An analysis of non-selfadjoint first-order differential operators with non-local point interactions

We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by $i{\partial }_x+V+k〈\delta ,\cdot 〉$. We give precise estimates on the location of the eigenvalues on the complex plane and prove that the root vectors of these operators form Riesz bases of $L^2(0,2\pi )$. Under the additional assumption that the operator is maximally dissipative, we prove that it can have at most one real eigenvalue, and given any $\lambda \in R$, we explicitly construct the unique operator realization such that λ is in its spectrum. We also investigate the time-evolution generated by these maximally dissipative operators.