[Paper Review] On Riesz Means of Eigenvalues

Evans M. Harrell, Lotfi Hermi|ArXiv.org|Dec 25, 2007

Spectral Theory in Mathematical Physics28 references4 citations

TL;DR

This paper establishes the equivalence of key eigenvalue inequalities for the Dirichlet Laplacian using integral transforms, including Laplace, Legendre, and Riemann-Liouville fractional transforms. It proves new universal inequalities for Riesz means, partition functions, and spectral zeta functions, and conjectures sharp bounds that improve upon classical results like Kac's and Berezin-Li-Yau.

ABSTRACT

In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5-3.7) and conjecture about additional bounds.

Motivation & Objective

• To unify and prove the equivalence of classical eigenvalue inequalities—such as Berezin-Li-Yau, Yang, and Kac—using integral transforms.
• To derive new universal inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian and Schrödinger operators with discrete spectra.
• To improve bounds on the partition function and spectral zeta function, and to conjecture sharp universal constants for these functions.
• To extend these results to Schrödinger operators with potentials growing at infinity, ensuring discrete, bounded-below spectra.
• To investigate the role of geometric quantities like the moment of inertia and domain volume in spectral estimates.

Proposed method

• Uses commutator identities and sum rules to derive eigenvalue inequalities, particularly for Riesz means of order $\rho > 0$.
• Applies integral transforms—Laplace, Legendre, Weyl, Mellin, and Riemann-Liouville fractional transforms—to relate different spectral inequalities.
• Employs the Riesz iteration formula: $ R_{\rho+\delta}(z) = \frac{\Gamma(\rho+\delta+1)}{\Gamma(\rho+1)\Gamma(\delta)} \int_0^\infty (z-t)_+^{\delta-1} R_\rho(t) dt $, linking Riesz means of different orders.
• Derives bounds via the Legendre transform of the Riesz mean, leading to improved estimates for the heat kernel trace.
• Applies the Weyl transform to the heat kernel bound to obtain estimates for the spectral zeta function $ \zeta_{\text{spec}}(\rho) $.
• Conjectures that the optimal bound replaces the geometric term $ M_d |\Omega| / I(\Omega) $ with $ |\Omega|^{-2/d} $, yielding a sharper universal constant.

Experimental results

Research questions

• RQ1 How are the Berezin-Li-Yau, Yang, and Kac inequalities for eigenvalues related via integral transforms?
• RQ2 Can new universal inequalities for Riesz means of eigenvalues be derived using commutator identities and sum rules?
• RQ3 What are the tightest possible bounds for the partition function and spectral zeta function of the Dirichlet Laplacian?
• RQ4 Is the conjectured bound $ \sum e^{-\lambda_i t} \leq \frac{|\Omega|}{(4\pi t)^{d/2}} e^{-t/|\Omega|^{2/d}} $ sharp for the heat kernel?
• RQ5 Can the spectral zeta function $ \zeta_{\text{spec}}(\rho) $ be universally bounded by a constant involving $ \Gamma(\rho - d/2)/\Gamma(\rho) $ for $ \rho > d/2 $?

Key findings

• The Berezin-Li-Yau inequality is equivalent to Kac’s inequality under the Legendre transform, and both are related via Riesz iteration and Laplace transforms.
• For $ \rho \geq 1 $, the Riesz mean satisfies $ R_\rho(z) \leq L_{\rho,d}^{\text{cl}} |\Omega| z^{\rho + d/2} $, with equality in the semiclassical limit.
• The heat kernel satisfies $ \sum e^{-\lambda_i t} \leq \frac{|\Omega|}{(4\pi t)^{d/2}} e^{-M_d |\Omega| t / I(\Omega)} $, derived from the Legendre transform of the Riesz mean.
• The spectral zeta function is bounded by $ \zeta_{\text{spec}}(\rho) \leq \frac{1}{(4\pi)^{d/2}} \frac{\Gamma(\rho - d/2)}{\Gamma(\rho)} |\Omega|^{2\rho/d} \left( M_d \frac{|\Omega|}{I(\Omega)} \right)^{d/2 - \rho} $ for $ \rho > d/2 $.
• The conjectured bound $ \sum e^{-\lambda_i t} \leq \frac{|\Omega|}{(4\pi t)^{d/2}} e^{-t/|\Omega|^{2/d}} $ would yield a sharper universal estimate for the spectral zeta function.
• Numerical evidence in Figure 3 supports Conjecture 4.2, showing that the proposed bound improves on Berezin-Li-Yau and Yang-type bounds for $ \rho > \rho_0 $, with $ \rho_0 $ depending on dimension $ d $.