Quantum singular operator limits of thin Dirichlet tubes via Γ-convergence

The Γ -convergence of lower bounded quadratic forms is used to study the singular operator limit of thin tubes (i.e. the vanishing of the cross-section diameter) of the Laplace operator with Dinchlet boundary conditions; a procedure to obtain the effective Schrödinger operator (in different subspaces) is proposed, generalizing recent results in case of compact tubes. Finally, after scaling curvature and torsion the limit of a broken line is briefly investigated.