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Full Description
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 Schrodinger 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.
Contents
Introduction
Distributions and function spaces
Spectral theory of nonselfadjoint operators
Linear stability of NLS solitary waves
Solitary waves of nonlinear Schrodinger equation
Limiting absorption principle
Carleman-Berthier-Georgescu estimates
The Dirac matrices
The Soler model
Bi-frequency solitary waves
Bifurcations of eigenvalues from the essential spectrum
Nonrelativistic asymptotics of solitary waves
Spectral stability in the nonrelativistic limit
Bibliography
Index
List of symbols.
