does asymptotic behavior guarantee the uniqueness?

Question

Suppose $w$ is 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 this solution unique?