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BIRS Workshop Lecture Videos

Algebraic Nevanlinna operator functions and applications to electromagnetics Engstrom, Christian


Operator functions with an algebraic dependence on a spectral parameter have applications in electromagnetic field theory, where the $\lambda$-dependent material properties are of Drude-Lorentz type. Let $M_\ell$, $\ell=1,2,\dots,L$ denote bounded linear operators in a Hilbert space $\mathcal H$ and denote by $A$ a self-adjoint operator with compact resolvent that is bounded from below. In this talk we consider operator functions of the form \[ \mathcal S(\lambda)=A-\lambda-\lambda\sum_{\ell=1}^L\dfrac{M_\ell}{c_\ell+id_{\ell}\sqrt{\lambda}-\lambda}, \quad \text{dom} \,\mathcal S(\lambda) = \text{dom} \,A, \quad \lambda \in \mathbb{C} \setminus P, \] where $P$ is the set of poles of the rational function and $c_\ell$, $d_{\ell}$ are non-negative constants. The operator function is self-adjoint if the damping $d_{\ell}$ is set to zero. Then we establish variational principles and provide optimal two-sided estimates for all the eigenvalues of $\mathcal S$. In the second part of the talk, I briefly present extensions to the non-self-adjoint case $d_{\ell}>0$ and ongoing work. The first part of the talk is based on a joint work with Heinz Langer and Christiane Tretter and the second part of the talk is based on a joint work with Axel Torshage.

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