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Quantum Graphs and Their Generalïzations
Lipovský, Jiří ; Exner, Pavel (advisor) ; Šeba, Petr (referee) ; Bolte, Jens (referee)
In the present theses we study spectral and resonance properties of quantum graphs. First, we consider graphs with rationally related lengths of the edges. In particular examples we study trajectories of resonances which arise if the ratio of the lengths of the edges is perturbed. We prove that the number of resonances under this perturbation is locally conserved. The main part is devoted to asymptotics of the number of resonances. We find a criterion how to distinguish graphs with non-Weyl asymptotics (i.e. constant in the leading term is smaller than expected). Furthermore, due to explicit construction of unitary equivalent operators we explain the non-Weyl behaviour. If the graph is placed into a magnetic field, the Weyl/non-Weyl characteristic of asymptotical behaviour does not change. On the other hand, one can turn a non-Weyl graph into another non-Weyl graph with different "effective size". In the final part of the theses, we describe equivalence between radial tree graphs and the set of halfline Hamiltonians and use this result for proving the absence of the absolutely continuous spectra for a class of sparse tree graphs.
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Quantum Graphs and Their Generalïzations
Lipovský, Jiří
In the present theses we study spectral and resonance properties of quantum graphs. First, we consider graphs with rationally related lengths of the edges. In particular examples we study trajectories of resonances which arise if the ratio of the lengths of the edges is perturbed. We prove that the number of resonances under this perturbation is locally conserved. The main part is devoted to asymptotics of the number of resonances. We find a criterion how to distinguish graphs with non-Weyl asymptotics (i.e. constant in the leading term is smaller than expected). Furthermore, due to explicit construction of unitary equivalent operators we explain the non-Weyl behaviour. If the graph is placed into a magnetic field, the Weyl/non-Weyl characteristic of asymptotical behaviour does not change. On the other hand, one can turn a non-Weyl graph into another non-Weyl graph with different "effective size". In the final part of the theses, we describe equivalence between radial tree graphs and the set of halfline Hamiltonians and use this result for proving the absence of the absolutely continuous spectra for a class of sparse tree graphs. Powered by TCPDF (www.tcpdf.org)
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Investigation of quantum reaction dynamics using semiclassical method.
Táborský, Jiří ; Čížek, Martin (advisor)
In the presented thesis we study quantum reaction dynamics of H2O- using semiclassical method. Using ab initio quantum potential evaluated on a fine grid we derive analytical formula for potential energy surface, which is used for solving classical equations of motion. A reaction path is analyzed using contour plots with a focus on a saddle point area. Reaction dynamics analysis is focused on properties of interaction probability and cross section depending on impact parameter, collision energy and initial vibrational state of interacting molecule. Final results are compared with experimental data.
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Investigation of quantum reaction dynamics using semiclassical method.
Táborský, Jiří ; Čížek, Martin (advisor)
In the presented thesis we study quantum reaction dynamics of H2O- using semiclassical method. Using ab initio quantum potential evaluated on a fine grid we derive analytical formula for potential energy surface, which is used for solving classical equations of motion. A reaction path is analyzed using contour plots with a focus on a saddle point area. Reaction dynamics analysis is focused on properties of interaction probability and cross section depending on impact parameter, collision energy and initial vibrational state of interacting molecule. Final results are compared with experimental data.
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Study of time evolution of metastable states in quantum mechanics
Gedeonová, Hedvika ; Kolorenč, Přemysl (advisor) ; Čížek, Martin (referee)
In this thesis, the metastable states are studied. The work focuses on a theoretical model of one or two metastable states decaying into a continuum of states which is bounded from below. We examine the time evolution of such systems and how it is affected by the energy of the metastable state(s) and by the position of the poles of the scattering matrix in the complex plane. We also look closely at the spectral line shape. Numerical integration of a system of differential equations is used for solving the problem of the time evolution and spectral line shape while Freshbach-Fano projection operator formalism is used for finding the position of the poles. The results are compared with first order perturbation theory and with semi-analytical formula obtained from adiabatic elimination of the continuum. The last part of the thesis is dedicated to an application of the model on neon-helium-neon cluster. 1
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Investigation of quantum reaction dynamics using semiclassical method.
Táborský, Jiří ; Čížek, Martin (advisor) ; Houfek, Karel (referee)
In the presented thesis we study quantum reaction dynamics of H2O- using semiclassical method. Using ab initio quantum potential evaluated on a fine grid we derive analytical formula for potential energy surface, which is used for solving classical equations of motion. A reaction path is analyzed using contour plots with a focus on a saddle point area. Reaction dynamics analysis is focused on properties of interaction probability and cross section depending on impact parameter, collision energy and initial vibrational state of interacting molecule. Final results are compared with experimental data.
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Difraction of particle on slit with internal structure
Hudec, Matěj ; Čížek, Martin (advisor) ; Houfek, Karel (referee)
In this thesis we study the difraction of a particle on a slit, focusing on the resonance effects. The two-dimensional model containing a delta-function barrier with variable height which represents the slit is solved on a belt with periodic boundary condition. In most cases we use the numerical solution of the problem using our program. The results are compared with elastic approximations. We study the characteristics of the resonances in detail, the dependence of their existence, energy and height on the parameters of the slit, and the relation between resonances and bound states. The last chapter is devoted to the problem of spreading the belt in order to get the results that describe the scattering process on the whole plain. We suggest two ap- proaches of description of this process and demonstrate that the appearance of the resonances is not limited to the given boundary condition. 1
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Multimode vibrational dynamics of electron scattering from molecule
Táborský, Jiří ; Čížek, Martin (advisor) ; Kolorenč, Přemysl (referee)
In the present work we study vibrational excitation of a generally polyatomic molecule by electron impact. We proceed from the theory of discrete state in continuum and the nonlocal resonance model and we approximate the molecule using two-dimensional harmonic oscillator. We have written numerical procedures in Fortran which compute a cross section of these collisions. The correctness of a transition from 1D model to 2D has been verified. We have performed several computations using different initial parameters and have studied the effect of changing selected parameters on the cross section depending on the energy of the incoming electron. Powered by TCPDF (www.tcpdf.org)
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Quantum Graphs and Their Generalïzations
Lipovský, Jiří
In the present theses we study spectral and resonance properties of quantum graphs. First, we consider graphs with rationally related lengths of the edges. In particular examples we study trajectories of resonances which arise if the ratio of the lengths of the edges is perturbed. We prove that the number of resonances under this perturbation is locally conserved. The main part is devoted to asymptotics of the number of resonances. We find a criterion how to distinguish graphs with non-Weyl asymptotics (i.e. constant in the leading term is smaller than expected). Furthermore, due to explicit construction of unitary equivalent operators we explain the non-Weyl behaviour. If the graph is placed into a magnetic field, the Weyl/non-Weyl characteristic of asymptotical behaviour does not change. On the other hand, one can turn a non-Weyl graph into another non-Weyl graph with different "effective size". In the final part of the theses, we describe equivalence between radial tree graphs and the set of halfline Hamiltonians and use this result for proving the absence of the absolutely continuous spectra for a class of sparse tree graphs. Powered by TCPDF (www.tcpdf.org)
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Generalization of the method of analytical continuation in coupling constant
Brožek, Pavel ; Horáček, Jiří (advisor) ; Čížek, Martin (referee)
In the thesis we study a method for determining resonance energies - gen- eralization of the method of analytical continuation in the coupling constant, which is based on continuation of the coupling constant λ as a function of the momentum k. A formula for λ(k) is derived for spherically symmetric potential consisting of finite number of δ-functions and its Taylor series is studied. Taylor series of λ(k) and its asymptotic behavior is studied for sep- arable potential. Proper choice of added potential parameters is studied on examples. A method for determining λ(k) poles is described for spherically symmetric potential with added δ-function. It is tested whether the knowl- edge of λ(k) poles can be useful to improve the accuracy of the determination of the resonance parameters of the original potential.
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