Nonlinear eigenvalues problems with a small parameter (Q799870)

In this work we study nonlinear eigenvalues problems like \([-\sigma^ 2d^ 2/dt^ 2+(t^ 2-\mu)^ 2+1]u=0\) where \(\mu \in {\mathbb{C}}\), \(\sigma >0\), \(u\in {\mathcal S}({\mathbb{R}}^ n)\). More precisely we study the spectrum of the operator \(Q(\sigma;\mu)=-\sigma^ 2d^ 2/dt^ 2+(t^ 2-\mu)^ 2+1\) when \(\sigma \to 0\), \(\sigma >0\). Our method of proof consists in replacing our problem by a linear eigenvalue problem about a non self adjoint system.