|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.
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.
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.
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 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.
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.