Journal of Statistical Physics

, Volume 85, Issue 5–6, pp 575–606 | Cite as

Upper bounds for the energy expectation in time-dependent quantum mechanics

  • Alain Joye


We consider quantum systems driven by Hamiltonians of the formH+W(t), where the spectrum ofH consists of an infinite set of bands andW(t) depends arbitrarily on time. Let 〈H〈φ(t) denote the expectation value ofH with respect to the evolution at timet of an initial state φ. We prove upper bounds of the type 〈H〉φ(t)=O(tδ), δ>0, under conditions on the strength ofW(t) with respect toH. Neither growth of the gaps between the bands nor smoothness ofW(t) is required. Similar estimates are shown for the expectation value of functions ofH. Sufficient conditions to have uniformly bounded expectation values are made explicit and the consequences on other approaches to quantum stability are discussed.

Key Words

Quantum stability energy expectations quantum diffusion 


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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Alain Joye
    • 1
  1. 1.Department of Mathematics and Center for Transport Theory and Mathematical PhysicsVirginia Polytechnic Institute and State UniversityBlacksburg

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