Spectral properties of the ladder-like Josephson junction array

In this paper theoretical analysis of the ladder-like multirow array of inductively coupled Josephson junctions is presented. An external dc current is applied at the top to each of the columns of the array and is extracted at the bottom of that column. The density of states of the Josephson plasma waves has a $\delta$-function term due to the flat band and $3N-2$ singularities where $N$ is the number of rows. The spatial distribution of the amplitudes of the plasmon wave is computed analytically for any given value of the wavenumber $q$. It is expressed through the orthogonal polynomials that are similar but not identical to the Chebyshev polynomials.