Conjectures on spectral numbers for upper triangular matrices and for singularities

Cecotti and Vafa proposed in 1993 a beautiful idea how to associate spectral numbers ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_n\in \mathbb{R}$ to real upper triangular $n\times n$matrices$S$with{1}^{\mathrm{\prime }}sonthediagonalandeigenvaluesof$S^{-1}S^t$ in the unit sphere. Especially, $\exp (-2\pi i{\alpha }_j)$ shall be the eigenvalues of $S^{-1}{S}^t$. We tried to make their idea rigorous, but we succeeded only partially. This paper fixes our results and our conjectures. For certain subfamilies of matrices their idea works marvellously, and there the spectral numbers fit well to natural (split) polarized mixed Hodge structures. We formulate precise conjectures saying how this should extend to all matrices $S$ as above. The idea might become relevant in the context of semiorthogonal decompositions in derived algebraic geometry. Our main interest are the cases of Stokes like matrices which are associated to holomorphic functions with isolated singularities (Landau-Ginzburg models). Also there we formulate precise conjectures (which overlap with expectations of Cecotti and Vafa). In the case of the chain type singularities, we have positive results. We hope that this paper will be useful for further studies of the idea of Cecotti and Vafa.