Non-self-adjoint Schrödinger operators with nonlocal one-point interactions

Abstract

We generalize and study, within the framework of quantum mechanics and working with $1$-dimensional, manifestly non-Hermitian Hamiltonians $H=-d^2/d{x}^2+V$, the traditional class of exactly solvable models with local point interactions $V=V\left(x\right)$. We discuss the consequences of the use of nonlocal point interactions such that $\left(Vf\right)\left(x\right)=\int K(x,s)f\left(s\right)ds$ by means of the suitably adapted formalism of boundary triplets.