## Small solutions of nonlinear Schrödinger equations near first excited states.(English)Zbl 1244.35136

Summary: Consider a nonlinear Schrödinger equation in $$\mathbb R^{3}$$ whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in $$H^{1}\cap L^{1}(\mathbb R^{3})$$ and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity. An essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states, around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35P05 General topics in linear spectral theory for PDEs

### Keywords:

nonlinear Schrödinger equation; first excited state
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### References:

 [1] Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Ann. sc. norm. super. Pisa ser. IV, 2, 151-218, (1975) · Zbl 0315.47007 [2] Agmon, S., Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators, (1982), Princeton University Press · Zbl 0503.35001 [3] Buslaev, V.S.; Perelʼman, G.S., Scattering for the nonlinear Schrödinger equations: states close to a solitary wave, Saint |St. Petersburg math. J., 4, 1111-1142, (1993) [4] Buslaev, V.S.; Perelʼman, G.S., On the stability of solitary waves for nonlinear Schrödinger equations. nonlinear evolution equations, (), 75-98 · Zbl 0841.35108 [5] Buslaev, V.S.; Sulem, C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 20, 3, 419-475, (2003) · Zbl 1028.35139 [6] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., 54, 9, 1110-1145, (2001) · Zbl 1031.35129 [7] Cuccagna, S., On asymptotic stability of ground states of NLS, Rev. math. phys., 15, 8, 877-903, (2003) · Zbl 1084.35089 [8] Cuccagna, S.; Mizumachi, T., On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. math. phys., 284, 1, 51-77, (2008) · Zbl 1155.35092 [9] Cycon, H.L.; Froese, R.G.; Kirsch, W.; Simon, B., Schrödinger operators with application to quantum mechanics and global geometry, (), x+319 pp · Zbl 0619.47005 [10] Gang, Z.; Sigal, I.M., Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. math., 216, 2, 443-490, (2007) · Zbl 1126.35065 [11] Gang, Z.; Weinstein, M.I., Dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations; mass transfer in systems with solitons and degenerate neutral modes, Anal. PDE, 1, 3, 267-322, (2008) · Zbl 1175.35136 [12] Gustafson, S.; Nakanishi, K.; Tsai, T.-P., Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves, Int. math. res. not., 2004, 66, 3559-3584, (2004) · Zbl 1072.35167 [13] Gustafson, S.; Phan, T.V., Stable directions for degenerate excited states of nonlinear Schrödinger equations, SIAM J. math. anal., 43, 4, 1716-1758, (2011) · Zbl 1230.35129 [14] Hunziker, W.; Sigal, I.M.; Soffer, A., Minimal escape velocities, Comm. partial differential equations, 24, 11-12, 2279-2295, (1999) · Zbl 0944.35014 [15] Jensen, A.; Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke math. J., 46, 3, 583-611, (1979) · Zbl 0448.35080 [16] Journé, J.-L.; Soffer, A.; Sogge, C.D., Decay estimates for Schrödinger operators, Comm. pure appl. math., 44, 5, 573-604, (1991) · Zbl 0743.35008 [17] Kirr, E.; Mizrak, Ö., Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases, J. funct. anal., 257, 3691-3747, (2009) · Zbl 1187.35238 [18] Krieger, J.; Schlag, W., Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. amer. math. soc., 19, 4, 815-920, (2006) · Zbl 1281.35077 [19] Mizumachi, T., Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. math. Kyoto univ., 47, 3, 599-620, (2007) · Zbl 1146.35085 [20] Mizumachi, T., Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential, J. math. Kyoto univ., 48, 3, 471-497, (2008) · Zbl 1175.35138 [21] Pillet, C.-A.; Wayne, C.E., Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. differential equations, 141, 2, 310-326, (1997) · Zbl 0890.35016 [22] Rauch, J., Local decay of scattering solutions to schrödingerʼs equation, Comm. math. phys., 61, 2, 149-168, (1978) · Zbl 0381.35023 [23] Schlag, W., Stable manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann. of math. (2), 169, 1, 139-227, (2009) · Zbl 1180.35490 [24] Skibsted, E., Propagation estimates for N-body schroedinger operators, Comm. math. phys., 142, 1, 67-98, (1991) · Zbl 0760.35035 [25] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering for nonintegrable equations, Comm. math. phys., 133, 1, 119-146, (1990) · Zbl 0721.35082 [26] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering for nonintegrable equations. II. the case of anisotropic potentials and data, J. differential equations, 98, 2, 376-390, (1992) · Zbl 0795.35073 [27] Soffer, A.; Weinstein, M.I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. math., 136, 1, 9-74, (1999) · Zbl 0910.35107 [28] Soffer, A.; Weinstein, M.I., Selection of the ground state for nonlinear Schrödinger equations, Rev. math. phys., 16, 8, 977-1071, (2004) · Zbl 1111.81313 [29] Tsai, T.-P., Asymptotic dynamics of nonlinear Schrödinger equations with many bounds states, J. differential equations, 192, 225-282, (2003) · Zbl 1038.35128 [30] Tsai, T.-P.; Yau, H.-T., Asymptotic dynamics of nonlinear Schrödinger equations: resonance dominated and dispersion dominated solutions, Comm. pure appl. math., 55, 153-216, (2002) · Zbl 1031.35137 [31] Tsai, T.-P.; Yau, H.-T., Relaxation of excited states in nonlinear Schrödinger equations, Int. math. res. not., 2002, 31, 1629-1673, (2002) · Zbl 1011.35120 [32] Tsai, T.-P.; Yau, H.-T., Stable directions for excited states of nonlinear Schrödinger equations, Comm. partial differential equations, 27, 11-12, 2363-2402, (2002) · Zbl 1021.35113 [33] Tsai, T.-P.; Yau, H.-T., Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. theor. math. phys., 6, 1, 107-139, (2002) · Zbl 1033.81034
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