## 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

### Keywords:

spectrum; numerical computation; gap property

Matlab; eigs
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