**Mathematical Surveys and Monographs**

Volume: 244;
2019;
297 pp;
Hardcover

MSC: Primary 35; 37; 47; 81;

**Print ISBN: 978-1-4704-4395-5
Product Code: SURV/244**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

**Electronic ISBN: 978-1-4704-5422-7
Product Code: SURV/244.E**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

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#### Supplemental Materials

# Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

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*Nabile Boussaïd; Andrew Comech*

This monograph gives a comprehensive treatment
of spectral (linear) stability of weakly relativistic solitary waves
in the nonlinear Dirac equation. It turns out that the instability is
not an intrinsic property of the Dirac equation that is only resolved
in the framework of the second quantization with the Dirac sea
hypothesis. Whereas general results about the Dirac-Maxwell and
similar equations are not yet available, we can consider the Dirac
equation with scalar self-interaction, the model first introduced in
1938. In this book we show that in particular cases solitary waves in
this model may be spectrally stable (no linear instability). This
result is the first step towards proving asymptotic stability of
solitary waves.

The book presents the necessary overview of the functional
analysis, spectral theory, and the existence and linear stability of
solitary waves of the nonlinear Schrödinger equation. It also presents
the necessary tools such as the limiting absorption principle and the
Carleman estimates in the form applicable to the Dirac operator, and
proves the general form of the Dirac-Pauli theorem. All of these
results are used to prove the spectral stability of weakly
relativistic solitary wave solutions of the nonlinear Dirac equation.

#### Readership

Graduate students and researchers interested in the analysis of non-linear PDEs.

#### Table of Contents

# Table of Contents

## Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

- Cover Cover11
- Title page i2
- Chapter I. Introduction 18
- Chapter II. Distributions and function spaces 916
- Chapter III. Spectral theory of nonselfadjoint operators 2532
- III.1. Basic theory of unbounded operators 2532
- III.2. Adjoint operators 3037
- III.3. Spectrum of a linear operator 3340
- III.4. Fredholm operators 3845
- III.5. Normal eigenvalues and the discrete spectrum 4350
- III.6. Operators in the Hilbert space: symmetric, normal, self-adjoint 4855
- III.7. Essential spectra and the Weyl theorem 5057
- III.8. The Schur complement 5562
- III.9. The Keldysh theory of characteristic roots 5764
- III.10. Quantum Mechanics examples 5966
- III.11. Spectrum of the Dirac operator 6269

- Chapter IV. Linear stability of NLS solitary waves 6774
- Chapter V. Solitary waves of nonlinear Schrödinger equation 7582
- Chapter VI. Limiting absorption principle 97104
- Chapter VII. Carleman–Berthier–Georgescu estimates 115122
- Chapter VIII. The Dirac matrices 141148
- Chapter IX. The Soler model 159166
- Chapter X. Bi-frequency solitary waves 183190
- Chapter XI. Bifurcations of eigenvalues from the essential spectrum 191198
- Chapter XII. Nonrelativistic asymptotics of solitary waves 203210
- XII.1. Main results 204211
- XII.2. Solitary waves in the nonrelativistic limit: the case 𝑓∈𝐶 208215
- XII.3. Positivity of 𝜙_{𝜔}*𝛽𝜙_{𝜔} and improved estimates 219226
- XII.4. Improved error estimates 229236
- XII.5. Solitary waves in the nonrelativistic limit: the case 𝑓∈𝐶¹ 232239
- XII.6. The Kolokolov condition for the nonlinear Dirac equation 239246

- Chapter XIII. Spectral stability in the nonrelativistic limit 243250
- Bibliography 279286
- Index 293300
- List of symbols 295302
- Back Cover Back Cover1306