Ueber Eigenwerte, Integrale und pi^2/6: Die Idee der Spurformel (On eigenvalues, integrals and pi^2/6: The idea of the trace formula)

TL;DR

This paper explores the analogy between eigenvalues and integrals through the trace formula, illustrating how daring analogies can lead to surprising mathematical results like the sum of inverse squares equaling pi^2/6.

Contribution

It introduces a novel analogy between matrix eigenvalues and integral formulas, demonstrating how this approach can derive classical results and connect different areas of mathematics.

Findings

Derivation of the sum of inverse squares as pi^2/6 using the trace formula analogy

Illustration of the connection between eigenvalues, integrals, and mathematical constants

Application of the analogy to physical systems like billiards and drums

Abstract

This is an expository article that results from a talk given to second year students at Oldenburg university. The aim of the talk was to show what beautiful and unexpected results may be obtained if one plays with daring analogies in a way that is usually not done in undergraduate education (unfortunately): We start from the fact that the sum of diagonal entries of a symmetric matrix equals the sum of its eigenvalues. We then guess an analogous formula where the matrix is replaced by a function of two real variables and sums are replaced by integrals in a systematic way. We show that this is indeed a worthwhile process: In a special case it yields that the sum of inverse squares of the positive integers is pi^2/6. Finally, an outline of the proof of the guessed formula is given, and further applications, for example to the connection between billiards and the frequencies of a drum, are explained.