Eigenvalues of one family of tridiagonal skew-self-adjoint Toeplitz matrices with complex perturbations on the corner

In this paper, we study the eigenvalues of the matrices $T_n(a)+\gamma E_{n,1,1}$ where $T_n(a)$ is the Toeplitz matrix with generating symbol $a(t)=t-t^{-1}$, $E_{n,1,1}$ is the $n\times n$ matrix whose upper left component is $1$ and the other components are zero, and $\gamma$ is a fixed complex number such that $0<\mathrm{\mid }\gamma \mathrm{\mid }<1$. As $n\to \mathrm{\infty }$, the eigenvalues of these matrices are asymptotically distributed as the function $2i\sin (x)$, $x\in [0,2\pi ]$. Our main result is an asymptotic formula for every eigenvalue with a residue of the order $O(1\mathrm{/}{n}^3)$.