General Properties of Two-dimensional Conformal Transformation in Electrostatics

Electrostatic properties of two-dimensional nanosystems can be described by their geometry resonances. In this paper we prove that these modes as well as the corresponding eigenvalues are invariant under any conformal transformation. This invariance further leads to a new way to studying the transformed structures. Namely, transforming a geometry is equivalent to modifying the strengths of these invariant eigenmodes excited by the external excitations.