Browsing by Subject "self-adjoint operators"

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Access status: Open Access ,
On one condition of absolutely continuous spectrum for self-adjoint operators and its applications
(Wydawnictwa AGH, 2018) Ânovič, Èduard Alekseevič
In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator $A$ by a sequence of operators $A_n$ with absolutely continuous spectrum on a given interval $[a,b]$ which converges to $A$ in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator $A$ spectrum on the finite interval $[a,b]$ and the condition for that the corresponding spectral density belongs to the class $L_{p}[a,b]$ ($p\geq 1$). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant $C\gt 0$ and a positive function $g(x)\in L_p[a,b]$ ($p\geq 1$) such that for all n sufficiently large and almost all $x\in [a,b]$ the estimate $\frac{1}{g(x)}\le b_n(P_{n+1}^2(x)+P_{n}^2(x))\le C$ holds, where $P_{n}(x)$ are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and $b_{n}$ is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on $[a,b]$ and for the corresponding spectral density $f(x)$ we have $f(x)\in L_p[a,b]$.
Access status: Open Access ,
Spontaneous decay of level from spectral theory point of view
(Wydawnictwa AGH, 2021) Ânovič, Èduard Alekseevič
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.