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Zbl 0752.35071
Bättig, Daniel; Guillot, Jean-Claude
The Fermi surface for the discretized Maxwell equations. (English)
[J] Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1991, No.XI, 6 p. (1991).

This investigation on the Fermi-surface for the discretized Maxwell equations is similar to that for the Schrödinger operator $-\Delta+V$ in dimensions 2 and 3 [{\it Ch. Peters}, Sémin. Bourbaki, Vol. 1989/90, 42 ème année, Asterisque 189-190, Exp. No. 723, 239-258 (1990; Zbl 0749.14027)]. For discretized systems the analogue of Fermi variety is an algebraic variety. Maxwell's equations (with $\mu=1)$, for $E$, lead to $$\varepsilon\sp{-1}\nabla\times\nabla\times E=\lambda E,\quad E(x+\gamma)=\exp i\langle k,\gamma\rangle\cdot E(x),$$ $\forall\gamma\in\Gamma\{\equiv a\sb 1Z+a\sb 2Z+a\sb 3Z\}$ a lattice in $\bbfR\sp 3$. This forms a self-adjoint boundary value problem yielding a discrete spectrum. The coordinates of $\gamma$ define the (physical) Fermi surface, ${\cal F}\sb{\text{phys}},\lambda(\varepsilon)$ for $\lambda\ne 0$ and the (complex) Fermi surface ${\cal F}\sb \lambda(\varepsilon)$ with coordinates of $\gamma$ in $(\bbfC\sp*)\sp 3$. The authors are interested in the questions: (i) Does ${\cal F}\sb{\text{phys}},\lambda(\varepsilon)$ determine ${\cal F}\sb \lambda(\varepsilon)$? (ii) Does the geometry of ${\cal F}\sb \lambda(\varepsilon)$ contains isospectral information? (iii) Does ${\cal F}\sb \lambda(\varepsilon)$ determine (generically) $\varepsilon$? In order to focus on these geometric aspects, a discrete approximation which consists of replacing the partial derivatives by the suitable shift operators is considered.
[N.D.Sengupta (Bombay)]

MSC 2000:: *35Q60 PDE of electromagnetic theory and optics
54D35 Compactifications
14J25 Special surfaces
78A99 Miscellaneous topics in optics and electromagnetic theory

Keywords: shift operator; algebraic variety; discrete spectrum

Citations: Zbl 0749.14027

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