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Spectral analysis for matrix Hamiltonian operators. (English) Zbl 1213.35371

The authors investigate the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schrödinger equation (NLS) in \(\mathbb{R}\times \mathbb{R}^d\) \[ i\psi_t+\Delta \psi+g(|\psi|^2)\psi=0,\,\,\,\psi(0,x)=\psi_0(x), \] about soliton solutions. They prove that the linearized operator of the focusing 3D cubic NLS has no embedded eigenvalues and no endpoint resonances. Numerical results for the solitons, index functions, solutions for the relevant boundary value problems (3D cubic NLS and 1D NLS) and the associated inner products are also presented.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35P20 Asymptotic distributions of eigenvalues in context of PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

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