Adiabatic Reduction of Schrodinger Dynamics on a Fibre with a Variable-Angle Conical Singularity

We construct a quantitatively controlled adiabatic (Born–Oppenheimer) reduction of a Schrödinger-type operator over an external space R cubed whose internal fibre is a compact surface carrying a conical singularity of angle 2 pi beta (x) that varies with the external point. The internal Laplacian is realized as a self-adjoint operator by isolating, at each conical point, the single radial s-wave that carries the self-adjoint extension freedom; the Friedrichs realization is selected and shown to be independent of the cone angle. Working in the Kondratiev (Fuchs-type) weighted scale, we compute the conormal symbol, fix the admissible weight window, and obtain a uniform elliptic isomorphism. The central estimate establishes that the band spectral projection is twice continuously differentiable in the external variable with bounded derivatives, despite the singular and x-dependent geometry: the conical deficit, through the threshold nu ell =| ell |/ beta >1, is precisely what places the singular modulus perturbation in the controllable range. The resulting effective dynamics is a magnetic Schrödinger operator with a position-dependent mass and a geometric scalar potential whose magnitude is fixed by the conical deficit; in a singular-defect extension of the smooth family, where the deficit develops a 1/ dist gradient profile near a codimension-two locus, this potential is inverse-square. It formally vanishes as beta to 1, the uniform estimates being established for beta maximum <1. The contribution is primarily analytic—the control of an adiabatic reduction across a variable conical singularity—rather than a claim of new physics; a Z 2 Aharonov–Bohm holonomy appears only as a secondary effect when a twist phase is superimposed.