Weyl formula for the negative dissipative eigenvalues of Maxwell's equations (Q1692351)

The authors analyze the Maxwell equations in an exterior domain when the domain is explicitly given by \(\{ x \in \mathbb{R}^3: |x|>1\}\). A dissipative boundary condition of the form \(E_{\text{tan}}-\gamma(x) (\nu \times B_{\text{tan}} ) = 0\) with \(\gamma \neq 1\) a constant is imposed, and a Weyl formula for the counting function of the negative real eigenvalues of a certain differential operator related to the Maxwell system is obtained. The proof involves properties of the spherical Hankel functions. Finally, a conjecture for the general case of strictly convex obstacles for certain \(\gamma\) is stated, which agrees with the authors' main result when the domain is the ball in \(\mathbb{R}^3\).