The Ćurgus Condition in Indefinite Sturm–Liouville Problems

For a finite signed measure $\mu$ on $(-1,1)$ changing its sign at zero, we study the Riesz basis property in the space $L_{2,|\mu|}$ of generalized eigenfunctions of the spectral problem $-u''(x)dx=\lambda u(x)d\mu(x)$, $-1<x<1$, $u(-1)=u(1)=0$. Primarily, our approach is based on the Ćurgus criterion. We present a criterion for the basis property in the case of an odd measure and sufficient conditions (in terms of $\mu$) known so far for a measure absolutely continuous with respect to the Lebesgue measure whose support is the whole interval. We prove the Riesz basis property for a degenerate discrete measure of a special form and a new necessary condition for this property. For a dense embedding $V\subset H=H'$ of a reflexive Banach space $V$ into a Hilbert space $H$ and a symmetric unitary (in $H$) operator $J$, we consider the interpolation equality $\bigl(V,(JV)'\bigr)_{1/2,2}=H$ applicable to nonlinear evolutionary equations of mixed type. We also exhibit conditions ensuring this equality and generalizing sufficient conditions for the basis property.