Suppose $w$ is a solution of 
$$\frac{d^2}{dx^2}w+\{u(x)+k^2\}w=0$$
with asymptotic condition
$$\lim\_{x\rightarrow \infty}w(x)e^{ikx}=1$$
and $u\in L^1\_1(\mathbb{R})=\{f:\int\_\mathbb{R}(1+|x|)|f|dx<\infty\}$, and $k>0$


Is the solution unique?



Does asymptotic behavior guarantee uniqueness?
Under your assumption on the potential, there is indeed such a unique solution,
(I assume you mean $x\to+\infty$ in your boundary condition. This is proved by reducing your differential equation to an integral one, which is solved by the method of successive approximations. See, for example,
V. A. Marchenko, Sturm-Liouville operators and applications,
(Translated from the Russian) Birkhauser 1986.
Chapter 3 Lemma 3.1.1.


