Inequalities between Dirichlet and Neumann Eigenvalues on the Heisenberg Group

Universal eigenvalue inequalities are a classical topic in the spectral theory of differential operators. Most relevant to our work here are comparison theorems between the Dirichlet and Neumann eigenvalues λj(−∆Ω ) and λj(−∆Ω ), j ∈ N, of the Laplacian in a smooth, bounded domain Ω ⊂ R. Note that λj(−∆Ω ) ≤ λj(−∆Ω ) for all j ∈ N by the variational characterization of eigenvalues. This trivial bound for j = 1 was strengthened by Pólya [Pól] who observed that λ2(−∆Ω ) < λ1(−∆Ω ) for d = 2. Payne [Pay], Aviles [Avi] and Levine and Weinberger [LevWei] obtained further results in this direction under suitable convexity assumptions on Ω. A breakthrough was made by Friedlander [Fri] who proved that