Counterexamples in indefinite Sturm--Liouville problems
The authors consider the spectral problem \[ -u''(x)=\lambda\omega(x) u(x),\quad x\in(-1,1), \quad u(-1)=u(1)=0, \] with \(\omega(x) \) being a sign-changing function. They describe some particular classes of functions \(\omega(x) \) for which the eigenfunctions to this problem do not form a~Riesz basis for the weighted space \(L_{2,\omega}(-1,1)\) under the norm \(\|u\|_{L_{2 ,\omega}(-1,1)}=\|\sqrt{|\omega|}u\|_{L_2 (-1,1)}\).
scientific article; zbMATH DE number 1431674