Quantum mechanics on the line with two origins

We study scalar and spinorial quantum dynamics on the standard line with two origins, \[
\Ltwo=(\R_1\sqcup\R_2)/\!\sim,
\qquad
(x,1)\sim(x,2)\quad\text{for }x\neq0, \] equipped with its identity-glued smooth structure and flat metric. We first show that the ordinary scalar theory is insensitive to the doubled origin: every continuous map from \(\Ltwo\) to a Hausdorff space factors through the quotient \(q:\Ltwo\to\R\), while, for the natural measure, \[
C^\infty(\Ltwo)\cong C^\infty(\R),
\qquad
L^2(\Ltwo)\cong L^2(\R). \] Accordingly, the natural scalar free Hamiltonian is unitarily equivalent to the free Laplacian on \(\R\).
The doubled origin remains visible, however, in the spinor line associated with the nontrivial spin structure. The oriented flat structure on \(\Ltwo\) admits two spin structures, distinguished by the relative transition sign on the two connected components of the chart overlap. For the nontrivial structure, these signs are opposite. This forces every continuous global spinor section to vanish at both origins and every smooth section to vanish there to infinite order. The associated first-order operator \[
-i\,\frac{\mathrm d}{\mathrm dx} \] on compactly supported smooth twisted sections is symmetric but not essentially self-adjoint: its deficiency indices are \((1,1)\), and a self-adjoint first-order evolution requires an additional transmission condition at the origin. The natural positive quadratic form associated with the flat twisted structure instead yields the direct sum of two Dirichlet half-line Laplacians and hence perfect reflection. Thus the ordinary scalar theory does not detect the doubled origin, whereas the nontrivially glued spinor line does, although the spin structure alone does not determine a unique unitary transmission law.