Eigenfunction expansions for the nonspectral differential operators determined by \(-D^ 2\) (Q1188251)

The authors study differential operators \(L\) in \(L^ 2[0,1]\) determined by \(-D^ 2\) and two-point boundary conditions such that the spectrum \(\sigma(L)\) is an infinite countable set of eigenvalues but \(L\) is no spectral operator. Sufficient conditions for a function \(u\in L^ 2[0,1]\) to have an \(L^ 2\) eigenfunction expansion are established. Under special assumptions both necessary and sufficient conditions are derived.