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?