arXiv subjects

E. A. Ianovich

Publications and source records attributed to E. A. Ianovich.

3 recordsLinked to original sources

arXiv2023

Eigenvalues asymptotics of unbounded operators. Two-photon quantum Rabi model

In this work the general results about asymptotics of eigenvalues of unbounded operators are obtained. We consider here different cases of compact, relatively compact, selfadjoint or nonselfadjoint perturbations. In particular we prove a generalization of Janas-Naboko lemma about eigenvalues asymptotics of unbounded operators at compact perturbation. A generalization of our previous result about noncompact perturbation of oscillator spectrum is also given. As an example we consider two-photon quantum Rabi model. We obtain tree-term asymptotic formula for large eigenvalues of the energy operator of this model. The asymptotics of related to this model polynomials is found. We give also an original proof of the Perelomov factorization theorem for contraction operator of quantum optics.

math.SP↗

arXiv2023

Spectral density in a Moszynski's class of Jacobi matrices. Spectral phase transition of 2nd type

In this paper it is considered a spectral density for a class of Jacobi matrices with absolutely continuous spectrum that was examined first by Moszynski. It is shown that the corresponding spectral density is equivalent to the positive continuous function everywhere except the point $x=0$. In the point $x=0$ the spectral density may be finite as well as essentially infinite depending on the parameter. So in this class we discover an example of 2nd type spectral phase transition.

math.SP↗

arXiv2019

Spontaneous decay of level from spectral theory point of view

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.

math.SP↗