Effective Hamiltonians for Thin Dirichlet Tubes with Varying Cross-Section

TL;DR

This paper extends recent methods for deriving effective Hamiltonians to quantum systems confined in thin Dirichlet tubes with varying cross-sections, highlighting differences in proof techniques despite similar structures.

Contribution

It adapts and applies the framework of effective Hamiltonians to a new class of quantum systems with Dirichlet boundary conditions and varying cross-sectional geometries.

Findings

Effective Hamiltonians can be derived for thin Dirichlet tubes.

The structure of the effective Hamiltonian remains consistent with previous results.

Proof techniques differ due to boundary conditions and geometry.

Abstract

We show how to translate recent results on effective Hamiltonians for quantum systems constrained to a submanifold by a sharply peaked potential to quantum systems on thin Dirichlet tubes. While the structure of the problem and the form of the effective Hamiltonian stays the same, the difficulties in the proofs are different.