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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?

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(a) It depends on what you mean by "variational methods". The stationary S. equation it is nothing but an elliptic equation, so you can use all the standard method appropriate for elliptic equations, existence of weak solutions and elliptic regularity (Weyl, Sobolev, Hoermander...). The temporal equation, to some extent is similar to the heat equation. At least for the pathological aspects like non-locality...

(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.

Using classes of those solutions, assuming to work in a so-called "nuclear Hilbert space", one may re-formulate a version of the spectral theorem due to Gelfand, equivalent (under some hypotheses) to the standard version due to von Neumann and based on projection valued measures.

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