Journal of Mathematical Chemistry

, Volume 45, Issue 3, pp 627–701 | Cite as

Exact treatment of open finite-dimensional quantum systems: I. Time-independent case

  • Tomislav P. Živković
Original Paper


Closed quantum systems that do not interact with the surrounding are described by an eigenvalue equation such as the Schrödinger equation. In particular, one can describe in this way a finite closed quantum system \({\bf S}_\rho^{a}\) that contains ρ eigenvalues and ρ eigenstates. Open quantum systems that interact with surrounding are usually treated within a perturbation expansion method. In a consistent quantum approach this “surrounding” should be treated as another (usually infinite) quantum system \({\bf S}_\infty^{\rm b}\) . In formal mathematical terms one has to find a solution of the combined system \({\bf S}_\infty \equiv {\bf S}_\rho^{\,a} \oplus {\bf S}_\infty^{\,b}\) with emphasize on the properties of the subsystem \({\bf S}_\rho^{a}\) . A new approach for the solution of this problem is presented. One finds that combined system S contains embedded eigenstates \(|\Psi (\varepsilon, \ldots)\rangle\) with continuous eigenvalues ε, and in addition it may contain isolated eigenstates \(|\Psi_r\rangle\) with discrete eigenvalues ε r . Two ρ ×  ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. In almost all cases those two equations produce a complete and exact description of the open quantum system \({\bf S}_\rho^{a}\) . The extremely rare exceptional cases can be also treated accordingly. The suggested method produces correct results, however strong the interaction between quantum systems \({\bf S}_\rho^{a}\) and \({\bf S}_\infty^{\rm b}\) . Two examples are presented in order to illustrate various aspects of this method.


Interaction of quantum systems Time-independent perturbation Open quantum systems 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Ru đ er Bošković InstituteZagrebCroatia

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