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

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.