The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices

Abstract

We consider a difference equation associated with a semi-infinite complex $(2N+1)$-diagonal transposition-antisymmetric matrix $J=(g_{k,l})_{k,l=0}^\infty$ with $g_{k,k+N} \not=0$, $k=0,1,2,\ldots ,$ ($g_{k,l}=-g_{l,k}$): $\sum_{j=-N}^N g_{k,k+j} y_{k+j} = \lambda^N y_k,\ k=0,1,2,\ldots ,$ where $y=(y_0,y_1,y_2,\ldots )$ is an unknown vector, $\lambda$ is a complex parameter, $g_{k,l}$ and $y_l$ with negative indices are equal to zero, $N\in\mathbb N$. We introduce a notion of the spectral function for this difference equation. We state and solve the direct and inverse problems for this equation.