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Nonlinear Dirac Equation: Spectral Stability of Solitary Waves [Kõva köide]

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  • Formaat: Hardback, 297 pages, kõrgus x laius: 254x178 mm, kaal: 760 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Jan-2020
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470443953
  • ISBN-13: 9781470443955
Teised raamatud teemal:
Nonlinear Dirac Equation: Spectral Stability of Solitary WavesSuurem pilt
  • Formaat: Hardback, 297 pages, kõrgus x laius: 254x178 mm, kaal: 760 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Jan-2020
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470443953
  • ISBN-13: 9781470443955
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470443955.html
Märksõnad:
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.
Chapter I Introduction
1(8)
Chapter II Distributions and function spaces
9(16)
II.1 Distributions
9(2)
II.2 Sobolev spaces
11(6)
II.3 Compactness of in Lq
17(2)
II.4 Cotlar-Stein almost orthogonality lemma
19(4)
II.5 The Polya-Szego inequality
23(1)
II.6 The Paley Wiener theorem
23(2)
Chapter III Spectral theory of nonselfadjoint operators
25(42)
III.1 Basic theory of unbounded operators
25(5)
III.2 Adjoint operators
30(3)
III.3 Spectrum of a linear operator
33(5)
III.4 Fredholm operators
38(5)
III.5 Normal eigenvalues and the discrete spectrum
43(5)
III.6 Operators in the Hilbert space: symmetric, normal, self-adjoint
48(2)
III.7 Essential spectra and the Weyl theorem
50(5)
III.8 The Schur complement
55(2)
III.9 The Keldysh theory of characteristic roots
57(2)
III.10 Quantum Mechanics examples
59(3)
III.11 Spectrum of the Dirac operator
62(5)
Chapter IV Linear stability of NLS solitary waves
67(8)
IV.1 Derrick's instability theorem vs. linear stability analysis
67(2)
IV.2 The Kolokolov stability criterion of NLS groundstates
69(6)
Chapter V Solitary waves of nonlinear Schrodinger equation
75(22)
V.1 The Pokhozhaev identity
75(3)
V.2 Existence of groundstates
78(4)
V.3 Decay and regularity of solitary waves
82(8)
V.4 Regularity and linear stability in pure power case
90(7)
Chapter VI Limiting absorption principle
97(18)
VI.1 Agmon's Appendix A
97(3)
VI.2 Improvement at the continuous spectrum
100(2)
VI.3 Limiting absorption principle for the Laplacian near the threshold
102(3)
VI.4 Limiting absorption principle for the Dirac operator
105(3)
VI.5 Analytic continuation of the free resolvent
108(7)
Chapter VII Carleman-Berthier-Georgescu estimates
115(26)
VII.1 Heuristics
115(2)
VII.2 Carleman-Berthier-Georgescu estimates for Dm + V
117(14)
VII.3 Carleman-Berthier-Georgescu estimates for J(Dm--- w + V)
131(2)
VII.4 Absence of embedded eigenstates
133(2)
VII.5 Exponential decay of eigenstates
135(6)
Chapter VIII The Dirac matrices
141(18)
VIII.1 The Dirac-Pauli theorem
143(7)
VIII.2 Possible number of real and imaginary Dirac matrices
150(9)
Chapter IX The Soler model
159(24)
IX.1 The well-posedness of the Cauchy problem in Sobolev spaces
160(3)
IX.2 Discrete symmetries
163(1)
IX.3 Continuous symmetries
164(5)
IX.4 Relation to the massive Thirring model
169(1)
IX.5 Solitary waves of the nonlinear Dirac equation
170(8)
IX.6 Linearization at a solitary wave
178(5)
Chapter X Bi-frequency solitary waves
183(8)
X.1 The Bogoliubov symmetry and associated charges
184(3)
X.2 Bi-frequency solitary waves
187(4)
Chapter XI Bifurcations of eigenvalues from the essential spectrum
191(12)
XI.1 Bifurcations of eigenvalues from the essential spectrum
194(1)
XI.2 Bifurcation of eigenvalues before the embedded threshold
194(2)
XI.3 Bifurcation of eigenvalues beyond the embedded thresholds
196(7)
Chapter XII Nonrelativistic asymptotics of solitary waves
203(40)
XII.1 Main results
204(4)
XII.2 Solitary waves in the nonrelativistic limit: the case ƒ $$C
208(11)
XII.3 Positivity of φωβφω and improved estimates
219(10)
XII.4 Improved error estimates
229(3)
XII.5 Solitary waves in the nonrelativistic limit: the case ƒ $$ C1
232(7)
XII.6 The Kolokolov condition for the nonlinear Dirac equation
239(4)
Chapter XIII Spectral stability in the nonrelativistic limit
243(36)
XIII.1 Main results
244(5)
XIII.2 The linearization operator
249(5)
XIII.3 Bifurcations from the origin
254(13)
XIII.4 Bifurcations from embedded thresholds
267(12)
Bibliography 279(14)
Index 293(2)
List of symbols 295
Nabile Boussaid, Universite de Franche-Comte, Besancon, France.

Andrew Comech, Texas A&M University, College Station, TX.