The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices - Zagorodnyuk, S. M. (Methods of Functional Analysis and Topology, 2008)
We consider a difference equation associated with a semi-infinite complex (2N+1)-diagonal transposition-antisymmetric matrix J=(g_k,l)_k,l=0^∞ with g_k,k+N ot=0, k=0,1,2,… , (g_k,l=-g_l,k): _j=-N^N g_k,k+j y_k+j = λ^N y_k, k=0,1,2,… , where y=(y_0,y_1,y_2,… ) is an unknown vector, λ is a complex parameter, g_k,l and y_l with negative indices are equal to zero, NN. We introduce a notion of the spectral function for this difference equation. We state and solve the direct and inverse problems for this equation.