claim: Backing vocalists provide vocal harmony.
evidence: In mathematics , the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation .. mathematics. mathematics. spectral theory. spectral theory. spectrum. Spectrum of an operator#Spectrum of unbounded operators. eigenfunction expansion. eigenfunction. ordinary differential equation. ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm -- Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval , possibly semi-infinite or infinite .. Hermann Weyl. Hermann Weyl. closed interval. closed interval. Unlike the classical case , the spectrum may no longer consist of just a countable set of eigenvalues , but may also contain a continuous part .. spectrum. Spectrum of an operator#Spectrum of unbounded operators. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure , given by the Titchmarsh -- Kodaira formula .. eigenfunction expansion. eigenfunction. spectral measure. spectral theory. Titchmarsh. E. C. Titchmarsh. Kodaira. Kunihiko Kodaira. formula. Titchmarsh-Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others , using von Neumann 's spectral theorem .. Kodaira. Kunihiko Kodaira. von Neumann. John von Neumann. spectral theorem. spectral theorem. It has had important applications in quantum mechanics , operator theory and harmonic analysis on semisimple Lie groups .. quantum mechanics. quantum mechanics. operator theory. operator theory. harmonic analysis. harmonic analysis