| eBook ISBN: | 978-1-4704-0151-1 |
| Product Code: | MEMO/119/572.E |
| List Price: | $48.00 |
| MAA Member Price: | $43.20 |
| AMS Member Price: | $28.80 |
| eBook ISBN: | 978-1-4704-0151-1 |
| Product Code: | MEMO/119/572.E |
| List Price: | $48.00 |
| MAA Member Price: | $43.20 |
| AMS Member Price: | $28.80 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 119; 1996; 148 ppMSC: Primary 35; 74
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.
ReadershipUndergraduates studying PDEs, graduate students, and research mathematicians interested in analysis with emphasis on inverse problems.
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Table of Contents
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Chapters
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Introduction
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1. Separation of eigenvalues for the Laplacian
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2. Eigenvalues for the finite dimensional problem
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3. Eigenfunctions for the finite dimensional problem
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4. Eigenvalues for $-\Delta + q$
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5. Eigenfunctions for $-\Delta + q$
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6. The inverse nodal problem
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7. The case $\intbar _R q \neq 0$
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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.
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.
Undergraduates studying PDEs, graduate students, and research mathematicians interested in analysis with emphasis on inverse problems.
-
Chapters
-
Introduction
-
1. Separation of eigenvalues for the Laplacian
-
2. Eigenvalues for the finite dimensional problem
-
3. Eigenfunctions for the finite dimensional problem
-
4. Eigenvalues for $-\Delta + q$
-
5. Eigenfunctions for $-\Delta + q$
-
6. The inverse nodal problem
-
7. The case $\intbar _R q \neq 0$
