## 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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### References:

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