{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:57:00Z","timestamp":1760230620899,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:00:00Z","timestamp":1659484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Supersymmetric quantum mechanics has wide applications in physics. However, there are few potentials that can be solved exactly by supersymmetric quantum mechanics methods, so it is undoubtedly of great significance to find more potentials that can be solved exactly. This paper studies the supersymmetric quantum mechanics problems of the Schr\u00f6dinger equation with a new kind of generalized trigonometric tangent superpotential: Atannpx+Btanmpx. We will elaborate on this new potential in the following aspects. Firstly, the shape invariant relation of partner potential is generated by the generalized trigonometric tangent superpotential. We find three shape invariance forms that satisfy the additive condition. Secondly, the eigenvalues and the eigenwave functions of the potential are studied separately in these three cases. Thirdly, the potential algebra of such a superpotential is discussed, and the discussions are explored from two aspects: one parameter\u2019s and two parameters\u2019 potential algebra. Through the potential algebra, the eigenvalue spectrums are given separately which are consistent with those mentioned earlier. Finally, we summarize the paper and give an outlook on the two-parameter shape-invariant potential.<\/jats:p>","DOI":"10.3390\/sym14081593","type":"journal-article","created":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T23:33:01Z","timestamp":1659569581000},"page":"1593","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A New Solvable Generalized Trigonometric Tangent Potential Based on SUSYQM"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7992-2424","authenticated-orcid":false,"given":"Lulin","family":"Xiong","sequence":"first","affiliation":[{"name":"College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 401331, China"}]},{"given":"Xin","family":"Tan","sequence":"additional","affiliation":[{"name":"Chongqing Fengjie Middle School, Chongqing 404699, China"}]},{"given":"Shikun","family":"Zhong","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 401331, China"}]},{"given":"Wei","family":"Cheng","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 401331, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3698-9311","authenticated-orcid":false,"given":"Guang","family":"Luo","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 401331, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1070\/PU1985v028n08ABEH003882","article-title":"Supersymmetry in quantum mechanics","volume":"28","author":"Gendenshtin","year":"1985","journal-title":"Sov. Phys. Uspekhi"},{"key":"ref_2","unstructured":"Junker, G. (2012). Supersymmetric Methods in Quantum and Statistical Physics, Springer Science & Business Media."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gangopadhyaya, A., Mallow, J.V., and Rasinariu, C. (2017). Supersymmetric Quantum Mechanics: An Introduction, World Scientific Publishing Company.","DOI":"10.1142\/10475"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1119\/1.1538576","article-title":"Supersymmetry in Quantum Mechanics","volume":"71","author":"Cooper","year":"2003","journal-title":"Am. J. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/0370-1573(94)00080-M","article-title":"Supersymmetry and quantum mechanics","volume":"251","author":"Cooper","year":"1995","journal-title":"Phys. Rep."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1063\/1.529954","article-title":"On supersymmetries in nonrelativistic quantum mechanics","volume":"33","author":"Beckers","year":"1992","journal-title":"J. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1497","DOI":"10.1088\/0305-4470\/9\/9\/010","article-title":"Supersymmetry and spin systems","volume":"9","author":"Nicolai","year":"1976","journal-title":"J. Phys. Math. Gen."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/0550-3213(81)90006-7","article-title":"Dynamical breaking of supersymmetry","volume":"188","author":"Witten","year":"1981","journal-title":"Nucl. Phys. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1142\/S0217751X90000647","article-title":"Supersymmetry in quantum mechanics","volume":"5","author":"Lahiri","year":"1990","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1142\/S0217751X97000232","article-title":"SUSUSY quantum mechanics","volume":"12","year":"1997","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bagchi, B.K. (2000). Supersymmetry in Quantum and Classical Mechanics, CRC Press.","DOI":"10.1201\/9780367801670"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ushveridze, A.G. (2017). Quasi-Exactly Solvable Models in Quantum Mechanics, CRC Press.","DOI":"10.1201\/9780203741450"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"126722","DOI":"10.1016\/j.physleta.2020.126722","article-title":"Exactness of SWKB for shape invariant potentials","volume":"384","author":"Gangopadhyaya","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.physletb.2011.06.075","article-title":"Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials","volume":"702","author":"Odake","year":"2011","journal-title":"Phys. Lett. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2180","DOI":"10.1016\/j.physleta.2015.06.058","article-title":"Generation of a novel exactly solvable potential","volume":"379","author":"Bougie","year":"2015","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Sukumar, C. (2004, January 4\u20135). Supersymmetric quantum mechanics and its applications. Proceedings of the AIP Conference Proceedings, Sacramento, CA, USA.","DOI":"10.1063\/1.1853202"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Dong, S.H. (2007). Factorization Method in Quantum Mechanics, Springer Science & Business Media.","DOI":"10.1007\/978-1-4020-5796-0"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/0022-247X(91)90267-4","article-title":"Exactly solvable supersymmetric quantum mechanics","volume":"158","author":"Arai","year":"1991","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1119\/1.15697","article-title":"Supersymmetry, shape invariance, and exactly solvable potentials","volume":"56","author":"Dutt","year":"1988","journal-title":"Am. J. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/0003-4916(84)90084-8","article-title":"A class of exactly solvable potentials. I. One-dimensional Schr\u00f6dinger equation","volume":"152","author":"Ginocchio","year":"1984","journal-title":"Ann. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/0550-3213(84)90321-3","article-title":"Supersymmetric quantum mechanics in one, two and three dimensions","volume":"244","author":"Khare","year":"1984","journal-title":"Nucl. Phys. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3707","DOI":"10.1088\/0305-4470\/22\/17\/035","article-title":"Supersymmetry, operator transformations and exactly solvable potentials","volume":"22","author":"Cooper","year":"1989","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1006\/aphy.1998.5856","article-title":"Conditionally exactly solvable potentials: A supersymmetric construction method","volume":"270","author":"Junker","year":"1998","journal-title":"Ann. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"103034","DOI":"10.1016\/j.rinp.2020.103034","article-title":"Exactly solvable new classes of potentials with finite discrete energies","volume":"17","author":"Benbourenane","year":"2020","journal-title":"Results Phys."},{"key":"ref_25","unstructured":"Benbourenane, J., Benbourenane, M., and Eleuch, H. (2021). Solvable Schrodinger Equations of Shape Invariant Potentials Having Superpotential W (x, A, B) = Atanh (px) + Btanh (6px). arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"105369","DOI":"10.1016\/j.rinp.2022.105369","article-title":"Shape invariance of solvable Schr\u00f6dinger equations with a generalized hyperbolic tangent superpotential","volume":"35","author":"Zhong","year":"2022","journal-title":"Results Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2458","DOI":"10.1103\/PhysRevD.36.2458","article-title":"Relationship between supersymmetry and solvable potentials","volume":"36","author":"Cooper","year":"1987","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"L901","DOI":"10.1088\/0305-4470\/26\/18\/003","article-title":"New shape-invariant potentials in supersymmetric quantum mechanics","volume":"26","author":"Khare","year":"1993","journal-title":"J. Phys. Math. Gen."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1007\/BF02065985","article-title":"Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics","volume":"104","author":"Bagrov","year":"1995","journal-title":"Theor. Math. Phys."},{"key":"ref_30","first-page":"7308","article-title":"Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schr\u00f6dinger equation","volume":"218","author":"Tian","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Hall, B.C. (2013). Lie groups, Lie algebras, and representations. Quantum Theory for Mathematicians, Springer.","DOI":"10.1007\/978-1-4614-7116-5_16"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3809","DOI":"10.1088\/0305-4470\/27\/11\/031","article-title":"Solvable potentials associated with su (1, 1) algebras: A systematic study","volume":"27","year":"1994","journal-title":"J. Phys. Math. Gen."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Zaitsev, V.F., and Polyanin, A.D. (2002). Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press.","DOI":"10.1201\/9781420035339"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"446","DOI":"10.14311\/AP.2017.57.0446","article-title":"Algebraic Description of Shape Invariance Revisited","volume":"57","author":"Ohya","year":"2017","journal-title":"Acta Polytech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"111","DOI":"10.2478\/s11534-007-0001-1","article-title":"Exactly solvable problems of quantum mechanics and their spectrum generating algebras: A review","volume":"5","author":"Rasinariu","year":"2007","journal-title":"Open Phys."},{"key":"ref_36","first-page":"516","article-title":"Faddeev-Skyrme Model and Rational Maps","volume":"40","author":"Su","year":"2002","journal-title":"Chin. J. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0065-3276(08)60613-9","article-title":"Lie algebraic methods and their applications to simple quantum systems","volume":"Volume 19","author":"Adams","year":"1988","journal-title":"Advances in Quantum Chemistry"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1593\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:01:41Z","timestamp":1760140901000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1593"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,3]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081593"],"URL":"https:\/\/doi.org\/10.3390\/sym14081593","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,3]]}}}