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Inverse Nodal Problems: Finding the Potential from Nodal Lines
Ole H. Hald, University of California, Berkeley, CA, and Joyce R. McLaughlin, Rensselaer Polytechnic Institute, Troy, NY

Memoirs of the American Mathematical Society
1996; 148 pp; softcover
Volume: 119
ISBN-10: 0-8218-0486-3
ISBN-13: 978-0-8218-0486-5
List Price: US$48
Individual Members: US$28.80
Institutional Members: US$38.40
Order Code: MEMO/119/572
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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.


  • 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.


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
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