Effective Hamiltonians for Complexes of Unstable Particles


Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for oneparticle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.

DOI: 10.1142/S1230161213400088

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@article{Urbanowski2013EffectiveHF, title={Effective Hamiltonians for Complexes of Unstable Particles}, author={Krzysztof Urbanowski}, journal={Open Syst. Inform. Dynam.}, year={2013}, volume={20} }