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 Patterns-Chaldni Patterns-are 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