[Paper Review] On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

Mark S. Ashbaugh, Lotfi Hermi|ArXiv.org|Dec 28, 2007

Spectral Theory in Mathematical Physics28 references18 citations

TL;DR

This paper presents a new, abstract proof of Harrell-Stubbe type inequalities for the discrete spectrum of self-adjoint operators using commutator algebra, the Rayleigh-Ritz principle, and an optimal application of the Cauchy-Schwarz inequality. It establishes parameter-free and projection-free universal eigenvalue estimates that generalize classical bounds such as Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang, with sharp inequalities valid for all $ p \leq 2 $ and $ p \geq 2 $, respectively.

ABSTRACT

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the Cauchy-Schwarz inequality--is used to produce ``parameter-free'', ``projection-free'' versions of their theorems. We also analyze the strength of the various inequalities that ensue. The results contain classical bounds for the eigenvalues. Extensions of a variety of inequalities à la Harrell-Stubbe are illustrated for both geometric and physical problems.

Motivation & Objective

• To provide a new, abstract derivation of Harrell-Stubbe type inequalities for the discrete spectrum of self-adjoint operators, avoiding domain-specific dependencies.
• To extend universal eigenvalue estimates beyond the original Harrell-Stubbe framework by employing an optimal usage of the Cauchy-Schwarz inequality.
• To unify and generalize classical eigenvalue bounds—such as Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang—within a single abstract framework.
• To analyze the strength and monotonicity of the derived inequalities across different values of the exponent $ p $, particularly proving that the $ p=2 $ case is the strongest for $ p \geq 2 $.
• To demonstrate the applicability of these inequalities to diverse geometric and physical problems, including Schrödinger operators, Sturm-Liouville problems, and elliptic operators on domains.

Proposed method

• Utilizes commutator algebra to define $ \rho_i = \sum_{k=1}^N \langle [A,B_k]u_i, B_k u_i \rangle $ and $ \Lambda_i = \sum_{k=1}^N \| [A,B_k]u_i \|^2 $, linking spectral gaps to operator commutators.
• Applies the Rayleigh-Ritz principle to bound eigenvalue gaps via trial functions, replacing algebraic identities with variational methods.
• Employs an 'optimal' form of the Cauchy-Schwarz inequality to derive sharp bounds without introducing free parameters or projections.
• Derives two families of inequalities: $ \sum_{i=1}^m (\lambda_{m+1} - \lambda_i)^p \leq \frac{4}{n} \sum_{i=1}^m \lambda_i (\lambda_{m+1} - \lambda_i)^{p-1} $ for $ p \leq 2 $, and $ \sum_{i=1}^m (\lambda_{m+1} - \lambda_i)^p \leq \frac{2p}{n} \sum_{i=1}^m \lambda_i (\lambda_{m+1} - \lambda_i)^{p-1} $ for $ p \geq 2 $, generalizing Harrell-Stubbe results.
• Applies the framework to physical and geometric settings, including the Dirichlet Laplacian, Schrödinger operators with magnetic potentials, and Sturm-Liouville problems.
• Transforms general second-order elliptic operators via change of variables and dependent variables to reduce them to the Laplacian case, where known inequalities apply.

Experimental results

Research questions

• RQ1 Can Harrell-Stubbe type inequalities be derived in a parameter-free and projection-free manner using abstract operator-theoretic tools?
• RQ2 How does the strength of the eigenvalue inequalities vary with the exponent $ p $, and is there a maximal inequality for $ p \geq 2 $?
• RQ3 To what extent can the classical Hile-Protter and H. C. Yang inequalities be unified and generalized within a single framework?
• RQ4 Can the derived inequalities be extended to physical and geometric problems such as Schrödinger operators and Sturm-Liouville systems?
• RQ5 What is the role of the optimal Cauchy-Schwarz inequality in improving the sharpness of universal eigenvalue estimates?

Key findings

• The inequality $ \sum_{i=1}^m (\lambda_{m+1} - \lambda_i)^p \leq \frac{4}{n} \sum_{i=1}^m \lambda_i (\lambda_{m+1} - \lambda_i)^{p-1} $ holds for all $ p \leq 2 $, with the $ p=0 $ case recovering the Hile-Protter inequality and the $ p=2 $ case yielding the H. C. Yang inequality.
• For $ p \geq 2 $, the inequality $ \sum_{i=1}^m (\lambda_{m+1} - \lambda_i)^p \leq \frac{2p}{n} \sum_{i=1}^m \lambda_i (\lambda_{m+1} - \lambda_i)^{p-1} $ is derived, and the $ p=2 $ case is shown to be the strongest among all $ p \geq 2 $, confirming it as the optimal bound in this range.
• The $ p=1 $ case yields the 'Yang 2' bound: $ \lambda_{m+1} \leq \left(1 + \frac{4}{n}\right) \frac{1}{m} \sum_{i=1}^m \lambda_i $, which is stronger than the classical Payne-Pólya-Weinberger inequality.
• The paper proves that the $ p=2 $ inequality (H. C. Yang) is the strongest for $ p \geq 2 $, and that the $ p \leq 2 $ family improves monotonically with increasing $ p $, with the $ p=2 $ case being the tightest in that range.
• For the Sturm-Liouville problem with $ Q(x) \geq M $, the inequalities $ \sum_{i=0}^m (\lambda_{m+1} - \lambda_i)^p \leq 4 \sum_{i=0}^m (\lambda_{m+1} - \lambda_i)^{p-1} (\lambda_i - M) $ for $ p \leq 2 $, and $ \sum_{i=0}^m (\lambda_{m+1} - \lambda_i)^p \leq 2p \sum_{i=0}^m (\lambda_{m+1} - \lambda_i)^{p-1} (\lambda_i - M) $ for $ p \geq 2 $, are established.
• The framework allows for transformation of general elliptic operators into the Laplacian case via change of variables and dependent variables, enabling the transfer of known inequalities to broader classes of operators with minimal assumptions.