Riemannian Manifolds as Quantum Mechanical Worlds: The Spectrum and Eigenfunctions of the Laplacian

Abstract

While harmonic analysis on domains in Euclidean space is a long established field, as seen in §1.8, the study of the Laplace operator on Riemannian manifolds (together with the heat and wave equations, and the spectrum and eigenfunctions) seems to have begun only quite recently. Some of the earliest accomplishments were the computation of the spectrum of ℂℙⁿ (see §§9.5.4) and Lichnerowicz’s inequality for the first eigenvalue (see §§9.10.1). The first paper to address the Laplacian on general Riemannian manifolds was Minakshisundaram & Pleijel 1949 [930]. More narrowly, Maaß 1949 [899] investigated the Laplacian on Riemann surfaces. Also, one can turn to Avakumović 1956 [90]. But a spark was lit in the 1960’s when Leon Green asked if a Riemannian manifold was determined by its spectrum (the complete set of eigenvalues of the Laplacian).