Unidirectionally invisible potentials as local building blocks of all scattering potentials

We give a complete solution of the problem of constructing a scattering potential v(x) that possesses scattering properties of one's choice at an arbitrary prescribed wave number. Our solution involves expressing v(x) as the sum of at most six unidirectionally invisible finite-range potentials for which we give explicit formulas. Our results can be employed for designing optical potentials. We discuss its application in modeling threshold lasers, coherent perfect absorbers, and bidirectionally and unidirectionally reflectionless absorbers, amplifiers, and phase shifters.