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Spectra of positive and negative energies in the linearized NLS problem. (English) Zbl 1064.35181

The authors study spectral properties of the linearized nonlinear Schrödinger equation (NLS) \[ L{\psi}=JH{\psi}=z{\psi},\qquad J=\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix},\qquad H=\begin{pmatrix} -\Delta+\omega+f(x)&g(x)\\ g(x)&-\Delta+\omega+f(x) \end{pmatrix}, \] \( x\in \mathbb{R}^3\), \(\omega >0\), \(f,g:\mathbb{R}^3 \to \mathbb{C}\) are exponentially decaying \(C^{\infty}\) functions, in the context of instabilities of excited states of the NLS equation \[ i\psi_t=-\Delta \psi +U(x)\psi+F(|\psi|^2)\psi,\quad (x,t)\in \mathbb{R}^3\times \mathbb{R},\quad \psi\in \mathbb{C}. \] The main results are based on separation of spectra of positive and negative energies with energy functional \(h=\langle{\psi},H {\psi}\rangle\) defined on \(H^1(\mathbb{R}^3,\mathbb{C}^2)\). It is proved that the spectrum of \(H\) with negative energy is related to a subset of isolated or embedded eigenvalues \(z\) of the point spectrum \(P_{\sigma}(L)\) corresponding to the eigenvectors \({\psi}(x)\). This part of the spectrum provides instabilities of the excited states, where the linearized NLS equation has eigenvalues \(z\) with Im\((z)>0.\) Sharp bounds on the number and type of unstable eigenvalues of \(L\) are given in terms of negative eigenvalues of the energy operator \(H\).
It is shown that the part of spectrum of \(H\) with positive energy, related to a nonsingular part of the essential spectrum of \(L\) or to another subset of isolated or embedded eigenvalues \(z\) with \(Im(z)>0,\) does not produce instabilities of excited states, but leads to instabilities when eigenvalues \(z\) with negative energy coalesce with the essential spectrum or eigenvalues \(z\) with positive energy. At the usage of Fermi golden rule the singular part of the essential spectrum is studied, where it is proved that embedded eigenvalues \(z\) with positive energy disappear under generic perturbations, while eigenvalues with negative energy bifurcate into isolated complex eigenvalues \(z\) of the \(P_{\sigma}(L)\). The instability bifurcations in the interior points of the essential spectrum of \(L\) are studied, which is supported in the spectrum of \(L\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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[1] Agmon, Ann Scuola Norm Sup Pisa Cl Sci (4) 2 pp 151– (1975)
[2] Comech, Comm Pure Appl Math 56 pp 1565– (2003)
[3] Cuccagna, Comm Pure Appl Math 54 pp 1110– (2001)
[4] ; Bifurcations from the end points of the essential spectrum in the linearized NLS problem. Preprint, 2004.
[5] Grillakis, Comm Pure Appl Math 43 pp 299– (1990)
[6] Grillakis, J Funct Anal 74 pp 160– (1987)
[7] Grillakis, J Funct Anal 94 pp 308– (1990)
[8] Howland, Arch Rational Mech Anal 39 pp 323– (1970)
[9] Howland, Pacific J Math 55 pp 157– (1974) · Zbl 0312.47010
[10] Jones, J Differential Equations 71 pp 34– (1988)
[11] Kapitula, SIAM J Math Anal 33 pp 1117– (2002)
[12] Kato, Math Ann 162 pp 258– (1965)
[13] Perturbation theory for linear operators. 2nd ed. Grundlehren der Mathematischen Wissenschaften, 132. Springer, Berlin-New York, 1976.
[14] Kevrekidis, Phys D (2004)
[15] McLeod, Trans Amer Math Soc 339 pp 495– (1993)
[16] Pelinovsky, Proc Roy Soc London Ser A (2004)
[17] Asymptotic stability of solitary waves for nonlinear Schrödinger equations. Seminaire: Équations aux Dérivées Partielles, 2002-2003, 18. École Polytechniques, Palaiseau, 2003.
[18] Rauch, Comm Math Phys 61 pp 149– (1978)
[19] ; Methods of modern mathematical physics. I. Functional analysis. Academic, New York-London, 1972.
[20] ; Methods of modern mathematical physics. IV. Analysis of operators. Academic, New York-London, 1978. · Zbl 0401.47001
[21] Rodnianski, Comm Pure Appl Math
[22] Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton Series in Physics. Princeton University, Princeton, N.J., 1971.
[23] Functional integration and quantum physics. Pure and Applied Mathematics, 86. Academic, New York-London, 1979. · Zbl 0434.28013
[24] Soffer, Geom Funct Anal 8 pp 1086– (1998)
[25] Steinberg, Arch Rational Mech Anal 31 pp 372– (1968)
[26] Nonlinear wave equations. CBMS Regional Conference Series in Mathematics, 73. American Mathematical Society, Providence, R.I., 1989.
[27] Tsai, J Differential Equations 192 pp 225– (2003)
[28] Tsai, Int Math Res Not pp 1629– (2002)
[29] Tsai, Comm Partial Differential Equations 27 pp 2363– (2002)
[30] ; Eigenvalues of zero energy in the linearized NLS problem. Preprint, 2004.
[31] Weinstein, Comm Pure Appl Math 39 pp 51– (1986)
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