**Time Asymmetry in Quantum Mechanics**A time asymmetric quantum theory can be achieved if one extends the Hilbert space of ordinary quantum mechanics (of measured systems) to Gel'fand triplets.

**Representations of Groups and Semigroups in Quantum Mechanics**Time evolution (and also the relativistic space-time symmetry group) extend to semigroups only, representing irreversibility that originates from causality (boundary) conditions.

**Quantum Geometric Phases and Applications to bound and resonance states**Scattering resonances and decaying states are described by Gamow states as elementary objects for which the decay probabilities have time ordering and obey the exponential decay law.

**Mathematical Physics in the Rigged Hilbert Space**A generalized spectral decomposition (generalization of Dirac's eigenket expansion) allows for a rigged Hilbert space formulation of chaotic maps.

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