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

TL;DR

This paper uses $ extGamma$-convergence of quadratic forms to analyze the limiting behavior of Laplace operators on thin Dirichlet tubes, deriving effective Schrödinger operators and exploring limits involving broken lines.

Contribution

It generalizes recent results by establishing a $ extGamma$-convergence framework for singular limits of thin tubes, including non-compact cases and geometric effects.

Findings

Derived effective Schrödinger operators for thin Dirichlet tubes

Extended analysis to non-compact tube geometries

Investigated limits involving scaled curvature and torsion

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 Schr\"odinger 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.