Schrödinger equation, deformation theory and $tt^*$ geometry

Abstract: I will explain my recent work on the deformation theory of
Schr\"odinger operator related to a strongly tame section-bundle system
$(M,g,f)$. This is a differential geometric description of
Landau-Ginzburg B model. We can construct the Hodge theory after proving
a key spectrum theorem of the form Schr\"odinger operators, then prove
the stability theorem, and finally we can construct the $tt^*$ geometry
structure on the Hodge bundle of the moduli space. As one application, we
can get Frobenius manifold structure via primitive vector which is given
by Oscillation integration of a harmonic form.