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Conservation of pseudo-norm in $${\mathcal P}{\mathcal T}$$ symmetric quantum mechanics. (English) Zbl 1052.46010
Slovák, Jan (ed.) et al., The proceedings of the 23th winter school “Geometry and physics”, Srní, Czech Republic, January 18–25, 2003. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 72, 211-218 (2004).
Summary: We show that the evolution of the wave functions in the $${\mathcal P}{\mathcal T}$$ symmetric quantum mechanics is pseudo-unitary. Their pseudo-norm $$\langle\psi |{\mathcal P}| \psi\rangle$$ remains time-independent. This persists even if the $${\mathcal P}{\mathcal T}$$ symmetry itself becomes spontaneously broken.
For the entire collection see [Zbl 1034.53002].
##### MSC:
 46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) 34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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