Quantum singular operator limits of thin Dirichlet tubes via $\Gamma$-convergence

Abstract

The $\Gamma$-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 Dirichlet 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.

28 pages; no figures; to appear in Rep. Math. Phys

References in corpus (3)

• Coupling in the singular limit of thin quantum waveguides
• Quantum graphs as holonomic constraints
• Mathematical justification of the Aharonov-Bohm hamiltonian

Cited by in corpus (5)

• The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions
• Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential
• Thin waveguides with Robin boundary conditions
• The adiabatic limit of Schrödinger operators on fibre bundles
• Lifshits tails for randomly twisted quantum waveguides