Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

Boussaïd, Nabile; Comech, Andrew

doi:10.1090/surv/244

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

About this Title

Nabile Boussaïd, Université de Franche-Comté, Besançon, France and Andrew Comech, Institute for Information Transmission Problems, Moscow, Russia

Publication: Mathematical Surveys and Monographs
Publication Year: 2019; Volume 244
ISBNs: 978-1-4704-4395-5 (print); 978-1-4704-5422-7 (online)
DOI: https://doi.org/10.1090/surv/244
MathSciNet review: 3971006
MSC: Primary 35-02; Secondary 35C08, 35Q55, 37K40, 47F05, 81Q05, 81Q10

PDF View full volume as PDF

Read more about this volume

Simonetta Abenda, Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998), no. 2, 229–244 (English, with English and French summaries). MR 1618672
Werner Amrein, Anne-Marie Berthier, and Vladimir Georgescu, Estimations du type Hardy ou Carleman pour des opérateurs différentiels à coefficients opératoriels, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 10, 575–578 (French, with English summary). MR 685028
A. Alvarez and B. Carreras, Interaction dynamics for the solitary waves of a nonlinear Dirac model, Phys. Lett. A 86 (1981), no. 6-7, 327–332. MR 637021, DOI 10.1016/0375-9601(81)90548-X
D. L. T. Anderson and G. H. Derrick, Stability of time-dependent particlelike solutions in nonlinear field theories. I, Journal of Mathematical Physics 11 (1970), pp. 1336–1346.
Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. MR 397194
D. L. T. Anderson, Stability of time-dependent particlelike solutions in nonlinear field theories. II, Journal of Mathematical Physics 12 (1971), pp. 945–952.
Constantin Apostol, On the closed range points in the spectrum of operators, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 8, 971–975. MR 425639
Henri Berestycki and Thierry Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489–492 (French, with English summary). MR 646873
G. Berkolaiko and A. Comech, On spectral stability of solitary waves of nonlinear Dirac equation in 1D, Math. Model. Nat. Phenom. 7 (2012), no. 2, 13–31. MR 2892774, DOI 10.1051/mmnp/20127202
Nabile Boussaid and Scipio Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations 37 (2012), no. 6, 1001–1056. MR 2924465, DOI 10.1080/03605302.2012.665973
Nikolaos Bournaveas and Timothy Candy, Global well-posedness for the massless cubic Dirac equation, Int. Math. Res. Not. IMRN 22 (2016), 6735–6828. MR 3632067, DOI 10.1093/imrn/rnv361
Nabile Boussaïd and Andrew Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal. 271 (2016), no. 6, 1462–1524. MR 3530581, DOI 10.1016/j.jfa.2016.04.013
Nabile Boussaïd and Andrew Comech, Nonrelativistic asymptotics of solitary waves in the Dirac equation with Soler-type nonlinearity, SIAM J. Math. Anal. 49 (2017), no. 4, 2527–2572. MR 3670258, DOI 10.1137/16M1081385
Nabile Boussaïd and Andrew Comech, Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models, Commun. Pure Appl. Anal. 17 (2018), no. 4, 1331–1347. MR 3842864, DOI 10.3934/cpaa.2018065
N. Boussaïd and A. Comech, Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity, J. Funct. Anal. 51 (2019), to appear, DOI 10.1016/j.jfa.2019.108289
Gregory Berkolaiko, Andrew Comech, and Alim Sukhtayev, Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields, Nonlinearity 28 (2015), no. 3, 577–592. MR 3311594, DOI 10.1088/0951-7715/28/3/577
James D. Bjorken and Sidney D. Drell, Relativistic quantum mechanics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0187641
James D. Bjorken and Sidney D. Drell, Relativistic quantum fields, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0187642
Anne Boutet de Monvel-Berthier, Dragos Manda, and Radu Purice, Limiting absorption principle for the Dirac operator, Ann. Inst. H. Poincaré Phys. Théor. 58 (1993), no. 4, 413–431 (English, with English and French summaries). MR 1241704
Marius Beceanu, New estimates for a time-dependent Schrödinger equation, Duke Math. J. 159 (2011), no. 3, 417–477. MR 2831875, DOI 10.1215/00127094-1433394
Jean-Marie Barbaroux, Maria J. Esteban, and Eric Séré, Some connections between Dirac-Fock and electron-positron Hartree-Fock, Ann. Henri Poincaré 6 (2005), no. 1, 85–102. MR 2119356, DOI 10.1007/s00023-005-0199-7
Volker Bach, Jürg Fröhlich, and Israel Michael Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (1998), no. 2, 299–395. MR 1639713, DOI 10.1006/aima.1998.1734
Anne Berthier and Vladimir Georgescu, On the point spectrum of Dirac operators, J. Funct. Anal. 71 (1987), no. 2, 309–338. MR 880983, DOI 10.1016/0022-1236(87)90007-3
Nabile Boussaid and Sylvain Golénia, Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies, Comm. Math. Phys. 299 (2010), no. 3, 677–708. MR 2718928, DOI 10.1007/s00220-010-1099-3
Henri Berestycki, Thierry Gallouët, and Otared Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 5, 307–310 (French, with English summary). MR 734575
E. Balslev and B. Helffer, Limiting absorption principle and resonances for the Dirac operator, Adv. in Appl. Math. 13 (1992), no. 2, 186–215. MR 1162140, DOI 10.1016/0196-8858(92)90009-L
P. Binding and R. Hryniv, Relative boundedness and relative compactness for linear operators in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2287–2290. MR 1756088, DOI 10.1090/S0002-9939-00-05729-4
Ioan Bejenaru and Sebastian Herr, The cubic Dirac equation: small initial data in $H^1(\Bbb {R}^3)$, Comm. Math. Phys. 335 (2015), no. 1, 43–82. MR 3314499, DOI 10.1007/s00220-014-2164-0
I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in $H^{\frac {1}{2}}\!(\mathbb {R}^2)$, Comm. Math. Phys. 343 (2016), pp. 515–562.
M. Sh. Birman, G. E. Karadzhov, and M. Z. Solomyak, Boundedness conditions and spectrum estimates for the operators $b(X)a(D)$ and their analogs, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 85–106. MR 1306510
Henri Berestycki and Pierre-Louis Lions, Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 7, A395–A398 (French, with English summary). MR 552061
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI 10.1007/BF00250555
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), no. 4, 347–375. MR 695536, DOI 10.1007/BF00250556
Haïm Brezis and Petru Mironescu, Gagliardo-Nirenberg inequalities and non-inequalities: the full story, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 5, 1355–1376. MR 3813967, DOI 10.1016/j.anihpc.2017.11.007
N. N. Bogoljubov, On a new method in the theory of superconductivity, Nuovo Cimento (10) 7 (1958), 794–805 (English, with Italian summary). MR 106740
N. Bohr, On the constitution of atoms and molecules, Phil. Mag. 26 (1913), pp. 1–25.
Nikolaos Bournaveas, Local existence for the Maxwell-Dirac equations in three space dimensions, Comm. Partial Differential Equations 21 (1996), no. 5-6, 693–720. MR 1391520, DOI 10.1080/03605309608821204
Nikolaos Bournaveas, Local and global solutions for a nonlinear Dirac system, Adv. Differential Equations 9 (2004), no. 5-6, 677–698. MR 2099976
Nabile Boussaid, Stable directions for small nonlinear Dirac standing waves, Comm. Math. Phys. 268 (2006), no. 3, 757–817. MR 2259214, DOI 10.1007/s00220-006-0112-3
Nikolaos Bournaveas, Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension, Discrete Contin. Dyn. Syst. 20 (2008), no. 3, 605–616. MR 2373206, DOI 10.3934/dcds.2008.20.605
Nabile Boussaid, On the asymptotic stability of small nonlinear Dirac standing waves in a resonant case, SIAM J. Math. Anal. 40 (2008), no. 4, 1621–1670. MR 2466169, DOI 10.1137/070684641
V. S. Buslaev and G. S. Perel′man, On nonlinear scattering of states which are close to a soliton, Astérisque 210 (1992), 6, 49–63. Méthodes semi-classiques, Vol. 2 (Nantes, 1991). MR 1221351
V. S. Buslaev and G. S. Perel′man, Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, Algebra i Analiz 4 (1992), no. 6, 63–102 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 6, 1111–1142. MR 1199635
V. S. Buslaev and G. S. Perel′man, On the stability of solitary waves for nonlinear Schrödinger equations, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75–98. MR 1334139, DOI 10.1090/trans2/164/04
I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, Vibrations and oscillatory instabilities of gap solitons, Phys. Rev. Lett. 80 (1998), pp. 5117–5120.
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
L. D. Broglie, Recherches sur la théorie des quanta, Ann. d. Phys. III 10 (1925), p. 22.
Friedemann Brock and Alexander Yu. Solynin, An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1759–1796. MR 1695019, DOI 10.1090/S0002-9947-99-02558-1
Vladimir S. Buslaev and Catherine Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 20 (2003), no. 3, 419–475 (English, with English and French summaries). MR 1972870, DOI 10.1016/S0294-1449(02)00018-5
Laura Burlando, On nilpotent operators, Studia Math. 166 (2005), no. 2, 101–129. MR 2109585, DOI 10.4064/sm166-2-1
Timothy Candy, Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension, Adv. Differential Equations 16 (2011), no. 7-8, 643–666. MR 2829499
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
John M. Chadam and Robert T. Glassey, On certain global solutions of the Cauchy problem for the (classical) coupled Klein-Gordon-Dirac equations in one and three space dimensions, Arch. Rational Mech. Anal. 54 (1974), 223–237. MR 369952, DOI 10.1007/BF00250789
Andrew Comech, Meijiao Guan, and Stephen Gustafson, On linear instability of solitary waves for the nonlinear Dirac equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 3, 639–654 (English, with English and French summaries). MR 3208458, DOI 10.1016/j.anihpc.2013.06.001
S. Coleman, V. Glaser, and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58 (1978), no. 2, 211–221. MR 468913
Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal. 39 (2007/08), no. 4, 1070–1111. MR 2368894, DOI 10.1137/050648389
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some non- linear Schrödinger equations, Comm. Math. Phys. 85 (1982), pp. 549–561.
Scipio Cuccagna and Tetsu Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys. 284 (2008), no. 1, 51–77. MR 2443298, DOI 10.1007/s00220-008-0605-3
Jesús Cuevas-Maraver, Panayotis G. Kevrekidis, Avadh Saxena, Andrew Comech, and Ruomeng Lan, Stability of solitary waves and vortices in a 2D nonlinear Dirac model, Phys. Rev. Lett. 116 (2016), no. 21, 214101, 6. MR 3613280, DOI 10.1103/PhysRevLett.116.214101
M. Cotlar, A combinatorial inequality and its applications to $L^2$-spaces, Rev. Mat. Cuyana 1 (1955), 41–55 (1956). MR 80263
Andrew Comech and Dmitry Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math. 56 (2003), no. 11, 1565–1607. MR 1995870, DOI 10.1002/cpa.10104
Marina Chugunova and Dmitry Pelinovsky, Block-diagonalization of the symmetric first-order coupled-mode system, SIAM J. Appl. Dyn. Syst. 5 (2006), no. 1, 66–83. MR 2217129, DOI 10.1137/050629781
Andres Contreras, Dmitry E. Pelinovsky, and Yusuke Shimabukuro, $L^2$ orbital stability of Dirac solitons in the massive Thirring model, Comm. Partial Differential Equations 41 (2016), no. 2, 227–255. MR 3462129, DOI 10.1080/03605302.2015.1123272
Andrew Comech, Tuoc Van Phan, and Atanas Stefanov, Asymptotic stability of solitary waves in generalized Gross-Neveu model, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 1, 157–196. MR 3592683, DOI 10.1016/j.anihpc.2015.11.001
Scipio Cuccagna, Dmitry Pelinovsky, and Vitali Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math. 58 (2005), no. 1, 1–29. MR 2094265, DOI 10.1002/cpa.20050
Andrew Comech and David Stuart, Small amplitude solitary waves in the Dirac-Maxwell system, Commun. Pure Appl. Anal. 17 (2018), no. 4, 1349–1370. MR 3842865, DOI 10.3934/cpaa.2018066
Scipio Cuccagna and Mirko Tarulli, On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential, J. Math. Anal. Appl. 436 (2016), no. 2, 1332–1368. MR 3447012, DOI 10.1016/j.jmaa.2015.12.049
Scipio Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), no. 9, 1110–1145. MR 1835384, DOI 10.1002/cpa.1018
Scipio Cuccagna, On asymptotic stability of ground states of NLS, Rev. Math. Phys. 15 (2003), no. 8, 877–903. MR 2027616, DOI 10.1142/S0129055X03001849
Scipio Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation, Trans. Amer. Math. Soc. 366 (2014), no. 6, 2827–2888. MR 3180733, DOI 10.1090/S0002-9947-2014-05770-X
T. Cazenave and L. Vázquez, Existence of localized solutions for a classical non- linear Dirac field, Comm. Math. Phys. 105 (1986), pp. 35–47.
E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825, DOI 10.1017/CBO9780511623721
S. De Bièvre, F. Genoud, and S. Rota Nodari, Orbital stability: analysis meets geometry, in Nonlinear optical and atomic systems, vol. 2146 of Lecture Notes in Math., pp. 147–273, Springer, Cham, 2015.
Stephan De Bièvre and Simona Rota Nodari, Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups, Arch. Ration. Mech. Anal. 231 (2019), no. 1, 233–284. MR 3894551, DOI 10.1007/s00205-018-1278-5
G. H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Mathematical Phys. 5 (1964), 1252–1254. MR 174304, DOI 10.1063/1.1704233
J. Derezinski, Open problems about many-body Dirac operators, Bulletin of International Association of Mathematical Physics (2012).
Piero D’Ancona and Luca Fanelli, Decay estimates for the wave and Dirac equations with a magnetic potential, Comm. Pure Appl. Math. 60 (2007), no. 3, 357–392. MR 2284214, DOI 10.1002/cpa.20152
P. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. Ser. A 117 (1928), pp. 610–624.
Thomas Duyckaerts, Hao Jia, Carlos Kenig, and Frank Merle, Soliton resolution along a sequence of times for the focusing energy critical wave equation, Geom. Funct. Anal. 27 (2017), no. 4, 798–862. MR 3678502, DOI 10.1007/s00039-017-0418-7
D. Darby and Th. W. Ruijgrok, A noncompact gauge group for the Dirac equation, Acta Phys. Polon. B 10 (1979), no. 11, 959–973. MR 570119
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
N. Dunford and J. T. Schwartz, Linear operators. II. Spectral Theory: self adjoint operators in Hilbert space, Interscience Publishers, Inc., New York, 1963.
Laurent Demanet and Wilhelm Schlag, Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation, Nonlinearity 19 (2006), no. 4, 829–852. MR 2214946, DOI 10.1088/0951-7715/19/4/004
S. Earnshaw, On the nature of the molecular forces which regulate the constitution of the luminiferous ether, Trans. Camb. Phil. Soc 7 (1842), pp. 97–112.
D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR 929030
Maria J. Esteban, Vladimir Georgiev, and Eric Séré, Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. Partial Differential Equations 4 (1996), no. 3, 265–281. MR 1386737, DOI 10.1007/BF01254347
W. D. Evans, R. T. Lewis, and A. Zettl, Nonselfadjoint operators and their essential spectra, in Ordinary differential equations and operators (Dundee, 1982), vol. 1032 of Lecture Notes in Math., pp. 123–160, Springer, Berlin, 1983.
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
M. J. Esteban and É. Séré, Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys. 171 (1995), pp. 323–350.
Maria J. Esteban and Eric Séré, An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 381–397. Current developments in partial differential equations (Temuco, 1999). MR 1897689, DOI 10.3934/dcds.2002.8.381
M. J. Esteban and É. Séré, Dirac-Fock models for atoms and molecules and related topics, in XIVth International Congress on Mathematical Physics, pp. 21–28, World Sci. Publ., Hackensack, NJ, 2005.
M. Escobedo and L. Vega, A semilinear Dirac equation in $H^s({\mathbf {R}}^3)$ for $s>1$, SIAM J. Math. Anal. 28 (1997), pp. 338–362.
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
B. V. Fedosov, Index theorems [ MR1135117 (92k:58262)], Partial differential equations, VIII, Encyclopaedia Math. Sci., vol. 65, Springer, Berlin, 1996, pp. 155–251. MR 1401125, DOI 10.1007/978-3-642-48944-0_{3}
R. Finkelstein, C. Fronsdal, and P. Kaus, Nonlinear spinor field, Phys. Rev. 103 (1956), pp. 1571–1579.
R. Finkelstein, R. LeLevier, and M. Ruderman, Non-linear spinor fields, Phys. Rev. (2) 83 (1951), 326–332. MR 42204
A. Galindo, A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento (2) 20 (1977), no. 6, 210–212. MR 503135
Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
A. Galtbayar, A. Jensen, and K. Yajima, Local time-decay of solutions to Schrödinger equations with time-periodic potentials, J. Statist. Phys. 116 (2004), no. 1-4, 231–282. MR 2083143, DOI 10.1023/B:JOSS.0000037203.79298.ec
I. C. Gohberg and M. G. Kreĭn, Fundamental aspects of defect numbers, root numbers and indexes of linear operators, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 2(74), 43–118 (Russian). MR 0096978
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), no. 9, 1794–1797. MR 460850, DOI 10.1063/1.523491
J. Ginibre and M. Moulin, Hilbert space approach to the quantum mechanical three-body problem, Ann. Inst. H. Poincaré Sect. A (N.S.) 21 (1974), 97–145. MR 368656
D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974), pp. 3235–3253.
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves, Int. Math. Res. Not. 66 (2004), 3559–3584. MR 2101699, DOI 10.1155/S1073792804132340
Vladimir Georgiev and Masahito Ohta, Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinger equations, J. Math. Soc. Japan 64 (2012), no. 2, 533–548. MR 2916078
Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0200692
Manoussos Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math. 41 (1988), no. 6, 747–774. MR 948770, DOI 10.1002/cpa.3160410602
Leonard Gross, The Cauchy problem for the coupled Maxwell and Dirac equations, Comm. Pure Appl. Math. 19 (1966), 1–15. MR 190520, DOI 10.1016/S0079-8169(08)61684-0
I. M. Gel′fand, D. A. Raĭkov, and G. E. Šilov, Commutative normed rings, Uspehi Matem. Nauk (N.S.) 1 (1946), no. 2(12), 48–146 (Russian). MR 0027130
I. C. Gohberg and E. I. Sigal, An operator generalization of the logarithmic residue theorem and Rouché’s theorem, Mat. Sb. (N.S.) 84(126) (1971), 607–629 (Russian). MR 0313856
Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. MR 901236, DOI 10.1016/0022-1236(87)90044-9
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
M. Guan, Solitary wave solutions for the nonlinear Dirac equations, ArXiv e-prints (2008), arXiv:0812.2273.
J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), no. 1, 163–188. MR 1151250
W. Heisenberg, Quantum theory of fields and elementary particles, Rev. Mod. Phys. 29 (1957), 269–278. MR 0090429, DOI 10.1103/revmodphys.29.269
Peter Hess, Zur Störungstheorie linearer Operatoren: Relative Beschränktheit und relative Kompaktheit von Operatoren in Banachräumen, Comment. Math. Helv. 44 (1969), 245–248 (German). MR 246156, DOI 10.1007/BF02564526
Peter Hess, Zur Störungstheorie linearer Operatoren in Banachräumen, Comment. Math. Helv. 45 (1970), 229–235 (German). MR 264432, DOI 10.1007/BF02567328
P. D. Hislop, Exponential decay of two-body eigenfunctions: a review, Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999) Electron. J. Differ. Equ. Conf., vol. 4, Southwest Texas State Univ., San Marcos, TX, 2000, pp. 265–288. MR 1785381
Hyungjin Huh and Bora Moon, Low regularity well-posedness for Gross-Neveu equations, Commun. Pure Appl. Anal. 14 (2015), no. 5, 1903–1913. MR 3359550, DOI 10.3934/cpaa.2015.14.1903
R. H. Hobart, On the instability of a class of unitary field models, Proc. Phys. Soc. 82 (1963), 201–203. MR 0155591
Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR 705278, DOI 10.1007/978-3-642-96750-4
Hyungjin Huh, Global strong solution to the Thirring model in critical space, J. Math. Anal. Appl. 381 (2011), no. 2, 513–520. MR 2802088, DOI 10.1016/j.jmaa.2011.02.042
Hyungjin Huh, Global solutions to Gross-Neveu equation, Lett. Math. Phys. 103 (2013), no. 8, 927–931. MR 3063953, DOI 10.1007/s11005-013-0622-9
Hyungjin Huh, Global strong solutions to some nonlinear Dirac equations in super-critical space, Abstr. Appl. Anal. , posted on (2013), Art. ID 602753, 8. MR 3070190, DOI 10.1155/2013/602753
Hyungjin Huh, On the growth rate of solutions to Gross-Neveu and Thirring equations, Commun. Korean Math. Soc. 29 (2014), no. 2, 263–267. MR 3206413, DOI 10.4134/CKMS.2014.29.2.263
A. Hurwitz, Ueber die Composition der quadratischen Formen von belibig vielen Variablen, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1898 (1898), pp. 309–316.
A. Hurwitz, Über die Komposition der quadratischen Formen, Math. Ann. 88 (1922), no. 1-2, 1–25 (German). MR 1512117, DOI 10.1007/BF01448439
V. P. Ilyin, Some integral inequalities and their applications in the theory of differentiable functions of several variables, Mat. Sb. (N.S.) 54 (96) (1961), pp. 331–380.
A. Iftimovici and M. Măntoiu, Limiting absorption principle at critical values for the Dirac operator, Lett. Math. Phys. 49 (1999), no. 3, 235–243. MR 1743451, DOI 10.1023/A:1007625918845
D. D. Ivanenko, Notes to the theory of interaction via particles, Zh. Éksp. Teor. Fiz 8 (1938), pp. 260–266.
Arne Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions results in $L^{2}(\textbf {R}^{m})$, $m\geq 5$, Duke Math. J. 47 (1980), no. 1, 57–80. MR 563367
David Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math. 62 (1986), no. 2, 118–134. MR 865834, DOI 10.1016/0001-8708(86)90096-4
A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time- decay of the wave functions, Duke Math. J. 46 (1979), pp. 583–611.
Arne Jensen and Gheorghe Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys. 13 (2001), no. 6, 717–754. MR 1841744, DOI 10.1142/S0129055X01000843
Konrad Jörgens, Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295–308 (German). MR 130462, DOI 10.1007/BF01180181
J.-L. Journé, A. Soffer, and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math. 44 (1991), no. 5, 573–604. MR 1105875, DOI 10.1002/cpa.3160440504
Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
M. V. Keldyš, On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 11–14 (Russian). MR 0041353
M. V. Keldyš, The completeness of eigenfunctions of certain classes of nonselfadjoint linear operators, Uspehi Mat. Nauk 26 (1971), no. 4(160), 15–41 (Russian). MR 0300125
Clayton Keller, Large-time asymptotic behavior of solutions of nonlinear wave equations perturbed from a stationary ground state, Comm. Partial Differential Equations 8 (1983), no. 10, 1073–1099. MR 709163, DOI 10.1080/03605308308820296
H. Kestelman, Anticommuting linear transformations, Canadian J. Math. 13 (1961), 614–624. MR 148679, DOI 10.4153/CJM-1961-050-2
Alexander Komech and Andrew Komech, Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Ration. Mech. Anal. 185 (2007), no. 1, 105–142. MR 2308860, DOI 10.1007/s00205-006-0039-z
A. Komech and E. Kopylova, Dispersion decay and Scattering Theory, John Wiley & Sons Inc., Hoboken, NJ, 2012.
Carlos Kenig, Andrew Lawrie, Baoping Liu, and Wilhelm Schlag, Stable soliton resolution for exterior wave maps in all equivariance classes, Adv. Math. 285 (2015), 235–300. MR 3406501, DOI 10.1016/j.aim.2015.08.007
Hideo Kozono and Tohru Ozawa, Relative bounds of closable operators in nonreflexive Banach spaces, Hokkaido Math. J. 19 (1990), no. 2, 241–247. MR 1059168, DOI 10.14492/hokmj/1381517358
A. A. Kolokolov, Stability of the dominant mode of the nonlinear wave equation in a cubic medium, J. Appl. Mech. Tech. Phys. 14 (1973), pp. 426–428.
A. I. Komech, On attractor of a singular nonlinear $\textrm {U}(1)$-invariant Klein-Gordon equation, Progress in analysis, Vol. I, II (Berlin, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 599–611. MR 2032730
Alexander Komech, Quantum mechanics: genesis and achievements, Springer, Dordrecht, 2013. MR 3012343, DOI 10.1007/978-94-007-5542-0
Hubert Kalf, Takashi Okaji, and Osanobu Yamada, The Dirac operator with mass $m_0\geq 0$: non-existence of zero modes and of threshold eigenvalues, Doc. Math. 20 (2015), 37–64. MR 3398708
Todd Kapitula and Björn Sandstede, Edge bifurcations for near integrable systems via Evans function techniques, SIAM J. Math. Anal. 33 (2002), no. 5, 1117–1143. MR 1897705, DOI 10.1137/S0036141000372301
Paschalis Karageorgis and Walter A. Strauss, Instability of steady states for nonlinear wave and heat equations, J. Differential Equations 241 (2007), no. 1, 184–205. MR 2356215, DOI 10.1016/j.jde.2007.06.006
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
S. T. Kuroda, An introduction to scattering theory, Lecture Notes Series, vol. 51, Aarhus Universitet, Matematisk Institut, Aarhus, 1978. MR 528757
Man Kam Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. MR 969899, DOI 10.1007/BF00251502
Tosio Kato and Kenji Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), no. 4, 481–496. MR 1061120, DOI 10.1142/S0129055X89000171
T. I. Lakoba, Numerical study of solitary wave stability in cubic nonlinear Dirac equations in 1D, Phys. Lett. A 382 (2018), no. 5, 300–308. MR 3739696, DOI 10.1016/j.physleta.2017.11.032
Antoine Levitt, Solutions of the multiconfiguration Dirac-Fock equations, Rev. Math. Phys. 26 (2014), no. 7, 1450014, 25. MR 3246964, DOI 10.1142/S0129055X14500147
S. Y. Lee and A. Gavrielides, Quantization of the localized solutions in two-dimen- sional field theories of massive fermions, Phys. Rev. D 12 (1975), pp. 3880–3886.
W. E. Lamb and R. C. Retherford, Fine structure of the hydrogen atom by a microwave method, Phys. Rev. 72 (1947), pp. 241–243.
B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I., 1975. Translated from the Russian by Amiel Feinstein. MR 0369797
A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, vol. 71, American Mathematical Society, Providence, RI, 1988. Translated from the Russian by H. H. McFaden; Translation edited by Ben Silver; With an appendix by M. V. Keldysh. MR 971506, DOI 10.1090/mmono/071
Kevin McLeod, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\textbf {R}^n$. II, Trans. Amer. Math. Soc. 339 (1993), no. 2, 495–505. MR 1201323, DOI 10.1090/S0002-9947-1993-1201323-X
J. Mehra, The Golden Age of Theoretical Physics, The Golden Age of Theoretical Physics, World Scientific, 2001.
F. Merle, Existence of stationary states for nonlinear Dirac equations, J. Differential Equations 74 (1988), no. 1, 50–68. MR 949625, DOI 10.1016/0022-0396(88)90018-6
F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys. 129 (1990), pp. 223–240.
Tetsu Mizumachi, Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ. 48 (2008), no. 3, 471–497. MR 2511047, DOI 10.1215/kjm/1250271380
Reinhard Mennicken and Manfred Möller, Non-self-adjoint boundary eigenvalue problems, North-Holland Mathematics Studies, vol. 192, North-Holland Publishing Co., Amsterdam, 2003. MR 1995773
Shuji Machihara, Makoto Nakamura, Kenji Nakanishi, and Tohru Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal. 219 (2005), no. 1, 1–20. MR 2108356, DOI 10.1016/j.jfa.2004.07.005
Shuji Machihara, Kenji Nakanishi, and Tohru Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation, Rev. Mat. Iberoamericana 19 (2003), no. 1, 179–194. MR 1993419, DOI 10.4171/RMI/342
Shuji Machihara, Kenji Nakanishi, and Kotaro Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math. 50 (2010), no. 2, 403–451. MR 2666663, DOI 10.1215/0023608X-2009-018
Shuji Machihara and Takayuki Omoso, The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation, Ric. Mat. 56 (2007), no. 1, 19–30. MR 2330348, DOI 10.1007/s11587-007-0002-9
Walter Moore, A life of Erwin Schrödinger, Canto, Cambridge University Press, Cambridge, 1994. MR 1302170
A. S. Markus and E. I. Sigal, The multiplicity of the characteristic number of an analytic operator function, Mat. Issled. 5 (1970), no. 3(17), 129–147 (Russian). MR 0312306
Cathleen S. Morawetz and Walter A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1–31. MR 303097, DOI 10.1002/cpa.3160250103
M. Naimark, Normed algebras, Wolters-Noordhoff, 1972, 1 edn.
Louis Nirenberg and Homer F. Walker, The null spaces of elliptic partial differential operators in $\textbf {R}^{n}$, J. Math. Anal. Appl. 42 (1973), 271–301. Collection of articles dedicated to Salomon Bochner. MR 320821, DOI 10.1016/0022-247X(73)90138-8
Susumu Okubo, Real representations of finite Clifford algebras. I. Classification, J. Math. Phys. 32 (1991), no. 7, 1657–1668. MR 1112690, DOI 10.1063/1.529277
H. Ounaies, Perturbation method for a class of nonlinear Dirac equations, Differential Integral Equations 13 (2000), no. 4-6, 707–720. MR 1750047
Tohru Ozawa and Kazuyuki Yamauchi, Structure of Dirac matrices and invariants for nonlinear Dirac equations, Differential Integral Equations 17 (2004), no. 9-10, 971–982. MR 2082456
W. Pauli, Contributions mathématiques à la théorie des matrices de Dirac, Ann. Inst. H. Poincaré 6 (1936), no. 2, 109–136 (French). MR 1508031
Dmitry Pelinovsky, Survey on global existence in the nonlinear Dirac equations in one spatial dimension, Harmonic analysis and nonlinear partial differential equations, RIMS Kôkyûroku Bessatsu, B26, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 37–50. MR 2883845
S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR 165 (1965), 36–39 (Russian). MR 0192184
S. Puntanen and G. Styan, Historical introduction: Issai Schur and the early development of the Schur complement,inF.Zhang,editor,The Schur Comp- lement and its Applications, vol. 4 of Numerical Methods and Algorithms, pp. 1–16, Springer, 2005.
Dmitry E. Pelinovsky and Atanas Stefanov, Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys. 53 (2012), no. 7, 073705, 27. MR 2985264, DOI 10.1063/1.4731477
Dmitry E. Pelinovsky and Yusuke Shimabukuro, Orbital stability of Dirac solitons, Lett. Math. Phys. 104 (2014), no. 1, 21–41. MR 3147657, DOI 10.1007/s11005-013-0650-5
Dmitry E. Pelinovsky, Andrey A. Sukhorukov, and Yuri S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials, Phys. Rev. E (3) 70 (2004), no. 3, 036618, 17. MR 2130334, DOI 10.1103/PhysRevE.70.036618
Claude-Alain Pillet and C. Eugene Wayne, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differential Equations 141 (1997), no. 2, 310–326. MR 1488355, DOI 10.1006/jdeq.1997.3345
J. Radon, Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922), no. 1, 1–14 (German). MR 3069384, DOI 10.1007/BF02940576
A. F. Rañada, Classical nonlinear Dirac field models of extended particles, in A. O. Barut, editor, Quantum theory, groups, fields and particles, vol. 4 of Mathematical Physics Studies, pp. 271–291, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1983.
A. F. Rañada,On nonlinear classical Dirac fields and quantum physics,inOld and new questions in physics, cosmology, philosophy, and theoretical biology, pp. 363–376, Plenum, New York, 1983.
Jeffrey Rauch, Local decay of scattering solutions to Schrödinger’s equation, Comm. Math. Phys. 61 (1978), no. 2, 149–168. MR 495958
Simona Rota Nodari, Perturbation method for particle-like solutions of the Einstein-Dirac-Maxwell equations, C. R. Math. Acad. Sci. Paris 348 (2010), no. 13-14, 791–794 (English, with English and French summaries). MR 2671162, DOI 10.1016/j.crma.2010.06.003
Simona Rota Nodari, Perturbation method for particle-like solutions of the Einstein-Dirac equations, Ann. Henri Poincaré 10 (2010), no. 7, 1377–1393. MR 2593110, DOI 10.1007/s00023-009-0015-x
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
Igor Rodnianski and Wilhelm Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), no. 3, 451–513. MR 2038194, DOI 10.1007/s00222-003-0325-4
F. Riesz and B. Sz.-Nagy, Functional Analysis, Blackie & Son Limited, 1956.
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
E. Schrödinger, Quantisierung als Eigenwertproblem, Ann. Phys. 384 (1926), pp. 361–376.
L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1949.
L. I. Schiff, Nonlinear meson theory of nuclear forces. I. Neutral scalar mesons with point-contact repulsion, Phys. Rev. 84 (1951), pp. 1–9.
L. I. Schiff, Nonlinear meson theory of nuclear forces. II. Nonlinearity in the meson-nucleon coupling, Phys. Rev. 84 (1951), pp. 10–11.
Martin Schechter, On the essential spectrum of an arbitrary operator. I, J. Math. Anal. Appl. 13 (1966), 205–215. MR 188798, DOI 10.1016/0022-247X(66)90085-0
Martin Schechter, Spectra of partial differential operators, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. MR 869254
I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France 91 (1963), 129–135. MR 153967
Irving Segal, Quantization and dispersion for nonlinear relativistic equations, Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) M.I.T. Press, Cambridge, Mass., 1966, pp. 79–108. MR 0217453
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Daniel B. Shapiro, Spaces of similarities. I. The Hurwitz problem, J. Algebra 46 (1977), no. 1, 148–170. MR 453636, DOI 10.1016/0021-8693(77)90397-0
J. Shatah, Stable standing waves of nonlinear Klein–Gordon equations, Comm. Math. Phys. 91 (1983), pp. 313–327.
Jalal Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985), no. 2, 701–710. MR 792821, DOI 10.1090/S0002-9947-1985-0792821-7
Daniel B. Shapiro, Compositions of quadratic forms, De Gruyter Expositions in Mathematics, vol. 33, Walter de Gruyter & Co., Berlin, 2000. MR 1786291, DOI 10.1515/9783110824834
D. S. Shirokov, Extension of Pauli’s theorem to the case of Clifford algebras, Dokl. Akad. Nauk 440 (2011), no. 5, 607–610 (Russian); English transl., Dokl. Math. 84 (2011), no. 2, 699–701. MR 2963685, DOI 10.1134/S1064562411060329
D. S. Shirokov, Pauli theorem in the description of $n$-dimensional spinors in the Clifford algebra formalism, Theoret. and Math. Phys. 175 (2013), no. 1, 454–474. Translation of Teoret. Mat. Fiz. 175 (2013), no. 1, 11–34. MR 3172130, DOI 10.1007/s11232-013-0038-9
I. M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993), no. 2, 297–320. MR 1218303
Barry Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Physics 97 (1976), no. 2, 279–288. MR 404846, DOI 10.1016/0003-4916(76)90038-5
V. A. Sloushch, Generalizations of the Cwikel estimate for integral operators, Proceedings of the St. Petersburg Mathematical Society. Vol. XIV, Amer. Math. Soc. Transl. Ser. 2, vol. 228, Amer. Math. Soc., Providence, RI, 2009, pp. 133–155. MR 2584397, DOI 10.1090/trans2/228/06
V. A. Sloushch, A Cwikel-type estimate as a consequence of some properties of the heat kernel, Algebra i Analiz 25 (2013), no. 5, 173–201 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 25 (2014), no. 5, 835–854. MR 3184610, DOI 10.1090/S1061-0022-2014-01318-0
A. Soffer, Soliton dynamics and scattering, in International Congress of Mathe- maticians. Vol. III, pp. 459–471, Eur. Math. Soc., Zürich, 2006.
Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579, DOI 10.1017/CBO9780511530029
M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D 1 (1970), pp. 2766–2769.
C. M. D. Sterke and J. Sipe, Gap solitons, Progress in Optics 33 (1994), pp. 203–260.
J. Shatah and W. Strauss, Spectral condition for instability, in Nonlinear PDE’s, dynamics and continuum physics (South Hadley, MA, 1998), vol. 255 of Contemp. Math., pp. 189–198, Amer. Math. Soc., Providence, RI, 2000.
Sigmund Selberg and Achenef Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations 23 (2010), no. 3-4, 265–278. MR 2588476
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
J. Strutt, The theory of sound, no. v. 1 in The theory of sound, Macmillan and Co., London, 1877.
J. Strutt, The theory of sound, no. v. 2 in The theory of sound, Macmillan and Co., London, 1878.
Walter A. Strauss, Decay and asymptotics for $cmu=F(u)$, J. Functional Analysis 2 (1968), 409–457. MR 0233062, DOI 10.1016/0022-1236(68)90004-9
Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 454365
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
Walter A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, vol. 73, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. MR 1032250
David Stuart, Existence and Newtonian limit of nonlinear bound states in the Einstein-Dirac system, J. Math. Phys. 51 (2010), no. 3, 032501, 13. MR 2647868, DOI 10.1063/1.3294085
A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), pp. 119–146.
A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Differential Equations 98 (1992), no. 2, 376–390. MR 1170476, DOI 10.1016/0022-0396(92)90098-8
A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), no. 1, 9–74. MR 1681113, DOI 10.1007/s002220050303
Terence Tao, A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations, Dyn. Partial Differ. Equ. 4 (2007), no. 1, 1–53. MR 2304091, DOI 10.4310/DPDE.2007.v4.n1.a1
Terence Tao, Why are solitons stable?, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 1, 1–33. MR 2457070, DOI 10.1090/S0273-0979-08-01228-7
Bernd Thaller, The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992. MR 1219537, DOI 10.1007/978-3-662-02753-0
Walter E. Thirring, A soluble relativistic field theory, Ann. Physics 3 (1958), 91–112. MR 91788, DOI 10.1016/0003-4916(58)90015-0
J. Thomson, Cathode rays, Philosophical Magazine 44 (1897), pp. 293–316.
J. Thomson, On the structure of the atom: an investigation of the stability and periods ofoscillationof a number of corpuscles arrangedatequalintervals around thecircumferenceofacircle;withapplicationoftheresults;to;thetheoryof atomic structure, Philosophical Magazine 7 (1904), pp. 237–265.
Tai-Peng Tsai and Horng-Tzer Yau, Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions, Comm. Pure Appl. Math. 55 (2002), no. 2, 153–216. MR 1865414, DOI 10.1002/cpa.3012
Tai-Peng Tsai and Horng-Tzer Yau, Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. Theor. Math. Phys. 6 (2002), no. 1, 107–139. MR 1992875, DOI 10.4310/ATMP.2002.v6.n1.a2
T.-P. Tsai and H.-T. Yau, Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not. (2002), pp. 1629–1673.
Tai-Peng Tsai and Horng-Tzer Yau, Stable directions for excited states of nonlinear Schrödinger equations, Comm. Partial Differential Equations 27 (2002), no. 11-12, 2363–2402. MR 1944033, DOI 10.1081/PDE-120016161
Luis Vazquez, Localised solutions of a non-linear spinor field, J. Phys. A 10 (1977), no. 8, 1361–1368. MR 456046
Bartel Leenert van der Waerden, Group theory and quantum mechanics, Die Grundlehren der mathematischen Wissenschaften, Band 214, Springer-Verlag, New York-Heidelberg, 1974. Translated from the 1932 German original. MR 0479090
N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in the medium with nonlinearity saturation, Radiophys. Quantum Electron. 16 (1973), pp. 783–789.
J. von Neumann and E. Wigner, Uber merkwürdige diskrete Eigenwerte. Uber das Verhalten von Eigenwerten bei adiabatischen prozessen, Physikalische Zeitschrift 30 (1929), pp. 467–470.
M.Wakano,Intensely localized solutions of the classical Dirac-Maxwell field equations, Progr. Theoret. Phys. 35 (1966), pp. 1117–1141.
Joachim Weidmann, Absolut stetiges Spektrum bei Sturm-Liouville-Operatoren und Dirac-Systemen, Math. Z. 180 (1982), no. 3, 423–427 (German). MR 664527, DOI 10.1007/BF01214182
Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044
Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. MR 783974, DOI 10.1137/0516034
Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 51–67. MR 820338, DOI 10.1002/cpa.3160390103
S. Weinberg, The Trouble with Quantum Mechanics, The New York Review of Books 64 (2017), pp. 51–53.
H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollstetig ist, Rendiconti del Circolo Matematico di Palermo 27 (1909), pp. 373–392.
František Wolf, On the essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math. 12 (1959), 211–228. MR 107750, DOI 10.1002/cpa.3160120202
D. R. Yafaev, Mathematical scattering theory, Mathematical Surveys and Monographs, vol. 158, American Mathematical Society, Providence, RI, 2010. Analytic theory. MR 2598115, DOI 10.1090/surv/158
Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945
Osanobu Yamada, On the principle of limiting absorption for the Dirac operator, Publ. Res. Inst. Math. Sci. 8 (1972/73), 557–577. MR 0320547, DOI 10.2977/prims/1195192961
Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
V. Zakharov, Instability of self-focusing of light, Zh. Éksp. Teor. Fiz 53 (1967), pp. 1735–1743.
E. Zeidler, Nonlinear functional analysis and its applications III. Variational methods and optimization, Springer, 1984, 1 edn.
Anton Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005. MR 2170950, DOI 10.1090/surv/121
V. Zakharov and V. Synakh, The nature of the self-focusing singularity, Zh. Éksp. Teor. Fiz 41 (1975), pp. 465–468, Russian original - ZhETF, Vol. 68, No. 3, p. 940, June 1975.
V. E. Zakharov, V. V. Sobolev, and V. S. Synakh, Character of singularity and stochastic phenomena in self focusing, ZhETF Letters 14 (1971), pp. 564–568.

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

About this Title

Table of Contents

References [Enhancements On Off] (What's this?)