**Memoirs of the American Mathematical Society**

1996;
148 pp;
Softcover

MSC: Primary 35; 74;

Print ISBN: 978-0-8218-0486-5

Product Code: MEMO/119/572

List Price: $48.00

Individual Member Price: $28.80

**Electronic ISBN: 978-1-4704-0151-1
Product Code: MEMO/119/572.E**

List Price: $48.00

Individual Member Price: $28.80

# Inverse Nodal Problems: Finding the Potential from Nodal Lines

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*Ole H. Hald; Joyce R. McLaughlin*

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.

#### Table of Contents

# Table of Contents

## Inverse Nodal Problems: Finding the Potential from Nodal Lines

- Contents v6 free
- Introduction 110 free
- 1 Separation of eigenvalues for the Laplacian 514 free
- 2 Eigenvalues for the finite dimensional problem 2534
- 3 Eigenfunctions for the finite dimensional problem 3140
- 4 Eigenvalues for … Δ+q 5160
- 5 Eigenfunctions for … Δ+q 7988
- 6 The inverse nodal problem 91100
- 7 The case f[sub(R)q≠0 103112
- Acknowledgment 105114
- References 106115
- Appendix A 109118
- Appendix B 117126
- Appendix C 121130
- Appendix D 141150

#### Readership

Undergraduates studying PDEs, graduate students, and research mathematicians interested in analysis with emphasis on inverse problems.