Geometric spectral theory for Kähler functions

We consider Kähler toric manifolds $N$ that are torifications of statistical manifolds $\mathcal{E}$ in the sense of [M. Molitor, “Kähler toric manifolds from dually flat spaces”, arXiv:2109.04839], and prove a geometric analogue of the spectral decomposition theorem in which Hermitian matrices are replaced by Kähler functions on $N$. The notion of “spectrum” of a Kähler function is defined, and examples are presented. This paper is motivated by the geometrization program of Quantum Mechanics that we pursued in previous works (see, e.g., [M. Molitor, “Exponential families, Kähler geometry and quantum mechanics”, J. Geom. Phys. 70, 54-80 (2013)]).