Memoirs of the American Mathematical Society 1996; 148 pp; softcover Volume: 119 ISBN10: 0821804863 ISBN13: 9780821804865 List Price: US$48 Individual Members: US$28.80 Institutional Members: US$38.40 Order Code: MEMO/119/572
 Can you hear the shape of a drum? No. In this book, the authors ask, "Can you see the force on a drum?" Hald and McLaughlin prove that for almost all rectangles the potential in a Schrödinger equation is uniquely determined (up to an additive constant) by a subset of the nodal lines. They derive asymptotic expansions for a rich set of eigenvalues and eigenfunctions. Using only the nodal line positions, they establish an approximate formula for the potential and give error bounds. The theory is appropriate for a graduate topics course in analysis with emphasis on inverse problems. Features:  The formulas that solve the inverse problem are very simple and easy to state.
 Nodal Line PatternsChaldni Patternsare shown to be a rich source of data for the inverse problem.
 The data in this book is used to establish a simple formula that is the solution of an inverse problem.
Readership Undergraduates studying PDEs, graduate students, and research mathematicians interested in analysis with emphasis on inverse problems. Table of Contents  Introduction
 Separation of eigenvalues for the Laplacian
 Eigenvalues for the finite dimensional problem
 Eigenfunctions for the finite dimensional problem
 Eigenvalues for \( \Delta + q\)
 Eigenfunctions for \( \Delta + q\)
 The inverse nodal problem
 The case \(f_R q\neq 0\)
 Acknowledgment
 References
 Appendices
