Complex $\Gamma$-convergence and magnetic Dirichlet Laplacian in bounded thin tubes

TL;DR

This paper extends $ ext{Γ}$-convergence techniques to complex Hilbert spaces and applies them to derive effective operators for magnetic Dirichlet Laplacians in thin tubular regions, demonstrating norm resolvent convergence as the tubes narrow.

Contribution

It adapts $ ext{Γ}$-convergence methods to complex spaces and provides a new approach to analyze magnetic Laplacians in thin tubes with weak regularity assumptions.

Findings

Established norm resolvent convergence of magnetic Laplacians in thin tubes

Derived effective operators for Dirichlet Laplacian with magnetic potentials

Applicable to tubes built along smooth closed curves with weak potential regularity

Abstract

The resolvent convergence of self-adjoint operators via the technique of $\Gamma$-convergence of quadratic forms is adapted to incorporate complex Hilbert spaces. As an application, we find effective operators to the Dirichlet Laplacian with magnetic potentials in very thin bounded tubular regions in space built along smooth closed curves; relatively weak regularity is asked for the potentials, and the convergence is in the norm resolvent sense as the cross sections of the tubes go uniformly to zero.