Abstract:
As a result of the so(2,1) of the hypergeometric Natanzon potential a set of potentials related to the given one is determined. The set arises as a result of the action of the so(2,1) generators.

Abstract:
Using the underlying algebraic structures of Natanzon potentials, we discuss conditions that generate shape invariant potentials. In fact, these conditions give all the known shape invariant potentials corresponding to a translational change of parameters. We also find that while the algebra for the general Natanzon potential is $SO(2,2)$, a subgroup $SO(2,1)$ suffices for all the shape invariant problems of Natanzon type.

Abstract:
Using the so(2,1) Lie algebra and the Baker, Campbell and Hausdorff formulas, the Green's function for the class of the confluent Natanzon potentials is constructed straightforwardly. The bound-state energy spectrum is then determined. Eventually, the three-dimensional harmonic potential, the three-dimensional Coulomb potential and the Morse potential may all be considered as particular cases.

Abstract:
the restricted class of natanzon potentials with two free parameters is studied within the context of supersymmetric quantum mechanics. the hierarchy of hamiltonians and a general form for the superpotential is presented. the first members of the superfamily are explicitly evaluated.

Abstract:
The restricted class of Natanzon potentials with two free parameters is studied within the context of Supersymmetric Quantum Mechanics. The hierarchy of Hamiltonians and a general form for the superpotential is presented. The first members of the superfamily are explicitly evaluated.

Abstract:
The restricted class of Natanzon potentials with two free parameters is studied within the context of Supersymmetric Quantum Mechanics. The hierarchy of Hamiltonians is indicated, where the first members of the superfamily are explicitly evaluated and a general form for the superpotential is proposed.

Abstract:
An algebraic method of constructing the confluent Natanzon potentials endowed with position-dependent mass is presented. This is possible by identifying the scaling resolvent operator (Green's function) to nonrelativistic position-dependent mass Schrodinger equation.

Abstract:
The ${\cal PT}$ symmetric version of the generalised Ginocchio potential, a member of the general exactly solvable Natanzon potential class is analysed and its properties are compared with those of ${\cal PT}$ symmetric potentials from the more restricted shape-invariant class. It is found that the ${\cal PT}$ symmetric generalised Ginocchio potential has a number of properties in common with the latter potentials: it can be generated by an imaginary coordinate shift $x\to x+{\rm i}\epsilon$; its states are characterised by the quasi-parity quantum number; the spontaneous breakdown of ${\cal PT}$ symmetry occurs at the same time for all the energy levels; and it has two supersymmetric partners which cease to be ${\cal PT}$ symmetric when the ${\cal PT}$ symmetry of the original potential is spontaneously broken.

Abstract:
A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this article we describe the Casson invariant of A'Campo's knots as a J^{+/-}-type invariant of the immersed curves. Thus we get an integral generalization of the Gusein-Zade--Natanzon theorem which says that the Arf invariant of a singularity is equal to J^{-}/2(mod 2) of the corresponding immersed curve. It turns out that this invariant is a second order invariant of the mixed J^{+}- and J^{-}-types.