[Paper Review] Semiclassical bounds for spectra of biharmonic operators

Davide Buoso, Luigi Provenzano+1 more|arXiv (Cornell University)|Apr 26, 2019

Nonlinear Partial Differential Equations22 references4 citations

TL;DR

This paper establishes sharp semiclassical bounds for the eigenvalues of biharmonic operators using the averaged variational principle (AVP), yielding two-sided estimates for Riesz means $ R_1(z) $ with a second-order term in the asymptotic expansion. The method provides eigenvalue bounds that are semiclassically sharp and enables comparisons between biharmonic and Laplacian spectra.

ABSTRACT

We provide complementary semiclassical bounds for the Riesz means $R_1(z)$ of the eigenvalues of various biharmonic operators, with a second term in the expected power of $z$. The method we discuss makes use of the averaged variational principle (AVP), and yields two-sided bounds for individual eigenvalues, which are semiclassically sharp. The AVP also yields comparisons with Riesz means of different operators, in particular Laplacians.

Motivation & Objective

• To derive sharp semiclassical bounds for the eigenvalues of biharmonic operators with various boundary conditions.
• To establish two-sided estimates for Riesz means $ R_1(z) $, including a second-order term in the asymptotic expansion.
• To apply the averaged variational principle (AVP) to obtain eigenvalue bounds that are semiclassically sharp.
• To compare the spectral behavior of biharmonic operators with that of Laplacians through Riesz means.
• To extend known Weyl asymptotics to include precise error terms and bounds for individual eigenvalues.

Proposed method

• Utilizes the averaged variational principle (AVP) to derive bounds on eigenvalues and Riesz means of biharmonic operators.
• Applies the AVP to compare eigenvalues of biharmonic operators with those of Laplacians, leveraging phase space volume estimates.
• Employs integral representations of Riesz means via the counting function $ N(z) $, related by $ R_ ho(z) = \rho \int_0^\infty (z-t)_+^{\rho-1} N(t) dt $.
• Derives asymptotic expansions for eigenvalues using the Weyl law, with principal symbol $ |p|^4 $ for the biharmonic operator.
• Uses summation estimates involving $ \sum_{n \geq 1} (R^4 - n^4)_+ $ and $ \sum_{n \geq 1} (R^4 - (n+1/2)^4)_+ $ to bound Riesz means with polynomial error terms.
• Applies concavity and extremal analysis on polynomial functions to derive tight upper and lower bounds for discrete sums, ensuring sharpness in the semiclassical limit.

Experimental results

Research questions

• RQ1 Can two-sided semiclassical bounds be established for the Riesz means $ R_1(z) $ of biharmonic operators with a second-order correction term?
• RQ2 How does the averaged variational principle (AVP) enable sharp eigenvalue estimates for biharmonic operators?
• RQ3 To what extent can the spectra of biharmonic operators be compared with those of Laplacians using Riesz means?
• RQ4 Are the derived bounds for eigenvalues and Riesz means semiclassically sharp, i.e., do they match the leading and subleading terms in the Weyl asymptotics?
• RQ5 What are the optimal polynomial bounds for sums of the form $ \sum (R^4 - n^4)_+ $ and $ \sum (R^4 - (n+1/2)^4)_+ $, and how do they contribute to spectral estimates?

Key findings

• The paper establishes two-sided bounds for $ R_1(z) $ of biharmonic operators with a second-order term, improving upon the leading Weyl term.
• For the biharmonic Dirichlet operator, the AVP yields lower bounds for Riesz means that are semiclassically sharp.
• The method produces asymptotically Weyl-sharp bounds on individual eigenvalues, meaning the error terms match the expected subleading order in the Weyl law.
• For the one-dimensional biharmonic problem, the authors derive explicit bounds for $ \sum_{n \geq 1} (R^4 - n^4)_+ $ and $ \sum_{n \geq 1} (R^4 - (n+1/2)^4)_+ $, with polynomial error terms.
• The bounds $ \sum (R^4 - n^4)_+ \leq \frac{4}{5}R^5 - \frac{1}{2}R^4 + \frac{1}{6}R^3 + \frac{1}{12}R^2 $ and $ \sum (R^4 - (n+1/2)^4)_+ \geq \frac{4}{5}R^5 - R^4 - \frac{11}{6}R^3 - \frac{3}{2}R^2 - \frac{127}{240}R $ are derived and shown to be sharp.
• The AVP enables comparison of Riesz means across different operators, particularly linking biharmonic and Laplacian spectra via phase space volume estimates.