Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2021
Volume
Vol. 41
Number
No. 6
Description
Special Issue - Spectral Theory and Applications
Journal Volume
Opuscula Mathematica
Vol. 41 (2021)
Projects
Pages
Articles
Generalized powers and measures
(Wydawnictwa AGH, 2021) Burdak, Zbigniew; Kosiek, Marek; Pagacz, Patryk; Rudol, Krzysztof; Słociński, Marek
Using the winding of measures on torus in »rational directions« special classes of unitary operators and pairs of isometries are defined. This provides nontrivial examples of generalized powers. Operators related to winding Szegö-singular measures are shown to have specific properties of their invariant subspaces.
Spectral properties of certain operators on the free Hilbert space F[H1,...,HN] and the semicircular law
(Wydawnictwa AGH, 2021) Cho, Ilwoo
In this paper, we fix $N$-many $l^2$-Hilbert spaces $H_k$ whose dimensions are $n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}$, for $k=1,\ldots,N$, for $N \in \mathbb{N}\setminus\{1\}$. And then, construct a Hilbert space $\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]$ induced by $H_{1},\ldots,H_{N}$, and study certain types of operators on $\mathfrak{F}$. In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by $\bigcup^N_{k=1} \mathcal{B}_{k}$, where $\mathcal{B}_{k}$ are the orthonormal bases of $H_k$, for $k=1,\ldots,N$.
The Krein-von Neumann extension of a regular even order quasi-differential operator
(Wydawnictwa AGH, 2021) Cho, Minsung; Hoisington, Seth; Nichols, Roger; Udall, Brian
We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl.
Corona theorem for strictly pseudoconvex domains
(Wydawnictwa AGH, 2021) Gwizdek, Sebastian
Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that M. Kosiek and K. Rudol obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved »abstract« Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries.
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.