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 ${i\partial_x+V+k\langle \delta,\cdot\rangle}$. 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 $\lambda\in\mathbb{R}$, we explicitly construct the unique operator realization such that $\lambda$ is in its spectrum. We also investigate the time-evolution generated by these maximally dissipative operators.