On the spectral behaviour of a non-self-adjoint operator with complex potential
A sequence of complex numbers \(\{v_n\}\), \(n\in\mathbb{Z}\), which takes values in a finite set \(M\) is called pseudo-ergodic if, for any positive integer \(N\) and an arbitrary set \(w_1,\dots,w_N\) from \(M\), there is \(q\in\mathbb{Z}\) such that \(v_{q+j}=w_j\), \(j=1,2,\dots,N\). The author studies the properties of the spectrum of a nonselfadjoint Jacobi operator \(J\) on \(\ell^2(\mathbb{Z})\) with constant off-diagonals \(\alpha,\beta\in\mathbb{R}\) and a pseudo-ergodic main diagonal \(v_n\) with the 4-point set \(M=\{\pm\gamma\pm i\delta\}\), \(\gamma,\delta>0\). By using the second order numerical range, the author obtains tight bounds on the spectrum and finds the bounds for the size of possible holes inside the spectrum.
scientific article; zbMATH DE number 5556070