The Harrell–Stubbe Gap Inequality for Dirichlet Eigenvalues Is Never Saturated

Let E_1 ≤ E_2 ≤ ⋯ be the eigenvalues of the Dirichlet Laplacian on a bounded open set Ω ⊂ R^n, and let M_p(J) = ((n+2p)/n · (1/J) Σ_{j≤J} E_j^p)^{1/p}. Harrell and Stubbe proved that M_1(J)^2 − M_2(J)^2 ≥ ¼(E_{J+1} − E_J)^2, and in a problem list of the 2009 Oberwolfach workshop on low eigenvalues of Laplace and Schrödinger operators they asked whether some Ω and J saturate this inequality. We show that the answer is no: the inequality is strict for every bounded open set and every J. Written in terms of the mean and the variance of E_1, …, E_J, the inequality is a gap estimate of Cheng and Yang, so that estimate is strict as well. The proof analyses the case of equality in the Harrell–Stubbe trace identity behind H. C. Yang's inequality. Equality would put each function x_k u_1, where u_1 ≥ 0 is a first eigenfunction, into a finite sum of eigenspaces. Then every partial derivative of u_1 would lie in H^1_0(Ω), so ∫_Ω Δu_1 = 0, which is impossible because Δu_1 = −E_1 u_1. The same argument shows that Yang's first inequality is strict whenever E_{J+1} > E_1, and hence for every J when Ω is connected; in 2002 Ashbaugh left the strictness of this inequality undecided. By contrast, for the harmonic oscillator and on spheres the analogous bounds are equalities at every spectral gap, that is, for every J with E_J < E_{J+1}. This is an unrefereed note.