Spectral Asymptotics in One-Dimensional Periodic Lattices with Geometric Interaction

We investigate the spectrum of one-dimensional periodic lattices with geometric interaction in the asymptotic regimes of vanishing attenuation and long lattice lengths. Our mathematical analysis is motivated by a number of real-world problems where chains of radiative elements are embedded in waveguides in order to enhance the interlattice interactions and collective resonances. Under certain assumptions the problem reduces to determining the eigenvalues of a complex-valued, symmetric Toeplitz matrix whose elements follow a geometric progression. A recurrence formula is derived for the characteristic polynomial and becomes the basis for developing spectral asymptotics and resolving the eigenvalue patterns on the complex plane. Analytical asymptotic formulas are derived for the eigenvalues and their accuracy is assessed numerically.