A Formula for Finding a Potential from Nodal Lines

In this announcement we consider an eigenvalue problem which arises in the study of rectangular membranes. The mathematical model is an elliptic equation, in potential form, with Dirichlet boundary conditions. We have shown that the potential is uniquely determined, up to an additive constant, by a subset of the nodal lines of the eigenfunctions. A formula is given which, when the additive constant is fixed, yields an approximation to the potential at a dense set of points. An estimate is presented for the error made by the formula. Introduction In this summary we consider the Dirichlet eigenvalue problem for the operator (A +q) on a rectangle. This problem arises in the study of rectangular, vibrating membranes. The goal here is to solve the inverse problem: find q from the nodal line positions of the eigenfunctions. We show that, for almost all rectangles and for sufficiently smooth q, a potential q whose integral average is zero can be uniquely determined from a subset of the nodal lines of the eigenfunctions. The theorems, which we present in this paper, extend the one-dimensional results of McLaughlin and Hald [8], [4], [5], where inverse nodal problems are defined. There the authors showed that the one-dimensional potential for the Sturm-Liouville problem with Dirichlet or mixed boundary conditions is uniquely determined, up to an additive constant, by a dense set of nodes. The analysis of the Sturm-Liouville problem depends on perturbation theory. The proofs of the perturbation results are reasonably straightforward. This is possible because the difference between consecutive eigenvalues increases as the order of the eigenvalues increases. The perturbation results provide asymptotic forms for large eigenvalues, the corresponding eigenfunctions and nodal positions. The uniqueness theorem follows. For the two-dimensional problem we also require asymptotic forms for the eigenvalues and eigenfunctions and approximate location of nodal lines. The difference between the one-dimensional and the higher dimensional cases is that in the higher dimensional case the eigenvalues are not well separated. Even for Received by the editors October 6, 1993, and, in revised form, September 9, 1994. 1991 Mathematics Subject Classification. Primary 35R30, 35P20, 73D50. Partial support for the first author's research came from ONR grant N00014-91J-1166 and from NSF grant VPW-8902967. Partial support for the second author's research came from NSF grant DMS-9003033. The research announced here was presented at the 1994 International Congress of Mathematicians in Zürich, Switzerland. ©1995 American Mathematical Society 0273-0979/95 $1.00 + $.25 per page