|
|
Quantum graphs with circulant vertex couplings
Pekař, Jan ; Exner, Pavel (advisor) ; Lipovský, Jiří (referee)
Motivated by recent investigation of several particular situations, we study various quantum graphs equipped with circulant vertex couplings and characterize their spectral properties. The case of a star graph is analyzed in full generality, and the same applies to the condition determining the spectrum of periodic rectangular lattices. Special attention is paid to permutation-invariant vertex conditions on a rectangular lattice, as well as to a coupling interpolating between the δ and 'rotational' coupling on a quantum chain, with the focus on low- and high-energy bands and the discrete spectrum. We describe not only their dependence on the topology and the vertex condition, but also provide detail of their behaviour with respect to the parameters involved. 1
|
|
|
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.
|
|
| |
|
| |
|
| |
|
|
Magnetic transport along translationally invariant obstacles
Grňo, Michal ; Exner, Pavel (advisor) ; Lotoreichik, Vladimir (referee)
We discuss the spectra of magnetic Schrödinger operators of the form (−i∇+ ⃗A(x))2 + V (x) on L2 (Ω), where Ω is either R2 , a half-plane, a strip, or a thin layer in R3 . This class of operators includes the Landau Hamiltonian, as well as the Iwatsuka Hamiltonian and other translationally invariant perturbations of it. We provide a comprehensive list of the known results regarding these operators, and inspect two Hamiltonians that have not yet been studied: the Landau Hamiltonian with a δ-interaction supported on a line and the Landau Hamiltonian in a half-plane with a Robin boundary condition. We prove that the spectra of these two Hamiltonians are purely absolute continuous and that the former has gaps between adjacent Landau levels. 1
|
|
|
Urban Structure Analysis
Vašata, D. ; Exner, Pavel ; Šeba, Petr
The built-up land represents an important type of overall landscape. We analyse the structure of built-up land in largest cities in the Czech Republic and selected cities in the USA using the framework of statistical physics. To do this, both the variance of the built-up area and the number variance of built-up landed plots in circles are calculated. In both cases the variance as a function of a circle radius follows a power law. The obtained value of the exponents are comparable to exponents typical for critical systems. The study is based on cadastral data in the Czech Republic and building footprints GIS data in the USA.
|
|
| |
|
|
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.
|
|
| |
guest ::
