Supersymmetric Quantum Mechanics of Hypergeometric-like Differential
Operators
- URL: http://arxiv.org/abs/2307.15948v2
- Date: Sat, 30 Sep 2023 04:28:43 GMT
- Title: Supersymmetric Quantum Mechanics of Hypergeometric-like Differential
Operators
- Authors: Tianchun Zhou
- Abstract summary: Systematic iterative algorithms of supersymmetric quantum mechanics (SUSYQM) type for solving the eigenequation of principal hypergeometric-like differential operator (HLDO)
These algorithms provide a simple SUSYQM answer to the question regarding why there exist simultaneously a series of principal as well as associated eigenfunctions for the same HLDO.
Due to their relatively high efficiency, algebraic elementariness and logical independence, the iterative SUSYQM algorithms developed in this paper could become the hopefuls for supplanting some traditional methods for solving the eigenvalue problems of principal HLDOs and their associated cousins.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Systematic iterative algorithms of supersymmetric quantum mechanics (SUSYQM)
type for solving the eigenequation of principal hypergeometric-like
differential operator (HLDO) and for generating the eigenequation of associated
HLDO itself as well its solutions are developed, without any input from
traditional methods. These are initiated by devising two types of active
supersymmetrization transformations and momentum operator maps, which work to
transform the same eigenequation of HLDO in its two trivial asymmetric
factorizations into two distinct supersymmetrically factorized Schr\"odinger
equations. The rest iteration flows are completely controlled by repeatedly
performing intertwining action and incorporating some generalized commutator
relations to renormalize the superpartner equation of the eigenequation of
present level into that of next level. These algorithms therefore provide a
simple SUSYQM answer to the question regarding why there exist simultaneously a
series of principal as well as associated eigenfunctions for the same HLDO,
which boils down to two basic facts: two distinct types of quantum momentum
kinetic energy operators and superpotentials are rooted in this operator; each
initial superpotential can proliferate into a hierarchy of descendant ones in a
shape-invariant fashion. The two active supersymmetrizations establish the
isomorphisms between the nonstandard and standard coordinate representations of
the SUSYQM algorithm either for principal HLDO or for its associated one, so
these algorithms can be constructed in either coordinate representation with
equal efficiency. Due to their relatively high efficiency, algebraic
elementariness and logical independence, the iterative SUSYQM algorithms
developed in this paper could become the hopefuls for supplanting some
traditional methods for solving the eigenvalue problems of principal HLDOs and
their associated cousins.
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