Conditions for localization of the limit spectrum of a model operator associated with the Orr-Sommerfeld equation (Q1761077)

Consider the operator \(L_\varepsilon\) generated in \(L^2(-1,1)\) by the differential expression \[ l_\varepsilon y:=i\varepsilon^2 y''+qy \] and the boundary conditions \(y(-1)=y(1)=0\). The author verifies the following hypothesis. The limit spectral graph of the operator \(L_\varepsilon\) with potential \(q\) has the \(l_c\)-property (for definition see the paper) only if \(q\) can be analytically continued to some neighborhood of the interval \([-1,1]\).