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

Abstract

We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by $i{\partial }_x+V+k⟨\hat{I}´,\cdot ⟩$. 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 $L^2(0,2\ddot{I}€)$. Under the additional assumption that the operator is maximally dissipative, we prove that it can have at most one real eigenvalue, and given any $\hat{I}»\in \mathbb{R}$, we explicitly construct the unique operator realization such that $\hat{I}»$ is in its spectrum. We also investigate the time-evolution generated by these maximally dissipative operators.

25 pages; added 1 figure illustrating Theorem 3.14