## On eigenfunction expansion of solutions to the Hamilton equations.(English)Zbl 1300.34195

The authors establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Their approach is a special version of M. Krein’s spectral theory of $$J$$-selfadjoint operators in Hilbert spaces with indefinite metric. Their main result is an application to the eigenfunction expansion for the linearized relativistic Ginzburg-Landau equation. It should be noted that the problem of eigenfunction expansion for the linearized relativistic Ginzburg-Landau equation has been studied early by the authors [Arch. Ration. Mech. Anal. 202, No. 1, 213–245 (2011; Zbl 1256.35146); Commun. Math. Phys. 302, No. 1, 225–252 (2011; Zbl 1209.35134)].

### MSC:

 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 83A05 Special relativity 34A30 Linear ordinary differential equations and systems

### Citations:

Zbl 1256.35146; Zbl 1209.35134; Zbl 0405.47007
Full Text:

### References:

 [1] Agmon, S., Spectral properties of Schrödinger operator and scattering theory, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 2, 151-218, (1975) · Zbl 0315.47007 [2] Azizov, T.Ya., Iokhvidov, I.S.: Linear Operators in Space with an Indefinite Metric. Wiley, Chichester (1989) · Zbl 0714.47028 [3] Bambusi, D.; Cuccagna, S., On dispersion of small energy solutions of the nonlinear Klein-Gordon equation with a potential, Am. J. Math., 133, 1421-1468, (2011) · Zbl 1237.35115 [4] Boussaid, N.; Cuccagna, S., On stability of standing waves of nonlinear Dirac equations, Commun. Partial Differ. Equ., 37, 1001-1056, (2012) · Zbl 1251.35098 [5] Buslaev, V.S.; Perelman, G.S., Scattering for the nonlinear Schrödinger equation: states close to a soliton, St. Petersburg Math. J., 4, 1111-1142, (1993) [6] Buslaev, V.S.; Perelman, G.S., On the stability of solitary waves for nonlinear Schrödinger equations, nonlinear evolution equations, No. 164, 75-98, (1995), Providence · Zbl 0841.35108 [7] Buslaev, V.S.; Sulem, C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 20, 419-475, (2003) · Zbl 1028.35139 [8] Cazenave, T., Haraux, A.: Semilinear Evolution Equations. Clarendon, Oxford (1998) · Zbl 0926.35049 [9] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Commun. Pure Appl. Math., 54, 1110-1145, (2001) · Zbl 1031.35129 [10] Cuccagna, S., On asymptotic stability of ground states of NLS, Rev. Math. Phys., 15, 877-903, (2003) · Zbl 1084.35089 [11] Cuccagna, S.; Pelinovsky, D.; Vougalter, V., Spectra of positive and negative energies in the linearized NLS problem, Commun. Pure Appl. Math., 58, 1-29, (2005) · Zbl 1064.35181 [12] Cuccagna, S., On scattering of small energy solutions of non-autonomous Hamiltonian nonlinear Schrödinger equations, J. Differ. Equ., 250, 2347-2371, (2011) · Zbl 1216.35134 [13] Cuccagna, S., The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Commun. Math. Phys., 305, 279-331, (2011) · Zbl 1222.35183 [14] Cuccagna, S.: On asymptotic stability of moving ground states of the nonlinear Schrödinger equation. Trans. Am. Math. Soc. (2012). To appear in arXiv:1107.4954 · Zbl 0721.35082 [15] Cuccagna, S.; Mizumachi, T., On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Commun. Math. Phys., 284, 51-77, (2008) · Zbl 1155.35092 [16] Cuccagna, S.; Tarulli, M., On asymptotic stability in energy space of ground states of NLS in 2D, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 26, 1361-1386, (2009) · Zbl 1171.35470 [17] Erdogan, M.B.; Schlag, W., Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. II, J. Anal. Math., 99, 199-248, (2006) · Zbl 1146.35324 [18] Gérard, C., Scattering theory for Klein-Gordon equations with non-positive energy, Ann. Henri Poincaré, 13, 883-941, (2012) · Zbl 1253.81049 [19] Gohberg, I.C., Krein, M.G.: Theory and Applications of Volterra Operators in Hilbert Space. AMS, Providence (1970) · Zbl 0194.43804 [20] Hörmander, L.: The Analysis of Linear Partial Differential Operators. IV: Fourier Integral Operators. Springer, Berlin (2009) · Zbl 1178.35003 [21] Iohvidov, I.S., Krein, M.G., Langer, H.: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Mathematical Research, vol. 9. Akademie Verlag, Berlin (1982) · Zbl 0506.47022 [22] Jonas, P., On the functional calculus and the spectral function for definitizable operators in Krein space, Beitr. Anal., 16, 121-135, (1981) · Zbl 0556.47019 [23] Jonas, P., On a class of unitary operators in Krein space, No. 17, 151-172, (1986), Basel [24] Jonas, P., On a class of selfadjoint operators in Krein space and their compact perturbations, Integral Equ. Oper. Theory, 11, 351-384, (1988) · Zbl 0651.47020 [25] Jonas, P., On the spectral theory of operators associated with perturbed Klein-Gordon and wave type equations, J. Oper. Theory, 29, 207-224, (1993) · Zbl 0818.35107 [26] Jonas, P., On bounded perturbations of operators of Klein-Gordon type, Glasnik Math., 35, 59-74, (2000) · Zbl 1017.47030 [27] Kako, T., Spectral and scattering theory for the J-selfadjoint operators associated with the perturbed Klein-Gordon type equations, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 23, 199-221, (1976) · Zbl 0367.47026 [28] Imaykin, V.; Komech, A.I.; Spohn, H., Scattering asymptotics for a charged particle coupled to the Maxwell field, J. Math. Phys., 52, (2011) · Zbl 1316.78002 [29] Komech, A.I.; Kopylova, E.A.; Spohn, H., Scattering of solitons for Dirac equation coupled to a particle, J. Math. Anal. Appl., 383, 265-290, (2011) · Zbl 1229.35228 [30] Kopylova, E.A.; Komech, A.I., On asymptotic stability of kink for relativistic Ginzburg-Landau equation, Arch. Ration. Mech. Anal., 202, 213-245, (2011) · Zbl 1256.35146 [31] Kopylova, E.A.; Komech, A.I., On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation, Commun. Math. Phys., 302, 225-252, (2011) · Zbl 1209.35134 [32] Komech, A., Kopylova, E.A.: Dispersion Decay and Scattering Theory. Wiley, Hoboken (2012) · Zbl 1317.35162 [33] Krein, M.G.; Langer, H.K., The spectral function of a selfadjoint operator in a space with indefinite metric, Sov. Math. Dokl., 4, 1236-1239, (1963) · Zbl 0131.12604 [34] Krein, M.G.; Shmul’jan, Yu., $$J$$-polar representations of plus-operators, Math. Oper.forsch. Stat., Ser. Stat., 1, 172-210, (1966) · Zbl 0201.16803 [35] Langer, H.; Butkovic, D. (ed.); Kraljevic, H. (ed.); Kurepa, S. (ed.), Spectral functions of definitizable operators in Krein spaces, 1-46, (1981), Berlin [36] Langer, H.; Najman, B., Perturbation theory for definitizable operators in Krein spaces, J. Oper. Theory, 9, 297-317, (1983) · Zbl 0516.47021 [37] Langer, H.; Najman, B.; Tretter, C., Spectral theory of the Klein-Gordon equation in Krein spaces, Proc. Edinb. Math. Soc., 51, 711-750, (2008) · Zbl 1152.81020 [38] Langer, H.; Najman, B.; Tretter, C., Spectral theory of the Klein-Gordon equation in Pontryagin spaces, Commun. Math. Phys., 267, 159-180, (2006) · Zbl 1114.47038 [39] Langer, H.; Tretter, C., Variational principles for eigenvalues of the Klein-Gordon equation, J. Math. Phys., 47, (2006) · Zbl 1112.78008 [40] Martel, Y.; Merle, F., Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341, 391-427, (2008) · Zbl 1153.35068 [41] Miller, J.; Weinstein, M., Asymptotic stability of solitary waves for the regularized long-wave equation, Commun. Pure Appl. Math., 49, 399-441, (1996) · Zbl 0854.35102 [42] Pego, R.L.; Weinstein, M.I., Asymptotic stability of solitary waves, Commun. Math. Phys., 164, 305-349, (1994) · Zbl 0805.35117 [43] Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, New York (1975) · Zbl 0308.47002 [44] Reed, M., Simon, B.: Methods of Modern Mathematical Physics III: Scattering Theory. Academic Press, New York (1979) · Zbl 0405.47007 [45] Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, New York (1978) · Zbl 0401.47001 [46] Rudin, W.: Functional Analysis. McGraw-Hill, New York (1991) · Zbl 0867.46001 [47] Sakurai, J.J.: Advanced Quantum Mechanics. Addison-Wesley, Reading (1967) · Zbl 0158.45604 [48] Schlag, W.; Bourgain, J. (ed.); etal., Dispersive estimates for Schrödinger operators: a survey, Princeton, NJ, USA, Princeton [49] Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, New York (1987) · Zbl 0616.47040 [50] Sigal, I.M., Nonlinear wave and Schrödinger equations. I: instability of periodic and quasiperiodic solutions, Commun. Math. Phys., 153, 297-320, (1993) · Zbl 0780.35106 [51] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering in nonintegrable systems, Commun. Math. Phys., 133, 119-146, (1990) · Zbl 0721.35082 [52] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering and stability II. the case of anisotropic and potential and data, J. Differ. Equ., 98, 376-390, (1992) · Zbl 0795.35073 [53] Soffer, A.; Weinstein, M.I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136, 9-74, (1999) · Zbl 0910.35107 [54] Soffer, A.; Weinstein, M.I., Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys., 16, 977-1071, (2004) · Zbl 1111.81313 [55] Spohn, H.: Dynamics of Charged Particles and Their Radiation Field. Cambridge University Press, Cambridge (2004) · Zbl 1078.81004 [56] Tsai, T.-P., Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Differ. Equ., 192, 225-282, (2003) · Zbl 1038.35128
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