Can One See the Shape of a Surface?

Abstract

Some ten years ago Professor Mark Kac published in these pages an article [3] entitled Can one hear the shape of a Drum?. Under this provocative title Kac raised a fascinating question: To what extent is the shape of a smooth closed surface (a drum) determined by the eigenvalues of the classical Dirichlet problem for the interior? Since these eigenvalues are just the resonance frequencies for the acoustic wave motion in the interior, the question asks: to what extent is the shape of a surface determined by the sound of its interior? This problem is a representative example of a class of problems, called inverse problems, which are quite different in character from the more familiar direct problems of partial differential equations. Generally speaking, direct problems ask us to develop a solution from the data, Whereas inverse problems ask us to recover the data from the solution. Inverse problems have attracted a consideKable and increasing interest in recent years because of their many immediate practical applications in our physical world. A splendid introduction may be found in Prof. Keller's recent article [20]. Typically they are very hard problems; Kac's problem, for example, remains unsolved today, though it has given rise to a considerable literature. Here we should like to raise a companion question: To what extent is the shape of a surface determined by the eigenfunctions of the classical Dirichlet problem for the exterior? We must note first of all that information about the shape of the drum is not to be found in the eigenvalues of the exterior problem, since, as we shall see, these consist of all negative real numbers, no matter what the shape. So whatever information about the shape can be recovered from the solution must be somehow contained in the eigenfunctions of the exterior problem. In particular, we expect on the basis of pract,ical experience that this information should be available in the asymptotic behavior of these eigenfunctions at great distances from the surface, since it is this behavior that determines what we when the surface is illuminated by a given source of light, radar, or sonar waves. That is, we expect to find that the shape of a surface is determined by the sight of its exterior. Thus we can see the shape of a drum, even if we can't hear it.