RESONANCES ON HEDGEHOG MANIFOLDS

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate this on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a ‘hedgehog’ manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis

Spectral determinant for the damped wave equation on an interval

We evaluate the spectral determinant for the damped wave equation on an interval of length $T$ with Dirichlet boundary conditions, proving that it does not depend on the damping. This is achieved by analysing the square of the damped wave operator using the general result by Burghelea, Friedlander, and Kappeler on the determinant for a differential operator with matrix coefficients.Comment: 11 page

Non-Weyl resonance asymptotics for quantum graphs in a magnetic field

We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the ``effective size'' of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field

Resonances from perturbations of quantum graphs with rationally related edges

We discuss quantum graphs consisting of a compact part and semiinfinite leads. Such a system may have embedded eigenvalues if some edge lengths in the compact part are rationally related. If such a relation is perturbed these eigenvalues may turn into resonances; we analyze this effect both generally and in simple examples.Comment: LaTeX source file with 10 pdf figures, 24 pages; a replaced version with minor improvements, to appear in J. Phys. A: Math. Theo

Magnetic square lattice with vertex coupling of a preferred orientation

We analyze a square lattice graph in a magnetic field assuming that the vertex coupling is of a particular type violating the time reversal invariance. Calculating the spectrum numerically for rational values of the flux per plaquette we show how the two effects compete; at the high energies it is the magnetic field which dominates restoring asymptotically the familiar Hofstadter's butterfly pattern.Comment: 19 pages, 6 figure

Non-Weyl asymptotics for quantum graphs with general coupling conditions

Inspired by a recent result of Davies and Pushnitski, we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices. We derive a criterion for the asymptotics to be of a non-Weyl character. We show that for balanced vertices with permutation-invariant couplings the asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions, while for graphs without permutation numerous examples of non-Weyl behaviour can be constructed. Furthermore, we present an insight helping to understand what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance point of view. Finally, we demonstrate a generalization to quantum graphs with nonequal edge weights.Comment: minor changes, to appear in Pierre Duclos memorial issue of J. Phys. A: Math. Theo

Kvantové grafy a jejich zobecnění

V předkládané práci studujeme spektrální a rezonanční vlastnosti kvantových grafů. Nejdříve uvažujeme grafy, délky jejichž některých hran jsou soudělné. V konkrétních případech studujeme trajektorie rezonancí, které vzniknou porušením poměru délek hran. Dokážeme, že počet rezonancí se při této perturbaci lokálně zachovává. Hlavní část práce se zabývá asymptotikou počtu rezonancí. Najdeme kritérium, jak rozlišit grafy s neweylovskou asymptotikou (konstanta u vedoucího členu je nižší, než se očekává). Navíc vysvětlíme toto neweylovské chování konstrukcí unitárně ekvivalentního grafu. Pokud umístíme graf do magnetického pole, jeho základní charakteristika (weylovskost/neweylovskost) se nezmění. Může se ale změnit "efektivní velikost" neweylovského grafu. V poslední části práce popíšeme ekvivalenci mezi radiálními stromovými grafy a množinou hamiltoniánů na polopřímkách. Tento výsledek využijeme pro důkaz absence absolutně spojitého spektra pro širokou třídu řídkých stromových grafů.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.Matematicko-fyzikální fakultaFaculty of Mathematics and Physic