Nonorthogonality constraints in open quantum and wave systems

Abstract

It is known that the squared modulus of the overlap (scalar product) of two energy eigenstates in a decaying quantum system is bounded from above by a function of the energy detuning and the individual decay rates. This is usually traced back to the positive definiteness of an appropriately defined decay operator. Here, we show that the weaker and more realistic condition of positive semidefiniteness is sufficient. We prove also that the bound becomes an equality for the case of single-channel decay. However, we show that the condition of positive semidefiniteness can be spoiled by quantum backflow. Hence, the overlap of quasibound quantum states subjected to outgoing-wave conditions can be larger than expected from the bound. A modified and less stringent bound, however, can be introduced. For electromagnetic systems, it turns out that a modification of the bound is not necessary due to the linear free-space dispersion relation. Finally, a geometric interpretation of the nonorthogonality bound is given which reveals that in this context the complex energy space can be seen as a surface of constant negative curvature.