Energy concentration and explicit Sommerfeld radiation condition for the electromagnetic Helmholtz equation

We study the electromagnetic Helmholtz equation(∇+ib(x))2u(x)+n(x)u(x)=f(x),x∈Rd, with the magnetic vector potential b(x)b(x) and n(x)n(x) a variable index of refraction that does not necessarily converge to a constant at infinity, but can have an angular dependence like n(x)→n∞(x|x|) as |x|→∞|x|→∞. We prove an explicit Sommerfeld radiation condition∫Rd|∇bu−in∞1/2x|x|u|2dx1+|x|<+∞ for solutions obtained from the limiting absorption principle and we also give a new energy estimate∫Rd|∇ωn∞(x|x|)|2|u|21+|x|dx<+∞, which explains the main physical effect of the angular dependence of n at infinity and deduces that the energy concentrates in the directions given by the critical points of the potential.