Spectral properties of coupled wave operators (Q1969071)

This very interesting paper is devoted to the mathematical model describing several pulsation mechanisms of distributed feedback (DFB for short) semiconductor lasers. It consists of a boundary value problem for a linear hyperbolic system of first-order complex-valued partial differential equations with piecewise constant coefficients (the so-called coupled wave equations), \[ \begin{aligned} \partial_tu_1(t,x) &= v_g(\partial_xu_1(t, x)+ c(x)u_1(t, x)+ d_1u_2(t, x)),\\ \partial_tu_2(t, x) &= v_g(\partial_xu_2(t, x)+ d_2u_1(t, x)+ c(x)u_2(t, x)),\;x\in(-l_1, l_2),\;t> 0,\end{aligned} \] where \(u_1\) and \(u_2\) describe the slowly varying complex amplitudes of the forward and backward traveling waves of the electric field, \(l_1,l_2>0\) are the lengths of the two laser sections, \(d_1\), \(d_2\) are complex coupling coefficients, \(v_g\) is the group velocity, and \(c(x)\) is a propagation coefficient (\(c= c_1\) for \(x\in(-l_1, 0)\) or \(c= c_2\) for \(x\in(0, l_2)\), \(c_1\), \(c_2\) are complex numbers). The boundary condition has the form \(u_1(t,-l_1)= r_1u_2(t,-l_1)\), \(u_2(t,l_2)= r_2 u_1(t,l_2)\),where \(r_1\) and \(r_2\) are given complex coefficients satisfying \(0<|r_j|< 1\) \((j= 1,2)\). Then the problem under consideration can be formulated as an abstract evolution equation \(du/dt= v_gHu\) in a suitable Hilbert space. Some spectral properties of \(H\) are investigated. The main result is that \(H\) generates a \(C_0\)-group of bounded operators in a suitable Hilbert space \({\mathcal U}\) consisting of root functions of \(H\), where all but finitely many of these root functions are eigenfunctions.