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• W3046065282 abstract "Abstract Let Γ be an arbitrary <?CDATA ${mathbb{Z}}^{n}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> -periodic metric graph, which does not coincide with a line. We consider the Hamiltonian <?CDATA ${mathcal{H}}_{varepsilon }$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msub> <mml:mrow> <mml:mi mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> </mml:math> on Γ with the action − ɛ −1 d 2 /d x 2 on its edges; here ɛ > 0 is a small parameter. Let <?CDATA $min mathbb{N}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant=double-struck>N</mml:mi> </mml:math> . We show that under a proper choice of vertex conditions the spectrum <?CDATA $sigma left({mathcal{H}}_{varepsilon }right)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:math> of <?CDATA ${mathcal{H}}_{varepsilon }$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:msub> <mml:mrow> <mml:mi mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> </mml:math> has at least m gaps as ɛ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of <?CDATA $sigma left({mathcal{H}}_{varepsilon }right)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:math> as ɛ → 0 can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) ɛ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers." @default.
• W3046065282 created "2020-08-03" @default.
• W3046065282 creator A5082759907 @default.
• W3046065282 date "2020-09-01" @default.
• W3046065282 modified "2023-09-27" @default.
• W3046065282 title "Periodic quantum graphs with predefined spectral gaps" @default.
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• W3046065282 doi "https://doi.org/10.1088/1751-8121/aba98b" @default.
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