Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group

TL;DR

This paper establishes strict inequalities between Neumann and Dirichlet eigenvalues of the sub-Laplacian on the Heisenberg group, also extending results to the Euclidean Laplacian with magnetic fields, revealing fundamental spectral differences.

Contribution

It proves a new inequality relating Neumann and Dirichlet eigenvalues on the Heisenberg group and extends these results to magnetic Laplacians in Euclidean space.

Findings

Neumann eigenvalues are strictly less than Dirichlet eigenvalues for the same index.

Established inequalities hold for any domain in the Heisenberg group.

Results include analogous inequalities for Euclidean Laplacian with magnetic fields.

Abstract

We prove that for any domain in the Heisenberg group the (k+1)'th Neumann eigenvalue of the sub-Laplacian is strictly less than the k'th Dirichlet eigenvalue. As a byproduct we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.