Nearby states in non-Hermitian quantum systems I: Two states
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The formalism for the description of open quantum systems (that are embedded into a common well-deﬁned environment) by means of a non-Hermitian Hamilton operator H is sketched. Eigenvalues and eigenfunctions are parametrically controlled. Using a 2 × 2 model, we study the eigenfunctions of H at and near to the singular exceptional points (EPs) at which two eigenvalues coalesce and the corresponding eigenfunctions diﬀer from one another by only a phase. Nonlinear terms in the Schr¨odinger equation appear nearby EPs which cause a mixing of the wavefunctions in a certain ﬁnite parameter range around the EP. The phases of the eigenfunctions jump by π at an EP. These results hold true for systems that can emit (“loss”) particles into the environment of scattering wavefunctions as well as for systems which can moreover absorb (“gain”) particles from the environment. In a parameter range far from an EP, open quantum systems are described well by a Hermitian Hamilton operator. The transition from this parameter range to that near to an EP occurs smoothly.