# weak solution of Schroedinger equation .. are they useful?

We know that Schroedinger equation can be deduced from a variational principle (non relativistic Schroedinger equation).

Assuming this I have 2 questions:

a) Using variational methods, could we prove that there exist weak solutions (distributions) to the Schroedinger equation under certain conditions?

b) Are 'weak solutions' of Schroedinger equation useful?

(b) Sure! Objects like $|x\rangle$ and $|p\rangle$ or "generalized" eigenvectors of the Hamiltonian operator associated to points in the continuous spectrum are, technically speaking, nothing but distributions, i.e. weak solutions of Schroedinger equation.