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For $T_2>T_1\ge -1$, the dual Strichartz estimate for the free Schrödinger equation givesThe assumed finite global $L_{t,x}^4$ norm makes the right side tend to zero as $T_1,T_2\to \infty$. The displayed family is therefore a Cauchy sequence in the complete space $L^2$ and has a strong limit $F_{\infty }$.

The Duhamel principle givesDefineSince the free Schrödinger group is unitary,This is scattering from a finite Strichartz norm.

The scaling in part 5 givesThus $u\in L_{t,x}^4([-1,\infty )\times {\mathbb{R}}^2)$. For $T<T^{\prime }$, the dual Strichartz estimate for the free Schrödinger equation giveswhich tends to zero as $T,T^{\prime }\to \infty$. The integrals in the question are therefore Cauchy in $L^2$.

The Duhamel principle in the interaction representation defines an $L^2$ limit $u_{\infty }$ and expresses $S(-t)u(t)-u_{\infty }$ as the tail integral from $t$ to infinity. The same estimate sends that tail to zero, proving the scattering from a finite Strichartz norm conclusion