Cardiff University Logo  Cardiff School of Mathematics
Prelude for Strings

THE ORIGINS OF SPECTRAL THEORY

The problems of spectral theory are among the very first questions in scientific history, and have served as a starting-point for the development of mathematics. The history of this discipline shows how the investigation of questions which, at a first glance, appear quite abstract and academic, can yield methods of enormous practical value, often after decades or centuries of research.

The Pythagoreans (6th - 5th century BC) studied the vibrations of a taut string, finding that harmonically consonant sounds are produced when the string is divided in simple numerical ratios. This observation corroborated their tenet that everything in the world is governed by relations of numbers. The Pythagorean Hippasos of Metapontum, credited, among other things, with the invention of the regular dodecahedron and the irrational numbers, is said to have investigated the vibrations of metal plates as well.
At the medieval universities, music - essentially the ancient theory of harmony - was taught in the quadrivium (fourfold way) along with arithmetic, geometry and astronomy; together with the trivium of grammar, logic and rhetoric, these constituted the seven liberal arts.

Johannes Kepler (1571-1630) tried to explain the radii of the planetary orbits in terms of geometrical ratios, the so-called harmony of the spheres. As a by-product of his speculations on this question (which is still unsolved today) he found Kepler's laws, the basis of the modern picture of the solar system.
John Wallis (1616-1703) described the relation between the harmonics of a vibrating string and the number of its nodes of vibration. In 1717, Brook Taylor (1685-1731) published the first mathematical paper on the vibrating string, but he did not yet know the differential equation of wave propagation.

In 1746, Jean le Rond d'Alembert (1717-1783) wrote down the one-dimensional wave equation and found a method of its solution for `arbitrary' initial data; the solution is a superposition of two waves, travelling to the right and left, resp.
The question how `arbitrary' the initial data can really be, i.e. which functions are admissible as solutions of the wave equation, was taken up by d'Alembert, Daniel Bernoulli (1700-1782) and Leonhard Euler (1707-1783). This was the beginning of the struggle for an exact definition of the concept of a function, which became one of the principal achievements of 19th century mathematics.

D. Bernoulli saw a general method of solving the wave equation in the superposition of (infinitely many) simple vibratory modes, an idea previously used by Euler in specific situations. This technique, along with separation of variables, became prominent in Joseph Fourier's (1768-1830) treatment of the heat equation (1822). However, the actual scope of the method remained obscure at first because of the limit process involved.
The idea of separation of variables, which reduces problems in two or more dimensions to a family of one-dimensional problems, is also the basis of tomography, a widely used tool of contemporary medicine.

The fundamental question of Fourier's method, viz. which functions can be expanded in their Fourier series, turned out to be very complicated and very fertile for the further development of mathematics. For piecewise continuous and piecewise monotonic functions and the trigonometric Fourier series, it was answered in 1829 by Peter Gustav Lejeune-Dirichlet (1805-1859); in its general form, it is the core of the mathematical field of harmonic analysis.
Fourier analysis is of great importance in many areas of science and technology; e.g. it is used in astronomy to enhance the optical resolution of telescopes.

Separation of variables, when applied to the fundamental equations of mathematical physics and differential geometry, often gives rise to ordinary differential equations of a general type studied by Charles-François Sturm (1803-1855) and Joseph Liouville (1809-1882). Liouville investigated the representation of solutions of this equation by generalised Fourier series. However, a satisfactory answer to this question could only be found in the framework of Hilbert space, named after David Hilbert (1862-1943), which is a corner-stone of functional analysis and the modern treatment of differential equations.

Hilbert space is also a central notion of quantum mechanics. The Schrödinger equation (1926), the fundamental equation of non-relativistic quantum mechanics, is closely related to the wave equation. Years before, Hermann Weyl (1885-1955) had paved the way for a study of this equation by his spectral analysis of the singular Sturm-Liouville equation.
The colour of the light emitted by a gas of atoms is determined by the spectrum of the associated Schrödinger equation, i.e. the frequencies of its simple vibratory modes. Thus Erwin Schrödinger (1887-1961) found the harmony of the spheres, sought in vain in the cosmos by Kepler, in the subatomic microcosmos.

<<< home

K. M. Schmidt fecit MMVIII