I am considering the Schrodinger equation on $[0,1]$: $$iu_t=-u_{xx}$$ I am looking for a particular solution $u(x,t)$ which satisfies the boundary condition $$\cos(t)\cdot u_x(0,t)=\sin(t)\cdot u(0,t)$$ (which is not $u=0$)
I know that this is an undetermined problem (since I am missing a boundary condition at $x=1$ and also an initial condition). But for my case, I only care about finding one particular solution which satisfies the condition above (and not some general solution), so I don't impose any more restrictions.
More generally (perhaps this will be easier), I am interested in an example of a solution to the Schrodinger equation such that its boundary condition at $x=0$ at time $t=0$ is Neumann ($u_x=0$), and at some other time it is Dirichlet ($u=0$), like in the example above, with $t=\pi/2$. My only restriction is that I want the ratio $u_x/u$ at the boundary to be real valued for all $t$ (such that $u\neq0$). If you can set it so that the same boundary condition also holds at $x=1$ - that would be even better.
Does anyone have an idea on how to find such an example?
Thanks in advance.
Here's one solution:
$$u(x,t) = \cos(x)e^{-it} + i \sin(x)e^{-it} + \cosh(x)e^{it} - i\sinh(x)e^{it}$$
Each term is independently a solution to the stated differential equation.
Note that it isn't exactly true that the problem isn't separable (I may be bending the definition here...). The differential equation's separable qualities are unaffected by your boundary condition: you can still use separation of variables to form elementary solutions to the differential equation. Normally before going further, one uses the boundary conditions to discard/constrain most of the elementary solutions, which is the part we can't perform here (at least not through the normal procedure).
By not discarding any elementary solutions, I was able to find the above solution which uses a superposition of both the sinusoid and hyperbolic sinusoid solutions.