”

Section: Floquet Decomposition Of Schrödinger Operators We Introducementioning

confidence: 99%

“…For example, discrete magnetic Schrödinger operators on periodic graphs were also considered in [HS99a], [HS99b]. Higuchi and Shirai [HS99a] obtained the relationship between the spectrum of the discrete magnetic Schrödinger operator on a periodic graph and that on the corresponding fundamental graph. Also they proved the analyticity of the bottom of the spectrum with respect to the magnetic flow and computed the second derivative of the bottom of the spectrum and represented it in terms of geometry of the graph.…”

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confidence: 99%

Magnetic Schrödinger operators on periodic discrete graphs

Korotyaev

Saburova

2017

Journal of Functional Analysis

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Abstract. We consider magnetic Schrödinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the spectrum in terms of the Betti numbers and show that these estimates become identities for specific graphs. We estimate a variation of the spectrum of the Schrödinger operators under a perturbation by a magnetic field in terms of magnetic fluxes. The proof is based on Floquet theory and a precise representation of fiber magnetic Schrödinger operators constructed in the paper.

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“…Specifically, eigenvalue behaviours of discrete Schrödinger operators on l 2 Z d are discussed in e.g. [2,8,10,12,13,15]. However, for discrete non-local Schrödinger operators only few results are known.…”

Section: Non-local Discrete Schrödinger Operators On Latticementioning

confidence: 99%

The spectrum of non-local discrete Schrödinger operators with a δ-potential

Hiroshima

Lőrinczi

2014

Pac. J. Math. Ind.

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The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schrödinger operators with a δ-potential. These operators arise by replacing the discrete Laplacian by a strictly increasing C 1 -function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete Schrödinger operator these behaviours are the same no matter which end of the continuous spectrum is considered, an asymmetry occurs for the non-local cases. A classification with respect to the spectral edge behaviour is also offered.

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The Spectrum of Magnetic Schrödinger Operators on a Graph with Periodic Structure

Cited by 25 publications

References 12 publications

Spectra of Schrödinger Operators on Equilateral Quantum Graphs

Spectra of Schrödinger Operators on Equilateral Quantum Graphs

Magnetic Schrödinger operators on periodic discrete graphs

The spectrum of non-local discrete Schrödinger operators with a δ-potential

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