Consider the partial differential equation $$\psi_t(t,x)=i\kappa \psi_{xx}(t,x) ~\mbox{for}~ 0<(t,x)\in\mathbb{R}\times\mathbb{R}$$ with boundary conditions $$\psi(0,x)=0 ~\mbox{for}~ x>0,$$ $$\psi(t,0)=\psi_0(t) ~\mbox{for}~ t\ge0.$$
Are these equation uniquely solvable whenever $\psi_0$ is sufficiently smooth?
Can one give an explicit expression for the linear operator mapping $\psi_0$ to the solution?
The equation under consideration is uniquely solvable in $H^s(\mathbb{R}^+)$, as soon as $\psi_0\in H^{(2s+1)/4}(\mathbb{R}^+)$, and there is a fairly explicit expression for the propagator. You can find these results, for example, in
J. L. Bona, S. Sun, B. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, https://doi.org/10.1016/j.matpur.2017.11.001,
where also the nonlinear problem is investigated.
