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Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation. (English) Zbl 1106.35044

The authors define two operators \[ L_{-}=-\Delta+\alpha^{2}-\phi^{2\beta},\qquad L_{+}=-\Delta+\alpha^{2}-(2\beta+1)\phi^{2\beta}, \] where \(\phi\) is the ground state for the supercritical nonlinear Schrödinger equation (NLS). The spectrum of these two operators is studied by a numerical Matlab computation. The gap property is defined to be the fact that \(L_{\pm}\) have no eigenvalues in \((0,\alpha^{2})\) and no resonance at \(\alpha^{2}\). The main result is that there exists a number \(\beta_{\star}\) such that \(\beta_{\star}<\beta\leq{1}\), and the gap property holds for NLS exponents of the form \(2\beta+1\).

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35-04 Software, source code, etc. for problems pertaining to partial differential equations

Software:

Matlab; eigs
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