General Properties of Two-dimensional Conformal Transformation in Electrostatics

TL;DR

This paper demonstrates that the electrostatic modes and eigenvalues of two-dimensional nanosystems are invariant under conformal transformations, offering a new approach to analyze transformed structures through eigenmode modifications.

Contribution

It proves the invariance of geometry resonances and eigenvalues under conformal transformations in 2D electrostatics, providing a novel method for studying transformed nanosystems.

Findings

Eigenvalues and modes are invariant under conformal transformations.

Transforming geometry equates to adjusting eigenmode strengths.

New analytical approach for 2D electrostatic structure analysis.

Abstract

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.