eBook ISBN:  9781470401511 
Product Code:  MEMO/119/572.E 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $28.80 
eBook ISBN:  9781470401511 
Product Code:  MEMO/119/572.E 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $28.80 

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.

Table of Contents

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 $\intbar _R q \neq 0$


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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 $\intbar _R q \neq 0$