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Spontaneous decay of level from spectral theory point of view

Ianovich Eduard

Opuscula Mathematica

2021

Vol. 41, no. 6

849-–859

In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent.

Non-linearity of the set of p-quasi-analytic vectors for some essentially self-adjoint operators

Rusinek J.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 3

287-292

We show that, for any p [is greater than or equal to] 1, there exists an essentially self-adjoint operator for which the set of p-quasi-analytic vectors is not linear.