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 \(L\).


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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