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Non-self-adjoint Hamiltonians with complex eigenvalues. (English) Zbl 1344.81084

Summary: Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R40 Symmetry breaking in quantum theory
81R12 Groups and algebras in quantum theory and relations with integrable systems
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