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N-fold Supersymmetry and Quasi-Solvability, pp. 621-679 $100.00
Authors:  (Toshiaki Tanaka, Department of Physics, National Cheng Kung University, Tainan, Taiwan, National Center for Theoretical Sciences, Taiwan)
In this chapter, we reveal underlying structures of quantum mechanical systems which admit exact solutions in view of N-fold supersymmetry and quasi-solvability. First, we introduce the definition of solvability in various degrees for linear differential
operators of a single variable. We then define N-fold supersymmetry and show its realization in one-body quantum mechanical systems. General aspects and consequences of the symmetry are reviewed. We present a systematic algorithm for the construction of N-fold supersymmetric systems and apply it to monomial spaces. In accordance with the existence of three inequivalent monomial spaces, there are three different types of N-fold supersymmetry which we shall call type A, B, and C, respectively. Type A N-fold supersymmetric models are essentially equivalent to the one-body quasi-solvable models constructed from the enveloping algebra of sl(2). We discuss underlying GL(2) symmetry of type A systems and utilize it to classify completely all the possible type A Hamiltonians. Type A N-fold superalgebra is associated with a set of so-called generalized Bender–Dunne polynomials generated by a four-term recursion relation. In addition, as a consequence of the GL(2) symmetry in type A, the polynomials of the critical degrees generate absolute algebraic invariants. Type B N-fold supersymmetry turns out to be peculiar in the sense that there is no underlying symmetry which prevents us from classifying the models and constructing polynomial systems of Bender–Dunne type. Type C N-fold supersymmetric systems preserve two linearly independent monomial spaces of type A. As a consequence,
type C models have various properties similar to those of type A models. The complete classification and the construction of associated polynomial systems for type C models are presented. Finally, we briefly refer to some further developments. 

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N-fold Supersymmetry and Quasi-Solvability, pp. 621-679