  We study the non-Hermitian skin effect in a three-dimensional system of
finitely many subwavelength resonators with an imaginary gauge potential. We
introduce a discrete approximation of the eigenmodes and eigenfrequencies of
the system in terms of the eigenvectors and eigenvalues of the so-called gauge
capacitance matrix $\mathcal{C}_N^\gamma$, which is a dense matrix due to
long-range interactions in the system. Based on translational invariance of
this matrix and the decay of its off-diagonal entries, we prove the
condensation of the eigenmodes at one edge of the structure by showing the
exponential decay of its pseudo-eigenvectors. In particular, we consider a
range-k approximation to keep the long-range interaction to a certain extent,
thus obtaining a k-banded gauge capacitance matrix $\mathcal{C}_{N,k}^\gamma$ .
Using techniques for Toeplitz matrices and operators, we establish the
exponential decay of the pseudo-eigenvectors of $\mathcal{C}_{N,k}^\gamma$ and
demonstrate that they approximate those of the gauge capacitance matrix
$\mathcal{C}_N^\gamma$ well. Our results are numerically verified. In
particular, we show that long-range interactions affect only the first
eigenmodes in the system. As a result, a tridiagonal approximation of the gauge
capacitance matrix, similar to the nearest-neighbour approximation in quantum
mechanics, provides a good approximation for the higher modes. Moreover, we
also illustrate numerically the behaviour of the eigenmodes and the stability
of the non-Hermitian skin effect with respect to disorder in a variety of
three-dimensional structures.
