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My final goal is to solve numerically Schrodinger's equation in 3D with some potential for the unbounded states, meaning that far away from the potential (at infinity) we may find a free wave and not something that goes to zero.

The basic idea is that I have a particle in (0,0,0) that receives an initial momentum (kx,ky,kz), which is equivalent to knowing the wave function derivatives at the origin. My idea was to create a very small sphere, and say that within this sphere we know the solution: its just a plane wave e^(i

**k * r**) with the given initial momentum

**k**(hbar = 1).

So basically I want to numerically solve the problem of Schrodinger's equation between two spheres (the outer sphere will be taken to be large enough to contain all the data we need), and to find the wave function in this region, with Dirichlet boundary condition and Neumann boundary condition

**just**on the inner sphere.

The problem: I don't know any boundary conditions on the outer sphere, only on the inner one.

The question is: is it solvable? Is knowing the function and its derivatives on the inner sphere enough to find a unique solution for the wave function everywhere?