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1
Content available remote Quenched asymptotics for symmetric Lévy processes interacting with Poissonian fields
Chen Zhi-He, Wang Jian
Probability and Mathematical Statistics
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2022
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Vol. 42, Fasc. 2
319--356
EN
We establish the quenched large time asymptotics for the Feynman-Kac functional [formula] associated with a pure-jump symmetric Lévy process (Zt)t⩾0 in general Poissonian random potentials V ω on Rd, which is closely related to the large time asymptotic behavior of solutions to the nonlocal parabolic Anderson problem with Poissonian interaction. In particular, when the density function with respect to the Lebesgue measure of the associated Lévy measure is given by [formula] for some α ∈ (0, 2), θ ∈ (0, ∞] and c > 0, an explicit quenched asymptotics is derived for potentials with the shape function given by φ(x) = 1 ∧ |x|−d−β for β ∈ (0, ∞] with β ̸ = 2, and it is completely different for β > 2 and β < 2. We also discuss the quenched asymptotics in the critical case (e.g., β = 2 in the example above). The work fills the gaps of the related work for pure-jump symmetric Lévy processes in Poissonian potentials, where only the case that the shape function is compactly supported (e.g., β = ∞ in the example above) has been handled in the literature.
2
Content available Spontaneous decay of level from spectral theory point of view
Ianovich Eduard
Opuscula Mathematica
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2021
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Vol. 41, no. 6
849-–859
EN
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.
3
Content available remote Diagonals of Self-adjoint Operators with Finite Spectrum
Bownik M., Jasper J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
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Vol. 63, no. 3
249--260
EN
Given a finite set X ⊆ R we characterize the diagonals of self-adjoint operators with spectrum X. Our result extends the Schur–Horn theorem from a finite-dimensional setting to an infinite-dimensional Hilbert space analogous to Kadison’s theorem for orthogonal projections (2002) and the second author’s result for operators with three-point spectrum (2013).
4
Content available The approximate location of imperfection in a unit circle using the spectrum of Laplace operator as a research tool
Brzęk M., Zagórowska M., Mitkowski W.
Automatyka / Automatics
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2015
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Vol. 19, no. 1
9--15
EN
In the following article we will try to find the dependence between the location of imperfections in a closed domain and the spectrum of the Laplace operator for this region. In the theoretical part we will define the spectral problem which is solved by eigenvalues. These eigenvalues are dependent on location and size of the imperfection. However, we are interested in the inverse task which consists in localizing the imperfection of the domain on a basis of the spectrum of the operator.
PL
W artykule przeanalizowano zastosowanie widma operatora Laplace'a jako narzędzia do przybliżonej lokalizacji uszkodzeń w kole jednostkowym. W części teoretycznej zdefiniowano zagadnienie spektralne rozwiązywane za pomocą wartości własnych. Znalezione wartości zależą od położenia i rozmiaru uszkodzeń. W artykule został zdefiniowany problem odwrotny, który polega na znalezieniu miejsca uszkodzenia na podstawie znanego widma.
5
Content available remote Unbounded Jacobi matrices with empty absolutely continuous spectrum
Cojuhari P., Janas J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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Vol. 56, no 1
39-51
EN
Sufficient conditions for the absence of absolutely continuous spectrum for unbounded Jacobi operators are given. A class of unbounded Jacobi operators with purely singular continuous spectrum is constructed as well.
6
Content available remote An algebraic approach to linear-quadratic optimization of second order dynamical systems
Respondek J.
Archives of Control Sciences
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2007
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Vol. 17, no. 1
105-116
7
Content available remote Interpolated subspaces of exponential vectors of the unbounded operators in Banach spaces
Dmytryshyn M., Lopushansky O.
Demonstratio Mathematica
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2004
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Vol. 37, nr 1
149--158
EN
It is shown that the spectral subspaces of the unbounded operators in Ba-nach spaces and also their integer degrees can be described with help of interpolation. The spectral subspaces of operators are described on the basis of abstract Bernstein inequality. The results are applied to research of the root subspaces of regular elliptic operators in a bounded domains.
8
Content available remote Absence of eigenvalues for integro-differential operators
Cojuhari P. A., Stanescu M.M.
Opuscula Mathematica
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2003
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Vol. 23
5-14
EN
Sufficient conditions for the absence of eigenvalues of integro-differential operators are presented.
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