Classical & weak solutions of Schrödinger equation Consider the problem of an infinite square well
$$ V(x) = \begin{equation}
     \begin{cases}
       0, \qquad {\rm if}\quad0 \le x \le L \\
       \infty, \qquad{\rm otherwise}
     \end{cases}
\end{equation}
$$
The time-independent Schrödinger equation is thus
$$
-\frac{\hbar^{2}}{2m} \frac{d^{2} \psi}{dx^{2}} = E \psi
$$
inside the well. We have a boundary condition $\psi(0) = 0 = \psi(L)$ and one can find that the solutions of this equation are
$$
\psi_{n}(x) = \sqrt{\frac{2}{L}} \sin(n \pi x/L).
$$
My question is:

*

*Is this a classical solution? What is a good definition for classical solutions for differential equations?


*Is it possible to get a weak solution for this equation (and boundary conditions)? What is a good definition for weak solutions?


*Why are weak solutions important in the context of Quantum Mechanics? Is it possible to get a solution (with physical meaning) that, for example, has a non-continuous second derivative?
 A: *

*That solution is classical since the function is smooth.


*All solutions of that equation are classical due to elliptic regularity of the (here $1D$) Laplace operator, since the equation does not contain non-smooth known functions.
A weak solution is a locally $L^2$ function which satisfies the  (linear) differential equation and where the derivatives of the function are computed in terms of "weak derivatives", i.e., interpreting the function as a distribution. I stress that a distributional solution is an even weaker notion of solution. A weak solution is in fact  a function (up to zero measure set), a distribution is not necessarily a function instead.


*Weak solution are of fundamental relevance in QM because the observables are selfadjoint operators that are extensions of the familiar differential operators and their domains are made of $L^2$ generally non-smooth functions. The stationary Schroedinger equation, if one insists on a differential equation, can only be intepreted in weak sense.
However, elliptic regularity-like theorems  (in particular some therems due to Weyl) finally prove that the found solutions have some regularity. For instance if there is a potential $U$ with finite discontinuities, the weak solutions must be $C^1$ and $C^2$ outside the singularitues of $U$.
