The eigenvalues and eigenfunctions of the toroidal dipole operator in a mesoscopic system

Abstract

We give analytical expressions for the eigenvalues and generalized eigenfunctions of ${\hat{T}}_3$, the $z$-axis projection of the toroidal dipole operator, in a system consisting of a particle confined in a thin film bent into a torus shape. We find the quantization rules for the eigenvalues, which are essential for describing measurements of ${\hat{T}}_3$. The eigenfunctions are not square-integrable, so they do not belong to the Hilbert space of wave functions, but they can be interpreted in the formalism of rigged Hilbert spaces as kernels of distributions. While these kernels appear to be problematic at first glance due to singularities, they can actually be used in practical computations. In order to illustrate this, we prescribe their action explicitly and we also provide a normalization procedure.

11 pages and 3 figures

References in corpus (6)

• Dielectric Metamaterials with Toroidal Dipolar Response
• Toroidal metamaterial
• Nonradiating anapole states in nanophotonics: from fundamentals to applications
• Adiabatic following of terahertz surface plasmon-polaritons based on tri-layered corrugated thin film coupler
• Anapole moment of the lightest neutralino in the cMSSM
• The self-adjoint toroidal dipole operator in nanostructures