### Abstract

The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem -Δu = λεu, where the dielectric constant ε(x) is a periodic function which assumes a large value ε near a periodic graph Σ in ℝ^{2} and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of the spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.

Original language | English (US) |
---|---|

Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Experimental Mathematics |

Volume | 8 |

Issue number | 1 |

State | Published - 1999 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Experimental Mathematics*,

*8*(1), 1-28.

**Spectral properties of high contrast band-gap materials and operators on graphs.** / Kuchment, Peter; Kunyansky, Leonid.

Research output: Contribution to journal › Article

*Experimental Mathematics*, vol. 8, no. 1, pp. 1-28.

}

TY - JOUR

T1 - Spectral properties of high contrast band-gap materials and operators on graphs

AU - Kuchment, Peter

AU - Kunyansky, Leonid

PY - 1999

Y1 - 1999

N2 - The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem -Δu = λεu, where the dielectric constant ε(x) is a periodic function which assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of the spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.

AB - The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem -Δu = λεu, where the dielectric constant ε(x) is a periodic function which assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of the spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.

UR - http://www.scopus.com/inward/record.url?scp=0033245667&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033245667&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033245667

VL - 8

SP - 1

EP - 28

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

IS - 1

ER -