Next to the harmonic oscillator and the Coulomb potential the class of two-body models with point interactions is the only one where complete solutions are available. All mathematical and physical quantities can be calculated explicitly which makes this field of research important also for more complicated and realistic models in quantum mechanics. The detailed results allow their implementation in numerical codes to analyse properties of alloys, impurities, crystals and other features in solid state quantum physics. This monograph presents in a systematic way the mathematical approach and unifies results obtained in recent years. The student with a sound background in mathematics will get a deeper understanding of Schrdinger Operators and will see many examples which may eventually be used with profit in courses on quantum mechanics and solid state physics. The book has textbook potential in mathematical physics and is suitable for additional reading in various fields of theoretical quantum physics.

In this paper we study the L p-Minkowski problem for p=鈭n鈭1, which corresponds to the critical exponent in the Blaschke鈥揝antalo inequality. We first obtain volume estimates for general solutions, then establish a priori estimates for rotationally symmetric solutions by using a Kazdan鈥揥arner type obstruction. Finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow-up analysis. We also include an existence result for the L p-Minkowski problem which corresponds to the super-critical case of the Blaschke鈥揝antalo inequality.

abstract

We construct a special principal series representation for the modular double Uqq∼(gR) of type Ar representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also relate q and. We use the cluster variables parameterization of the positive unipotent matrices to derive the formulas in the classical case. Then we quantize them after applying the Mellin transform. Our construction is inspired by the previous results for gR =sl(2,R) and can be generalized to all other types of simple split real Lie algebra. We conjecture that our positive representations are closed under the tensor product and we discuss the future perspectives of the new representation theory following the parallel with the established developments of the finite-dimensional representation theory of quantum groups.

Using the complex coloring method, we present the graphs of the quantum dilogarithm function Gb(z) and visualize its analytic and asymptotic behaviours. In particular we demonstrate the limiting process when the modified Gb(z)→Γ(z) as b→0. We also survey the relations of Gb(z) with different variants of the quantum dilogarithm function.