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Functional model for generalised resolvents and its application to time-dispersive media

Kirill D. Cherednichenko, Yulia Yu. Ershova, Sergey N. Naboko

Abstract

Motivated by recent results concerning the asymptotic behaviour of differential operators with highly contrasting coefficients, which have involved effective descriptions involving generalised resolvents, we construct the functional model for a typical example of the latter. This provides a spectral representation for the generalised resolvent, which can be utilised for further analysis, in particular the construction of the scattering operator in related wave propagation setups.

Functional model for generalised resolvents and its application to time-dispersive media

Abstract

Motivated by recent results concerning the asymptotic behaviour of differential operators with highly contrasting coefficients, which have involved effective descriptions involving generalised resolvents, we construct the functional model for a typical example of the latter. This provides a spectral representation for the generalised resolvent, which can be utilised for further analysis, in particular the construction of the scattering operator in related wave propagation setups.

Paper Structure

This paper contains 12 sections, 6 theorems, 93 equations, 1 figure.

Table of Contents

  1. From resonant composites to generalised resolvents
  2. Motivation for the problem to be analysed
  3. The operator $A$ as the dilation of a generalised resolvent
  4. Infinite-graph setup and Gelfand transform
  5. An example of operator on a graph and it norm-resovent approximation
  6. Abstract boundary triples
  7. The boundary triple for the prototype dilation operator
  8. Spectral form of the functional model for the Štraus-Neumark dilation
  9. Explicit functional model representation
  10. Construction of a Clark-type measure for the model representation
  11. The resolvent as an operator of multiplication by the independent variable
  12. Application to high-contrast homogenisation: an explicit functional model representation

Key Result

Theorem 2.1

Denote $H:=\oplus_{j=1}^3 L^2(0, l_j).$ There exists $C>0,$ independent of $\varepsilon$ and $\tau,$ such that where $\Psi$ is a partial isometry from $H$ to $L^2(0,l_2)\oplus\mathbb C.$

Figures (1)

  • Figure 1: Example of a periodic graph with contrast.The infinite graph $\mathbb G_\infty$ and the "period" $\mathbb G_\varepsilon$ are outlined on the left; the graph unit cell $\mathbb G$ obtained after applying the Gelfand transform is shown on the right. The soft component is drawn in blue.

Theorems & Definitions (10)

  • Theorem 2.1: GrandePreuve
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • Theorem 4.1
  • Remark 1
  • Theorem 4.2