Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

AbstractWe investigate boundedness of the evolutioneitH in the sense ofL2(ℝ3→L2(ℝ3) as well asL1(ℝ3→L∞(ℝ3) for the non-selfadjoint operator $$\mathcal{H} = \left[ \begin{gathered} - \Delta + \mu - V_1 \\ V_2 \\ \end{gathered} \right. \left. \begin{gathered} V_2 \\ \Delta - \mu + V_1 \\ \end{gathered} \right],$$ where μ>0 andV1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)–A4) below, but without imposing any restrictions on the edges±μ of the essential spectrum. Our goal is to develop an “axiomatic approach,” which frees the linear theory from any nonlinear context in which it may have arisen.

[1]  Arne Jensen,et al.  Spectral properties of Schrödinger operators and time-decay of the wave functions , 1979 .

[2]  Dmitry E. Pelinovsky,et al.  Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem , 2005 .

[3]  A. Soffer,et al.  Decay estimates for Schrödinger operators , 1991 .

[4]  Dispersive Estimates for Schrödinger Operators in Dimensions One and Three , 2003, math/0306108.

[5]  Walter A. Strauss,et al.  Nonlinear Wave Equations , 1990 .

[6]  Dmitry E. Pelinovsky,et al.  Purely nonlinear instability of standing waves with minimal energy , 2003 .

[7]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[8]  J. Shatah.  Stable standing waves of nonlinear Klein-Gordon equations , 1983 .

[9]  J. Shatah,et al.  Instability of nonlinear bound states , 1985 .

[10]  Walter A. Strauss,et al.  Existence of solitary waves in higher dimensions , 1977 .

[11]  M. Grillakis,et al.  Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system , 1990 .

[12]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[13]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[14]  P. Couturier.  Japan , 1988, The Lancet.

[15]  A. Jensen,et al.  A unified approach to resolvent expansions at thresholds , 2001 .

[16]  S. Cuccagna.  Stabilization of solutions to nonlinear Schrödinger equations , 2001 .

[17]  The $W^{k,p}$-continuity of wave operators for Schrödinger operators , 1995 .

[18]  A. Soffer,et al.  Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data , 1992 .

[19]  M. Weinstein.  Lyapunov stability of ground states of nonlinear dispersive evolution equations , 1986 .

[20]  F. Gesztesy,et al.  A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr , 1999, solv-int/9907017.

[21]  Dmitry E. Pelinovsky,et al.  Spectra of positive and negative energies in the linearized NLS problem , 2005 .

[22]  Israel Michael Sigal,et al.  Introduction to Spectral Theory , 1996 .

[23]  P. Lions,et al.  Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

[24]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[25]  Israel Michael Sigal,et al.  Introduction to Spectral Theory: With Applications to Schrödinger Operators , 1995 .

[26]  W. Schlag,et al.  Time decay for solutions of Schrödinger equations with rough and time-dependent potentials , 2001, math/0110098.

[27]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[28]  K. Yajima.  Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue , 2005 .

[29]  Wilhelm Schlag.  Stable manifolds for an orbitally unstable NLS , 2004 .

[30]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[31]  James Serrin,et al.  Uniqueness of positive radial solutions of Δu+f(u)=0 in ℝn , 1987 .

[32]  J. Rauch.  Local decay of scattering solutions to Schrödinger's equation , 1978 .

[33]  K. Yajima.  The $W^{k,p}$-continuity of wave operators for Schrödinger operators , 1995 .

[34]  W. Schlag,et al.  The nonlinear Schrödinger equation , 2008 .

[35]  G. Perelman.  Asymptotic Stability of Multi-soliton Solutions for Nonlinear Schrödinger Equations , 2003 .

[36]  G. Perelman.  On the Formation of Singularities in Solutions of the Critical Nonlinear Schrödinger Equation , 2001 .

[37]  M. Grillakis,et al.  Linearized instability for nonlinear Schr?odinger and Klein-Gordon equations , 1988 .

[38]  PRINCETON UNIVERSITY PRESS , 2005 .

[39]  Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials , 2004, math/0409327.

[40]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[41]  Vladimir S. Buslaev,et al.  On the stability of solitary waves for nonlinear Schr?odinger equations , 1995 .

[42]  I. Rodnianski,et al.  Dispersive analysis of charge transfer models , 2003, math/0309112.

[43]  Wilhelm Schlag,et al.  Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I , 2004 .

[44]  H.-G. Schöpf.  V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . , 1970 .

[45]  V. I. Arnolʹd,et al.  Ergodic problems of classical mechanics , 1968 .

[46]  C. V. Coffman.  Uniqueness of the ground state solution for Δu−u+u3=0 and a variational characterization of other solutions , 1972 .

[47]  Michael I. Weinstein,et al.  Multichannel nonlinear scattering for nonintegrable equations , 1990 .

[48]  M. Murata.  Asymptotic expansions in time for solutions of Schrödinger-type equations , 1982 .

[49]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, II existence of infinitely many solutions , 1983 .

[50]  Shmuel Agmon,et al.  Spectral properties of Schrödinger operators and scattering theory , 1975 .