  In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and
Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] the
instabilities of the Gabor phase retrieval problem, i.e., the problem of
reconstructing a function $f$ from its spectrogram $|\mathcal{G}f|$, where $$
\mathcal{G}f(x,y)=\int_{\mathbb{R}^d} f(t) e^{-\pi|t-x|^2} e^{-2\pi i t\cdot y}
dt, \quad x,y\in \mathbb{R}^d, $$ have been completely classified in terms of
the disconnectedness of the spectrogram. These findings, however, were
crucially restricted to the onedimensional case ($d=1$) and therefore not
relevant for many practical applications. In the present paper we not only
generalize the aforementioned results to the multivariate case but also
significantly improve on them. Our new results have comprehensive implications
in various applications such as ptychography, a highly popular method in
coherent diffraction imaging.
