Spectral orthogonality of special flows

In this paper, we study the spectral orthogonality problem for special flows built over irrational rotations under two different types of roof functions: 1) the roof functions are real analytic. 2) the roof functions are piecewise $C^1$ with one discontinuity. These flows are also known as von-Neumann flows. We show that if $\{T_{\alpha }^f\}$ is as in 1) and weak mixing, then for a $G_{\delta }$ dense set of $\beta$, we have that $\{T_{\beta }^f\}$ is weak-mixing and is spectrally orthogonal to $\{T_{\alpha }^f\}$. On the other hand, if $\{T_{\alpha }^f\}$ is as in 2), then for a full measure set of $\beta$, the flows $\{T_{\alpha }^f\}$ and $\{T_{\beta }^f\}$ are spectrally orthogonal.