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Man'ko S.S.,Ivan Franko National University of Lviv
Journal of Physics A: Mathematical and Theoretical | Year: 2010

We discuss the potential scattering on the noncompact star graph. The Schrödinger operator with the short-range potential localized in a neighborhood of the graph vertex is considered. We study the asymptotic behavior of the corresponding scattering matrix in the zero-range limit. It has been known for a long time that in dimension 1 there is no non-trivial Hamiltonian with the distributional potential δ', i.e. the δ' potential acts as a totally reflecting wall. Several authors have, in recent years, studied the scattering properties of the regularizing potentials αε -2Q(x/ε) approximating the first derivative of the Dirac delta function. A non-zero transmission through the regularized potential has been shown to exist as ε → 0. We extend these results to star graphs with the point interaction, which is an analog of the δ' potential on the line. We prove that generically such a potential on the graph is opaque. We also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs. This set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary Sturm-Liouville problem associated with Q on the graph. © 2010 IOP Publishing Ltd.

Tkachuk V.M.,Ivan Franko National University of Lviv
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2012

Studies in string theory and quantum gravity lead to the generalized uncertainty principle (GUP) and suggest the existence of a fundamental minimal length which, as was established, can be obtained within the deformed Heisenberg algebra. The first look on the classical motion of bodies in a space with corresponding deformed Poisson brackets in a uniform gravitational field can give an impression that bodies of different mass fall in different ways and, thus, the equivalence principle is violated. Analyzing the kinetic energy of a composite body, we find that the motion of its center of mass in the deformed space depends on some effective parameter of deformation. It gives a possibility to recover the equivalence principle in the space with deformed Poisson brackets and, thus, GUP is reconciled with the equivalence principle. We also show that the independence of kinetic energy on composition leads to the recovering of the equivalence principle in the space with deformed Poisson brackets. © 2012 American Physical Society.

Rovenchak A.,Ivan Franko National University of Lviv
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2014

A two-parameter fractional statistics is proposed, which can be used to model a weakly interacting Bose system. It is shown that the parameters of the introduced weakly nonadditive Polychronakos statistics can be linked to the effects of interactions as well as to finite-size corrections. Calculations are made of the specific heat and condensate fraction of the model system corresponding to harmonically trapped Rb-87 atoms. The behavior of the specific heat of three-dimensional isotropic harmonic oscillators with respect tothe statistics parameters is studied in the temperature domain including the BEC-like phase-transition point. © 2014 American Physical Society.

Gnatenko K.P.,Ivan Franko National University of Lviv
Physics Letters, Section A: General, Atomic and Solid State Physics | Year: 2013

The motion of a composite system made of N particles is examined in a space with a canonical noncommutative algebra of coordinates. It is found that the coordinates of the center-of-mass position satisfy noncommutative algebra with effective parameter. Therefore, the upper bound of the parameter of noncommutativity is re-examined. We conclude that the weak equivalence principle is violated in the case of a non-uniform gravitational field and propose the condition for the recovery of this principle in noncommutative space. Furthermore, the same condition is derived from the independence of kinetic energy on the composition. © 2013 Elsevier B.V.

Stetsko M.M.,Ivan Franko National University of Lviv
International Journal of Modern Physics A | Year: 2013

We investigate a microscopic black hole in the case of modified generalized uncertainty principle with a minimal uncertainty in position as well as in momentum. We calculate thermodynamical functions of a Schwarzschild black hole such as temperature, entropy and heat capacity. It is shown that the incorporation of minimal uncertainty in momentum leads to minimal temperature of a black hole. Minimal temperature gives rise to appearance of a phase transition. Emission rate equation and black hole's evaporation time are also obtained. © World Scientific Publishing Company.

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