On an algorithm for two-term spectral asymptotic formulas

TL;DR

This paper challenges a previously established algorithm for calculating two-term spectral asymptotic formulas, providing a counter-example in the elastic eigenvalues case that questions the validity of earlier conclusions.

Contribution

It presents a counter-example demonstrating the failure of an existing algorithm for spectral asymptotics in elastic eigenvalues, questioning prior theoretical results.

Findings

The established algorithm does not apply to elastic eigenvalues.

Previous spectral asymptotic formulas based on this algorithm are incorrect.

Counter-example invalidates key conclusions in the referenced book.

Abstract

In the book [Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, Amer. Math. Soc., Providence, RI, 1997], a key and central ``algorithm'' was established, by which the coefficients of two-term asymptotic expansions of the eigenvalue counting functions can be explicitly calculated for many partial differential operators under an additional geometric assumption. In this paper, we give a counter-example to this ``algorithm'' by discussing the case of elastic eigenvalues. This implies that the most conclusions in the above book written by Yu. Safarov and D. Vassiliev are fundamentally wrong because they are based on the erroneous ``algorithm''.