In the article an integral representation of Mathieu (a,λ)-series S(%, p,a,λ) = ∑∞ n=0 a(n)(λ(n)+%) −p is obtained generalizing certain results by Guo, Tomovski, Tomovski and Trenčevski, and Qi. Bilateral bounding inequalities are given for S(%, p,a,λ) using this integral expression.

The notes are based on a series of six lectures, given during my stay at the CRC 701 in June/July 2008. The lecture series intended to give a survey of some of the results for the almost Mathieu operator that have been obtained since the early 1980’s. Specifically, the metalinsulator transition is discussed in detail, along with its relation to the ten Martini problem via duality and reducibility.

The Very High Energy Gamma Ray Astronomy (VHE) is a rapidly evolving branch of modern astronomy, which covers the range from about 50 GeV to several tens of TeV from the ground. In the past years, the second generation instruments firmly established a growing and varied list of sources including plerions, supernova remnants and active galactic nuclei, and started to study some fundamental quest...

The t-expansion, a nonperturbative analytic method for calculating the ground-state expectation values of arbitrary operators of the Hamiltonian, is applied to the Mathieu equation. Connrming previous results in other systems, a t-expansion up to order t 2 is suucient to recover the behaviour of the exact solution. A very good performance is observed when a variational wave function of the form...