Supersymmetric Construction of three-dimensional isospectral systems

The concept of supersymmetry arose in the study of elementary particle physics. The term “supersymmetry” was originally used in reference to the symmetry of bosons and fermions, which behave according to different statistical laws. Supersymmetry is widely regarded as a necessary concept for unifying all the elementary forces. The concept of supersymmetry was originally formulated in the framework of field theory. However, it is not easy to analyze the results from supersymmetry (that is, whether or not a model exhibits the breaking of supersymmetry) in field theory. For this reason, supersymmetric quantum mechanics was proposed.1) Although the original purpose of formulating supersymmetric quantum mechanics was to help obtain an understanding of supersymmetric quantum field theory, once this formulation of quantum mechanics was made, many interesting applications were found. Supersymmetric quantum mechanics is actually a reformulation of Darboux-Crum transformations and the factorization method.2) Darboux-Crum transformations are techniques of generating a new Hamiltonian and its eigenfunctions from an existing Hamiltonian and its known eigenfunctions. The new spectrum is almost identical to the original.3), 4) The factorization method is a technique of solving Schrödinger equation by factorizing the Hamiltonian, which is the second order differential operator with respect to the spatial variables in position space, into the first order operators.5) In quantum mechanics, supersymmetry refers to a relationship of isospectrality of two systems. More precisely, supersymmetric quantum mechanics is a formulation of quantum mechanics involving pairs of Hamiltonians (so called “partner Hamiltonians”) whose eigenvalues are identical (except for a possible ground state). The existence of such (non-trivial) pairs is possible because there is not a one-to-one correspondence between a spectrum and a potential. Indeed, for a given spectra there will be many potentials from which it can be derived. Supersymmetric quantum mechanics provides a method to construct pairs of isospectral systems.6) An operator called intertwiner plays a central role in constructing isospectral