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?



Schrödinger equation with nonstandard boundary conditions
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.


