Spectral properties of a nonlocal second-order difference operator (Q483745)

The Samarskii-Ionkin problem with parameter is of the following form \[ \begin{aligned} \frac{\partial u}{\partial t} &=\frac{\partial}{\partial x} \left(k(x) \frac{\partial u}{\partial z}\right)+f(x,t), \qquad 0<x<1, \; t>0, \\ u(x,0)& =\varphi(x), \qquad \leq x \leq 1, \qquad u(0,t)=0, \; t>0,\\ \gamma k(0)\frac{\partial u}{\partial x}(0,t)& =k(1) \frac{\partial u}{\partial x}(1,t), \qquad t>0, \end{aligned} \] where \(\gamma \in \mathbb{C}\) and the coefficient \(k(x)\) ranges in the interval \([\kappa_1,\kappa_2], \kappa_1>0.\) The algebraic and geometric multiplicity of the eigenvalues and the sign of their real part are studied. The characteristics of the spectrum are established for the general case in which the coefficients of a difference scheme approximating the problem \(a_k=k(x_k-0.5h)\) are arbitrary positive numbers and the parameter \(\gamma\) belongs to \(\mathbb{C}\).