Eigenvalue Problems for Linear Compact Symmetric Operators

Abstract

In this chapter we want to study the following eigenvalue problem: $Au\, = \lambda u,\;\quad u \in X,\quad \lambda \in \mathbb{K},\quad u \ne 0,$on the Hilbert space X over K,along with applications to integral equations and boundary-value problems.

The validity of theorems on eigenfunctions can be made plausible by the following observation made by Daniel Bernoulli (1700–1782). A mechanical system of n degrees of freedom possesses exactly n eigensolutions. A membrane is, however, a system with an infinite number of degrees of freedom. This system will, therefore, have an infinite number of eigenoscillations. Arnold Sommerfeld, 1900

In 1900 Fredholm had proved the existence of solutions for linear integral equations of the second kind. His result was sufficient to solve the boundary-value problems of potential theory. But Fred-holm’s theory did not include the eigenoscillations and the expansion of arbitrary functions with respect to eigenfunctions. Only Hilbert solved this problem by using finite-dimensional approximations and a passage to the limit. In this way he obtained a generalization of the classical principal-axis transformation for symmetric matrices to infinite-dimensional matrices. The symmetry of the matrices corresponds to the symmetry of the kernels of integral equations, and it turns out that the kernels appearing in oscillation problems are indeed symmetrical. Otto Blumenthal, 1932

A great master of mathematics passed away when Hilbert died in Göttingen on February 14, 1943, at the age of eighty-one. In retrospect, it seems that the era of mathematics upon which he impressed the seal of his spirit, and which is now sinking below the horizon, achieved a more perfect balance than has prevailed before or since, between the mastering of single concrete problems and the formation of general abstract concepts. Hermann Weyl, 1944