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This paper is concerned with the Schr\"odinger operators $\Delta_{f_0}$ and$\Delta_f$ attached to a pair $(\mathbb{C}^n, f_0)$ and its deformation$(\mathbb{C}^n, f)$, where $f_0$ is a non-degenerate and quasi-homogeneouspolynomial on $\mathbb{C}^n$ and $f$ is its relevant or marginal deformation.We give the $tt^*$ geometry structure on the Hodge bundle associated to$\Delta_f$, which describes the genus 0 anomaly. Next we study thecorresponding singularity torsion type invariants and give the anomaly formulasfor the 2nd torsion type invariant.

$tt^*$ Geometry, Singularity Torsion and Anomaly Formulas