# New implicitly solvable potential produced by second order shape invariance

## Abstract

The procedure proposed recently by Bougie et al. (2010) to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions. - Highlights: • New potential with 2nd order irreducible shape invariance was constructed. • The connection conditions at the singularity of potential were obtained. • The explicit expressions for all wave functions were derived. • The implicit equation for the energy spectrum was obtained.

- Authors:

- INFN, Via Irnerio 46, 40126 Bologna (Italy)
- Saint Petersburg State University, 198504 Saint-Petersburg (Russian Federation)
- Akaki Tsereteli State University, 4600 Kutaisi, Georgia (United States)
- (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22451175

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 356; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ENERGY SPECTRA; HYPERGEOMETRIC FUNCTIONS; ONE-DIMENSIONAL CALCULATIONS; QUANTUM MECHANICS; SCHROEDINGER EQUATION; SUPERSYMMETRY; WAVE FUNCTIONS

### Citation Formats

```
Cannata, F., E-mail: cannata@bo.infn.it, Ioffe, M.V., E-mail: m.ioffe@spbu.ru, Kolevatova, E.V., E-mail: e.v.krup@yandex.ru, Nishnianidze, D.N., E-mail: cutaisi@yahoo.com, and Saint Petersburg State University, 198504 Saint-Petersburg.
```*New implicitly solvable potential produced by second order shape invariance*. United States: N. p., 2015.
Web. doi:10.1016/J.AOP.2015.03.020.

```
Cannata, F., E-mail: cannata@bo.infn.it, Ioffe, M.V., E-mail: m.ioffe@spbu.ru, Kolevatova, E.V., E-mail: e.v.krup@yandex.ru, Nishnianidze, D.N., E-mail: cutaisi@yahoo.com, & Saint Petersburg State University, 198504 Saint-Petersburg.
```*New implicitly solvable potential produced by second order shape invariance*. United States. doi:10.1016/J.AOP.2015.03.020.

```
Cannata, F., E-mail: cannata@bo.infn.it, Ioffe, M.V., E-mail: m.ioffe@spbu.ru, Kolevatova, E.V., E-mail: e.v.krup@yandex.ru, Nishnianidze, D.N., E-mail: cutaisi@yahoo.com, and Saint Petersburg State University, 198504 Saint-Petersburg. Fri .
"New implicitly solvable potential produced by second order shape invariance". United States. doi:10.1016/J.AOP.2015.03.020.
```

```
@article{osti_22451175,
```

title = {New implicitly solvable potential produced by second order shape invariance},

author = {Cannata, F., E-mail: cannata@bo.infn.it and Ioffe, M.V., E-mail: m.ioffe@spbu.ru and Kolevatova, E.V., E-mail: e.v.krup@yandex.ru and Nishnianidze, D.N., E-mail: cutaisi@yahoo.com and Saint Petersburg State University, 198504 Saint-Petersburg},

abstractNote = {The procedure proposed recently by Bougie et al. (2010) to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions. - Highlights: • New potential with 2nd order irreducible shape invariance was constructed. • The connection conditions at the singularity of potential were obtained. • The explicit expressions for all wave functions were derived. • The implicit equation for the energy spectrum was obtained.},

doi = {10.1016/J.AOP.2015.03.020},

journal = {Annals of Physics},

issn = {0003-4916},

number = ,

volume = 356,

place = {United States},

year = {2015},

month = {5}

}