The authors treat singular dissipative operators with finite impulsive conditions of the form \[ T y(x) = w^{-1}(x)\left[-(p(x)y')' + q(x)y\right], \] \(x \in \Lambda = (c_0,c_{n+1})\setminus\{c_1,\ldots,c_n\}\), where \(-\infty < c_0 < c_1 < \ldots < c_{n+1} \leq \infty\), \(p\), \(q\), \(w\) are real-valued Lebesgue measurable functions, \(p^{-1}\), \(q\) can be singular at \(c_i\), \(i=0,\ldots,n+1\). The domain of the operator \(T\) is the set of functions satisfying certain boundary and impulsive conditions (at fixed times \(c_i\)) on \(\Lambda\).
From the abstract: After passing to the inverse operators, it is obtained that imaginary parts of the inverse operators are nuclear. Finally, using Krein’s theorem, it is proved that all root vectors of the singular dissipative operators with finite impulsive conditions are complete in the Hilbert space.