Decay Semigroups for the Resonances of Quantum Mechanical Scattering Systems

Details Decay Semigroups for the Resonances of Quantum Mechanical Scattering Systems Decay Semigroups for the Resonances of Quantum Mechanical Scattering Systems

2009-09-07

arxiv.org/abs/0909.1174v2

Physics

Mathematical Physics

25 pages; Supplements for section 1, added references; simplified argument in subsection 3.2, result unchanged; section 6 supp

Scientific paper

For selected classes of quantum mechanical Hamiltonians a canonical association of a decay semigroup is presented. The spectrum of the generator of this semigroup is a pure eigenvalue spectrum and it coincides with the set of all resonances. The essential condition for the results is the meromorphic continuability of the scattering matrix onto $\Bbb{C}\setminus(-\infty,0]$ and the rims $\Bbb{R}_{-}\pm i0$. Further finite multiplicity is assumed. The approach is based on an adaption of the Lax-Phillips scattering theory to semi-bounded Hamiltonians. It is applied to trace class perturbations with analyticity conditions. A further example is the potential scattering for central-symmetric potentials with compact support and angular momentum 0.

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