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Nfold Supersymmetry and QuasiSolvability, pp. 621679 
$100.00 

Authors: (Toshiaki Tanaka, Department of Physics, National Cheng Kung University, Tainan, Taiwan, National Center for Theoretical Sciences, Taiwan)

Abstract: In this chapter, we reveal underlying structures of quantum mechanical systems which admit exact solutions in view of Nfold supersymmetry and quasisolvability. First, we introduce the definition of solvability in various degrees for linear differential operators of a single variable. We then define Nfold supersymmetry and show its realization in onebody quantum mechanical systems. General aspects and consequences of the symmetry are reviewed. We present a systematic algorithm for the construction of Nfold supersymmetric systems and apply it to monomial spaces. In accordance with the existence of three inequivalent monomial spaces, there are three different types of Nfold supersymmetry which we shall call type A, B, and C, respectively. Type A Nfold supersymmetric models are essentially equivalent to the onebody quasisolvable 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 Nfold superalgebra is associated with a set of socalled generalized Bender–Dunne polynomials generated by a fourterm 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 Nfold 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 Nfold 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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