The present article discusses the connection between exactly-solvable Schrödinger equations and the Liouville transformation. This transformation yields a large class of exactly-solvable potentials, which includes the relatively recently introduced potentials of Natanzon and Ginocchio. As well, this class is shown to contain two new families of exactly solvable potentials. AMS classification sc...

In this paper, all semigroups are assumed to be finite unless otherwise stated. If H is a pseudovariety of groups (that is, a class of groups closed under formation of finite direct products, subgroups and quotients groups), then it is natural to define a group to be Hsolvable if it has a subnormal series each of whose quotients belongs to H. For instance a group is solvable in the classical se...

An application of solvable structures to the reduction ODEs with a lack local symmetries is given. Solvable considered here are all defined in nonlocal extension, or covering space, given ODE. Examples procedure provided.

Using supersymmetric quantum mechanics we develop a new method for constructing quasi-exactly solvable (QES) potentials with two known eigenstates. This method is extended for constructing conditionally-exactly solvable potentials (CES). The considered QES potentials at certain values of parameters become exactly solvable and can be treated as CES ones.

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is induced from a character φ of U with solvable factor group U/ker(φ). Using the classification of finite simple groups, we prove that these groups are also s...

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulat...