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Differentiate the spatial equation with respect to $t$ and the temporal equation with respect to $x$. Substitution ofinto ${\psi }_{xt}-ik{\psi }_t$ cancels every term involving $\psi$, $u$, and $u_x$, leavingThe left side is $u_t$ by differentiating ${\psi }_x-ik\psi =u$, so consistency is exactly $u_t=i{u}_{xx}$. This is a scalar Lax pair formulation.

Taking the half-line Fourier transform and integrating twice by parts givesMultiplication by $e^{i{k}^{2}t}$ and integration in time yields

Replace $k$ by $-k$ and subtract. The unknown Neumann boundary value cancels:For $x>0$, Fourier inversion of the zero extension givesInverting the preceding identity therefore givesThusThis elimination is the global relation for the half-line free Schrodinger equation.