Does Schrödinger equation have dual-property with Heat equation? I have experimental data that Schödinger equation maintains high frequencies, while heat equation low.
Does Schrödinger equation have some duality property with heat equation?
 A: "I have experimental data that Schödinger equation maintains high frequencies, while heat equation low."
The heat equation is of the form
$\partial_t f = \nabla^2 f$
so eigenfunctions of $\nabla^2$ decay exponentially with time, and the higher the spatial frequency (eigenvalue), the faster the decay. Therefore low spatial frequencies decay more slowly.
The Schödinger equation, as far as I understand, doesn't have quite this behaviour because all eigenfunctions of the Hamiltonian merely oscillate in time. It maintains both high and low frequencies, so this isn't exactly a duality.
There are lots of what seem to be deep connections between the two equations, however. These are best studied from the point of view of the path integral and the partition function. The Schödinger equation is what happens when a system samples all paths weighted by a phase factor $e^{-\frac{i}{\hbar} S}$. The heat equation is what happens when a system samples all microstates weighted by a factor that leads to maximum entropy.
A: Both equations are of the type $\partial_{t} f = C \partial_{x}^{2} f$.
They are related by Wick rotation. However, there is a slight difference. Statistical mechanics n-point functions satisfy positivity whereas Wick-rotated quantum field theories satisfy reflection positivity. TODO more info about this is needed.
