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Surgery of Graphs: M-Function and Spectral Gap

100%

Kurasov P.

Acta Physica Polonica A

2017

tom 132

nr 6

1666-1671

We discuss behaviour of the spectral gap for quantum graphs when two metric graphs are glued together. It appears that precise answer to this question can be given using a natural generalisation of the Titchmarsh-Weyl M-functions.

Interpolated subspaces of exponential vectors of the unbounded operators in Banach spaces

80%

Dmytryshyn M. , Lopushansky O.

2004

tom Vol. 37, nr 1

149--158

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.

An algebraic approach to linear-quadratic optimization of second order dynamical systems

80%

Respondek J.

2007

tom Vol. 17, no. 1

105-116

Diagonals of Self-adjoint Operators with Finite Spectrum

70%

Bownik M. , Jasper J.

2015

tom Vol. 63, no. 3

249--260

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).

Spontaneous decay of level from spectral theory point of view

70%

Ianovich E.

tom 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.

Absence of eigenvalues for integro-differential operators

70%

Cojuhari P. A. , Stanescu M. M.

tom Vol. 23

5-14

Sufficient conditions for the absence of eigenvalues of integro-differential operators are presented.

Quenched asymptotics for symmetric Lévy processes interacting with Poissonian fields

70%

Chen Z.-H. , Wang J.

tom Vol. 42, Fasc. 2

319--356

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.