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A GL(3) Kuznetsov Trace Formula and the Distribution of Fourier Coefficients of Maass FormsGuerreiro, João Leitão January 2016 (has links)
We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl’s law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications.
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Kvantové grafy a jejich zobecnění / Quantum Graphs and Their GeneralïzationsLipovský, Jiří January 2011 (has links)
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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Asymptotiques spectrales et géométrie des nombresLagacé, Jean 06 1900 (has links)
No description available.
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