http://cds.cern.ch/record/262851/files/9405029.pdf

Supersymmetry and quantum mechanics

In contrast to ordinary symmetries, supersymmetry (SUSY) interchanges bosons and fermions. Originally proposed as a symmetry of our universe, it still awaits experimental verification. Here, we theoretically show that SUSY emerges naturally in condensed matter systems known as topological superconductors. We argue that the quantum phase transitions at the boundary of topological superconductors in both two and three dimensions display SUSY when probed at long distances and times. Experimental consequences include exact relations between quantities measured in disparate experiments and, in some cases, exact knowledge of the universal critical exponents. The topological surface states themselves may be interpreted as arising from spontaneously broken SUSY, indicating a deep relation between topological phases and SUSY.

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Emergent Space-Time Supersymmetry at the Boundary of a Topological Phase

The author investigates a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions. It turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions, satisfying linear homogeneous second-order differential equations, can be solutions of the Schrodinger equation with potentials of the form V(x)=W2(x)-W'(x). The author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape-invariant potentials.

http://iopscience.iop.org/article/10.1088/0305-4470/22/6/020/pdf

A search for shape-invariant solvable potentials

New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same charac- teristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are con- structed. In the linear case, they contain ( + 1)th-degree polynomials with = 0,1,2,..., which are shown to beX1-Laguerre orX1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of ( + 2)th-degree Laguerre-type polynomials and a single one of ( + 2)th-degree Jacobi-type polynomials with = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.

/pdf/solvable-rational-potentials-and-exceptional-orthogonal-1uy72zitiw.pdf

Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Polynomial SUSY in quantum mechanics and second derivative Darboux transformations

A pedagogical review on supersymmetry in quantum mechanis is presented which provides a comprehensive coverage of the subject. First, the key ingredients on the quantization of the systems with anticommuting variables are discussed. The supersymmetric Hamiltotian in quantum mechanics is then constructed by emphasizing the role of partner potentials and the superpotentials. We also make explicit the mathematical formulation of the Hamiltonian by considering in detail the N=1 and N=2 supersymmetric (quantum) mechanics. Supersymmetry is then discussed in the context of one-dimensional problems and the importance of the factorization method is highlighted. We treat in detail the technique of constructing a hierarchy of Hamiltonians employing the so-called ‘shape-invariance’ of potentials. To make transparent the relationship between supersymmetry and solvable potentials, we also solve several examples. We then go over to the formulation of supersymmetry in radial problems, paying a special attention to the Co...

Supersymmetry in Quantum Mechanics

A correct procedure for constructing supersymmetry in three dimensions is presented. The degeneracies are found between states of the same l but different n and Z and the previous results on the Coulomb and the three-dimensional isotropic oscillator problems are reestablished. The authors also consider the hydrogen-helium problem and find supersymmetry to hold to a good approximation.

http://iopscience.iop.org/article/10.1088/0305-4470/20/12/030/pdf

Supersymmetry in atomic physics and the radial problem

By transforming the radial equation of the isotropic oscillator to the Coulomb form, the authors show that the energy degeneracy of the three-dimensional oscillator may be understood on account of supersymmetry.

http://iopscience.iop.org/article/10.1088/0305-4470/20/15/052/pdf

Supersymmetry and the three-dimensional isotropic oscillator problem

It is shown that the eigenvalue problem of the L operator for the sine-Gordon equation can be put in a supersymmetric form. We comment on the connection between the conserved quantities of the Korteweg--de Vries and sine-Gordon systems.

Conservation laws, Korteweg-de Vries and sine-Gordon systems, and the role of supersymmetry.

We enquire into the consequences of the nonuniqueness of the factorizability of a quantum mechanical Hamiltonian in one dimension. This leads to a hierarchy of potentials, a particular class of which is endowed with the energy spectrum of the harmonic oscillator.

A. Lahiri

Papers

Supersymmetry in Quantum Mechanics

Supersymmetry in atomic physics and the radial problem

Supersymmetry and the three-dimensional isotropic oscillator problem

Conservation laws, Korteweg-de Vries and sine-Gordon systems, and the role of supersymmetry.

Nonuniqueness of the factorization scheme in quantum mechanics