Explanation of Nikiforov-Uvarov method in quantum mechanics – with a view toward algebraic geometry

In this essay, we review the Nikiforov-Uvarov method which is used to solve Schro¨dinger equation. We shed light on the algorithm from the viewpoint of algebraic geometry so that we shall the ideas of the latter (such as the resolution of singularity, normalization, primary decomposition of ideal) are lurking behind the algorithm. Besides, we study the application of the introductory D-module theory in this sort of eigenvalue problem and we present an algorithm alternative to the Nikiforov-Uvarov method.

G-GRAPHS AND SPECIAL REPRESENTATIONS FOR BINARY DIHEDRAL GROUPS IN GL(2,ℂ)

Given a finite subgroup G⊂GL(2,ℂ), it is known that the minimal resolution of singularity ℂ2/G is the moduli space Y=G-Hilb(ℂ2) of G-clusters ⊂ℂ2. The explicit description of Y can be obtained by calculating every possible distinguished basis for as vector spaces. These basis are the so-called G-graphs. In this paper we classify G-graphs for any small binary dihedral subgroup G in GL(2,ℂ), and in the context of the special McKay correspondence we use this classification to give a combinatorial description of special representations of G appearing in Y in terms of its maximal normal cyclic subgroup H ⊴ G.

BERGE'S KNOTS IN THE FIBER SURFACES OF GENUS ONE, LENS SPACE AND FRAMED LINKS

In 1990, John Berge described several families of knots in the three-dimensional sphere which have non-trivial Dehn surgeries yielding lens spaces. We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces, Kirby calculus in topology of four-dimensional manifolds and A'Campo's divide knot theory.