I'm trying to solve the initial value problem for the Schrödinger equation (, f Schwartz).
I know that a fundamental solution is given by .
How do I interpret here?
I'm trying to show that if I convolve the above fundamental solution K with the initial data f (convolution in the spatial variable x), then I obtain the solution to the initial value problem.
Specifically, how do I prove that as ? More generally, what are the differences between this problem and the analogous problem for the heat equation ?
[I know that the Schrödinger equation and fundamental solution are obtained from their heat counterparts via
Why is the Schrödinger equation time reversible (i.e. why can it be solved both forwards and backwards in time), while the heat equation isn't? The total integral of the heat kernel (with respect to x) is 1; is the total integral of the "Schrödinger kernel" K also equal to 1?