Schr\"odinger operator in the limit of shrinking wave-guide cross section and singularity scaled twisting

TL;DR

This paper investigates the asymptotic behavior of Schr"odinger operators in twisted wave-guides with shrinking cross-sections and singular twisting, showing convergence to a one-dimensional harmonic oscillator with boundary conditions.

Contribution

It introduces a novel analysis of Schr"odinger operators in wave-guides with simultaneous shrinking and twisting, establishing their norm-resolvent convergence to a harmonic oscillator model.

Findings

Schr"odinger operators converge to a harmonic oscillator in the limit.

The limit operator includes a Dirichlet boundary condition at the twisting point.

The analysis applies to wave-guides with non-circular cross-sections.

Abstract

Motivated by the method of self-similar variables for the study of the large time behavior of the heat equation in twisted wave-guides whose non circular cross-section and the support of twisting diminushing simutaneously to zero. Since in this limit the strength of the twisting increases to infinity and its support shrinks to the point, we show that the corresponding Schr\"odinger operator converges in a suitable norm-resolvent sense to a one-dimensional harmonic oscillator on the reference line of the wave-guide, subject to some extra Dirichlet boundary condition at the twisting point support.