{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T11:39:49Z","timestamp":1774611589176,"version":"3.50.1"},"reference-count":101,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,5]],"date-time":"2022-06-05T00:00:00Z","timestamp":1654387200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Universe"],"abstract":"<jats:p>In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of N=2 supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schr\u00f6dinger\u2013Wheeler\u2013DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersymmetric partner Hamiltonians, H1 and H2. Moreover, we point out how the shape invariance property can be employed to bring a relation among several factor orderings choices for our Schr\u00f6dinger\u2013Wheeler\u2013DeWitt equation. The ground state is retrieved, and the excited states easily written. Finally, the Hamiltonians, H1 and H2, are explicitly presented within a N=2 supersymmetric quantum mechanics framework.<\/jats:p>","DOI":"10.3390\/universe8060316","type":"journal-article","created":{"date-parts":[[2022,6,5]],"date-time":"2022-06-05T10:47:11Z","timestamp":1654426031000},"page":"316","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Shape Invariant Potentials in Supersymmetric Quantum Cosmology"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4854-2960","authenticated-orcid":false,"given":"Shahram","family":"Jalalzadeh","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal de Pernambuco, Recife 52171-900, PE, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3455-1954","authenticated-orcid":false,"given":"Seyed Meraj M.","family":"Rasouli","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, Rua Marqu\u00eas d\u2019Avila e Bolama, 6200-001 Covilh\u00e3, Portugal"},{"name":"Department of Physics, Qazvin Branch, Islamic Azad University, Qazvin 341851416, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7170-8952","authenticated-orcid":false,"given":"Paulo","family":"Moniz","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, Rua Marqu\u00eas d\u2019Avila e Bolama, 6200-001 Covilh\u00e3, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,5]]},"reference":[{"key":"ref_1","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"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BFb0105329","article-title":"Shape invariance and its connection to potential algebra","volume":"Volume 502","author":"Aratyn","year":"1998","journal-title":"Supersymmetry and Integrable Models"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4188","DOI":"10.1103\/PhysRevA.57.4188","article-title":"Algebraic approach to shape invariance","volume":"57","author":"Balantekin","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/S0375-9601(01)00266-3","article-title":"Broken supersymmetric shape invariant systems and their potential algebras","volume":"283","author":"Gangopadhyaya","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.physleta.2005.12.024","article-title":"Exactly solvable potentials of the Klein Gordon equation with the supersymmetry method","volume":"352","author":"Chen","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1008","DOI":"10.1119\/1.17698","article-title":"Supersymmetry, shape invariance and exactly solvable noncentral potentials","volume":"62","author":"Khare","year":"1994","journal-title":"Am. J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Quesne, C. (2020). Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of \u210f. Symmetry, 12.","DOI":"10.3390\/sym12111853"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"435304","DOI":"10.1088\/1751-8113\/47\/43\/435304","article-title":"A relation between Z3-graded symmetry and shape invariant supersymmetric systems","volume":"47","author":"Oikonomou","year":"2014","journal-title":"J. Phys. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.physletb.2012.07.047","article-title":"Supersymmetry, shape invariance and the Legendre equations","volume":"715","author":"Bazeia","year":"2012","journal-title":"Phys. Lett. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1088\/0305-4470\/22\/8\/019","article-title":"Remarks on the equivalence between the shape-invariance condition and the factorisation condition","volume":"22","author":"Stahlhofen","year":"1989","journal-title":"J. Phys. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/S0375-9601(97)00161-8","article-title":"Supersymmetry and shape invariance in differential equations of mathematical physics","volume":"230","author":"Jafarizadeh","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.5506\/APhysPolB.43.1795","article-title":"Supersymmetry Approach and Shape Invariance for Pseudo-harmonic Potential","volume":"43","author":"Amani","year":"2012","journal-title":"Acta Phys. Pol. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1140\/epjc\/s10052-016-4373-5","article-title":"Classical Universe emerging from quantum cosmology without horizon and flatness problems","volume":"76","author":"Fathi","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1140\/epjc\/s10052-014-3241-4","article-title":"On the relation between boundary proposals and hidden symmetries of the extended pre-big bang quantum cosmology","volume":"75","author":"Jalalzadeh","year":"2015","journal-title":"Eur. Phys. J. C"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"023526","DOI":"10.1103\/PhysRevD.92.023526","article-title":"Quantum cosmological intertwining: Factor ordering and boundary conditions from hidden symmetries","volume":"92","author":"Rostami","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"083504","DOI":"10.1103\/PhysRevD.89.083504","article-title":"Dirac observables and boundary proposals in quantum cosmology","volume":"89","author":"Jalalzadeh","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1630009","DOI":"10.1142\/S0218271816300093","article-title":"Quantum cosmology: From hidden symmetries towards a new (supersymmetric) perspective","volume":"25","author":"Jalalzadeh","year":"2016","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Moniz, P.V. (2010). Quantum Cosmology\u2014The Supersymmetric Perspective\u2014Vol. 1, Springer. Lecture Notes in Physics.","DOI":"10.1007\/978-3-642-11575-2"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Moniz, P.V. (2010). Quantum Cosmology\u2014The Supersymmetric Perspective\u2014Vol. 2, Springer. Lecture Notes in Physics.","DOI":"10.1007\/978-3-642-11570-7"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"R7071","DOI":"10.1103\/PhysRevD.57.R7071","article-title":"Origin of structure in supersymmetric quantum cosmology","volume":"57","author":"Moniz","year":"1998","journal-title":"Phys. Rev. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"045006","DOI":"10.1103\/PhysRevD.72.045006","article-title":"Semiclassical approximation to supersymmetric quantum gravity","volume":"72","author":"Kiefer","year":"2005","journal-title":"Phys. Rev. D"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"185002","DOI":"10.1088\/0264-9381\/28\/18\/185002","article-title":"Novel Complete Non-Compact Symmetries for the Wheeler\u2013DeWitt Equation in a Wormhole Scalar Model and Axion-Dilaton String Cosmology","volume":"28","author":"Cordero","year":"2011","journal-title":"Class. Quant. Grav."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1140\/epjp\/s13360-020-00114-3","article-title":"New exact supersymmetric wave functions for a massless scalar field and axion\u2013dilaton string cosmology in a FRWL metric","volume":"135","author":"Cordero","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"D\u00edaz, J.S.G., Reyes, M.A., Mora, C.V., and Pozo, E.C. (2019). Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable Potentials. Panorama of Contemporary Quantum Mechanics-Concepts and Applications, IntechOpen.","DOI":"10.5772\/intechopen.82254"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"L183","DOI":"10.1088\/0305-4470\/38\/11\/L01","article-title":"Shape invariant potentials in SUSY quantum mechanics and periodic orbit theory","volume":"38","author":"Bhaduri","year":"2005","journal-title":"J. Phys. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"275307","DOI":"10.1088\/1751-8113\/44\/27\/275307","article-title":"Method for generating additive shape-invariant potentials from an Euler equation","volume":"44","author":"Bougie","year":"2011","journal-title":"J. Phys. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1238\/Physica.Regular.064a00548","article-title":"Generalized Ladder Operators for Shape-invariant Potentials","volume":"64","author":"Filho","year":"2001","journal-title":"Phys. Scripta"},{"key":"ref_28","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_29","doi-asserted-by":"crossref","first-page":"452","DOI":"10.3390\/sym4030452","article-title":"Supersymmetric Quantum Mechanics and Solvable Models","volume":"4","author":"Bougie","year":"2012","journal-title":"Symmetry"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3467","DOI":"10.1088\/0305-4470\/33\/17\/305","article-title":"Shape-invariant potentials depending on n parameters transformed by translation","volume":"33","author":"Ramos","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4303\/jpm\/P110502","article-title":"Shape invariant potentials in second-order supersymmetric quantum mechanics","volume":"3","author":"Su","year":"2011","journal-title":"J. Phys. Math."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Dong, S.H. (2007). Factorization Method in Quantum Mechanics, Springer.","DOI":"10.1007\/978-1-4020-5796-0"},{"key":"ref_33","unstructured":"Nasuda, Y., and Sawado, N. (2021). SWKB Quantization Condition for Conditionally Exactly Solvable Systems and the Residual Corrections. arXiv."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gangopadhyaya, A., Mallow, J.V., and Rasinariu, C. (2017). Supersymmetric Quantum Mechanics: An Introduction, World Scientific.","DOI":"10.1142\/10475"},{"key":"ref_35","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_36","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_37","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. Rept."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/0370-2693(84)90339-3","article-title":"Relationship Between Supersymmetry and the Inverse Method in Quantum Mechanics","volume":"145","author":"Nieto","year":"1984","journal-title":"Phys. Lett. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2267","DOI":"10.1103\/PhysRevD.33.2267","article-title":"Isometric operators, isospectral Hamiltonians, and supersymmetric quantum mechanics","volume":"33","author":"Pursey","year":"1986","journal-title":"Phys. Rev. D"},{"key":"ref_40","first-page":"309","article-title":"On the determination of a differential equation from its spectral function","volume":"15","author":"Levitan","year":"1951","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1103\/PhysRevA.22.1333","article-title":"Changes in potentials due to changes in the point spectrum: Anharmonic oscillators with exact solutions","volume":"22","author":"Abraham","year":"1980","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/BF01609166","article-title":"A sketch of Lie superalgebra theory","volume":"53","author":"Kac","year":"1977","journal-title":"Comm. Math. Phys."},{"key":"ref_43","first-page":"356","article-title":"Derivation of Exact Spectra of the Schrodinger Equation by Means of Supersymmetry","volume":"38","author":"Gendenshtein","year":"1983","journal-title":"JETP Lett."},{"key":"ref_44","first-page":"9","article-title":"A Method of Determining Quantum-Mechanical Eigenvalues and Eigenfunctions","volume":"46","year":"1940","journal-title":"Proc. R. Ir. Acad. A Math. Phys. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1103\/RevModPhys.23.21","article-title":"The Factorization Method","volume":"23","author":"Infeld","year":"1951","journal-title":"Rev. Mod. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/BF02856337","article-title":"Supersymmetry in quantum mechanics","volume":"49","author":"Khare","year":"1997","journal-title":"Pramana-J. Phys."},{"key":"ref_47","first-page":"183","article-title":"Further Studies on Solving Eigenvalue Problems by Factorization","volume":"46","year":"1940","journal-title":"Proc. R. Ir. Acad., A Math. phys. sci."},{"key":"ref_48","first-page":"1456","article-title":"On a proposition relative to linear equations","volume":"94","author":"Darboux","year":"1882","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1103\/PhysRevD.33.431","article-title":"New Schr\u00f6dinger equations for old: Inequivalence of the Darboux and Abraham-Moses constructions","volume":"33","author":"Luban","year":"1986","journal-title":"Phys. Rev. D"},{"key":"ref_50","first-page":"065025","article-title":"Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit","volume":"83","author":"Arancibia","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Sonnenschein, J., and Tsulaia, M. (2022). A Note on Shape Invariant Potentials for Discretized Hamiltonians arXiv 2022, arXiv. arXiv.","DOI":"10.1142\/S021773232250153X"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"105022","DOI":"10.1103\/PhysRevD.94.105022","article-title":"Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator","volume":"94","author":"Plyushchay","year":"2016","journal-title":"Phys. Rev. D"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"026017","DOI":"10.1103\/PhysRevD.98.026017","article-title":"Rational deformations of conformal mechanics","volume":"98","author":"Inzunza","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"105009","DOI":"10.1103\/PhysRevD.92.105009","article-title":"Chiral asymmetry in propagation of soliton defects in crystalline backgrounds","volume":"92","author":"Arancibia","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/0375-9601(93)91182-5","article-title":"Shape-invariant potentials and an associated coherent state","volume":"180","author":"Fukui","year":"1993","journal-title":"Phys. Lett. A"},{"key":"ref_56","unstructured":"Jalalzadeh, S., and Moniz, P.V. (2022). Challenging Routes in Quantum Cosmology, World Scientific."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1016\/j.aop.2018.04.014","article-title":"Inflationary universe in deformed phase space scenario","volume":"393","author":"Rasouli","year":"2018","journal-title":"Ann. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"100446","DOI":"10.1016\/j.dark.2019.100446","article-title":"Late time cosmic acceleration in modified S\u00e1ez\u2013Ballester theory","volume":"27","author":"Rasouli","year":"2020","journal-title":"Phys. Dark Univ."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Rasouli, S.M.M. (2022). Noncommutativity, Saez-Ballester Theory and Kinetic Inflation. Universe, 8.","DOI":"10.3390\/universe8030165"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"2895","DOI":"10.1007\/s10714-011-1208-4","article-title":"Horizon problem remediation via deformed phase space","volume":"43","author":"Rasouli","year":"2011","journal-title":"Gen. Rel. Grav."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"083533","DOI":"10.1103\/PhysRevD.90.083533","article-title":"Noncommutative minisuperspace, gravity-driven acceleration, and kinetic inflation","volume":"90","author":"Rasouli","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.aop.2016.09.007","article-title":"Non-singular Brans\u2013Dicke collapse in deformed phase space","volume":"375","author":"Rasouli","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_63","first-page":"19","article-title":"Gravity-Driven Acceleration and Kinetic Inflation in Noncommutative Brans-Dicke Setting","volume":"29","author":"Rasouli","year":"2016","journal-title":"Odessa Astron. Pub."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"100269","DOI":"10.1016\/j.dark.2019.100269","article-title":"Kinetic inflation in deformed phase space Brans\u2013Dicke cosmology","volume":"24","author":"Rasouli","year":"2019","journal-title":"Phys. Dark Univ."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1088\/0264-9381\/10\/2\/018","article-title":"Signature dynamics in general relativity","volume":"10","author":"Dereli","year":"1993","journal-title":"Class. Quant. Grav."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1088\/0264-9381\/10\/8\/005","article-title":"Signature transitions in quantum cosmology","volume":"10","author":"Dereli","year":"1993","journal-title":"Class. Quant. Grav."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"012","DOI":"10.1088\/1126-6708\/2003\/08\/012","article-title":"Multidimensional classical and quantum cosmology: Exact solutions, signature transition and stabilization","volume":"2003","author":"Jalalzadeh","year":"2003","journal-title":"J. High Energy Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s10714-007-0429-z","article-title":"Quantum noncommutative multidimensional cosmology","volume":"39","author":"Khosravi","year":"2007","journal-title":"Gen. Rel. Grav."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physletb.2007.11.098","article-title":"Quantum cosmology with varying speed of light: Canonical approach","volume":"660","author":"Pedram","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"1354","DOI":"10.1007\/s10773-007-9577-x","article-title":"Noncommutativity, generalized uncertainty principle and FRW cosmology","volume":"47","author":"Bina","year":"2008","journal-title":"Int. J. Theor. Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s10773-011-0904-x","article-title":"Quantization of the interior Schwarzschild black hole","volume":"51","author":"Jalalzadeh","year":"2012","journal-title":"Int. J. Theor. Phys."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1088\/1126-6708\/2006\/01\/134","article-title":"Non-commutative multi-dimensional cosmology","volume":"2006","author":"Khosravi","year":"2006","journal-title":"J. High Energy Phys."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.physletb.2010.03.016","article-title":"Late time acceleration in a deformed phase space model of dilaton cosmology","volume":"687","author":"Vakili","year":"2010","journal-title":"Phys. Lett. B"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0375-9601(99)00409-0","article-title":"Large scale\u2014Small scale duality and cosmological constant","volume":"259","author":"Darabi","year":"1999","journal-title":"Phys. Lett. A"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1007\/s10714-006-0315-0","article-title":"A quantum cosmology and discontinuous signature changing classical solutions","volume":"38","author":"Darabi","year":"2006","journal-title":"Gen. Rel. Grav."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"100741","DOI":"10.1016\/j.dark.2020.100741","article-title":"Classical universe arising from quantum cosmology","volume":"30","author":"Jalalzadeh","year":"2020","journal-title":"Phys. Dark Univ."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1002\/andp.200751907-801","article-title":"A new look at the quantum mechanics of the harmonic oscillator","volume":"519","author":"Kastrup","year":"2007","journal-title":"Ann. Phys."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"11002","DOI":"10.1209\/0295-5075\/98\/11002","article-title":"Supersymmetric Oscillator: Novel Symmetries","volume":"98","author":"Kumar","year":"2012","journal-title":"EPL"},{"key":"ref_79","unstructured":"D\u2019Eath, P.D. (1996). Supersymmetric Quantum Cosmology, Cambridge University Press. Cambridge Monographs on Mathematical Physics."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"4321","DOI":"10.1142\/S0217751X96002017","article-title":"Supersymmetric quantum cosmology: Shaken, not stirred","volume":"11","author":"Moniz","year":"1996","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Garc\u00eda-Parrado, A., Mena, F.C., Moura, F., and Vaz, E. (2014). Quantum Cosmology: Meeting SUSY. Progress in Mathematical Relativity, Gravitation and Cosmology, Springer.","DOI":"10.1007\/978-3-642-40157-2"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Garc\u00eda-Compe\u00e1n, H., Obreg\u00f3n, O., and Ram\u00edrez, C. (2021). Topics in Supersymmetric and Noncommutative Quantum Cosmology. Universe, 7.","DOI":"10.3390\/universe7110434"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"1618","DOI":"10.1007\/s10714-013-1618-6","article-title":"Supersymmetric quantum cosmology: A \u2018Socratic\u2019 guide","volume":"46","author":"Moniz","year":"2014","journal-title":"Gen. Rel. Grav."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"235014","DOI":"10.1088\/0264-9381\/32\/23\/235014","article-title":"Supersymmetric quantum matrix cosmology","volume":"32","year":"2015","journal-title":"Class. quant. grav."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1103\/PhysRevD.57.1015","article-title":"Dirac-like formulation of quantum supersymmetric cosmology","volume":"57","author":"Obregon","year":"1998","journal-title":"Phys. Rev. D"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1103\/PhysRevD.49.799","article-title":"Supersymmetric homogeneous quantum cosmologies coupled to a scalar field","volume":"49","author":"Bene","year":"1994","journal-title":"Phys. Rev. D"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"4129","DOI":"10.1103\/PhysRevLett.74.4129","article-title":"Supersymmetric minisuperspace with nonvanishing fermion number","volume":"74","author":"Csordas","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"061701","DOI":"10.1103\/PhysRevD.80.061701","article-title":"Supersymmetric quantum cosmological billiards","volume":"80","author":"Kleinschmidt","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1142\/S0218271898000462","article-title":"Supersymmetric quantum cosmology for Bianchi class A models","volume":"7","author":"Mielke","year":"1998","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"162001","DOI":"10.1088\/0264-9381\/30\/16\/162001","article-title":"Quantum supersymmetric cosmology and its hidden Kac\u2013Moody structure","volume":"30","author":"Damour","year":"2013","journal-title":"Class. Quant. Grav."},{"key":"ref_91","unstructured":"Kiefer, C. (2004). Quantum Gravity, Clarendon."},{"key":"ref_92","unstructured":"Esposito, G. (2009). Quantum Gravity, Quantum Cosmology and Lorentzian Geometries, Springer. Lecture Notes in Physics Monographs."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Calcagni, G. (2017). Classical and Quantum Cosmology, Springer.","DOI":"10.1007\/978-3-319-41127-9"},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Bojowald, M. (2011). Quantum Cosmology: A Fundamental Description of the Universe, Springer. Lecture Notes in Physics.","DOI":"10.1007\/978-1-4419-8276-6"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"6829","DOI":"10.1103\/PhysRevD.51.6829","article-title":"Quantum cosmology of generalized two-dimensional dilaton-gravity models","volume":"51","author":"Lidsey","year":"1995","journal-title":"Phys. Rev. D"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"4823","DOI":"10.1088\/0264-9381\/17\/23\/303","article-title":"Supersymmetric quantization of anisotropic scalar-tensor cosmologies","volume":"17","author":"Lidsey","year":"2000","journal-title":"Class. Quant. Grav."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1103\/PhysRevLett.67.1381","article-title":"Supersymmetric Bianchi type IX cosmology","volume":"67","author":"Graham","year":"1991","journal-title":"Phys. Rev. Lett."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"R5407","DOI":"10.1103\/PhysRevD.52.R5407","article-title":"Scale factor duality and hidden supersymmetry in scalar-tensor cosmology","volume":"52","author":"Lidsey","year":"1995","journal-title":"Phys. Rev. D"},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"2349","DOI":"10.1088\/0264-9381\/13\/9\/002","article-title":"Supersymmetric action for Bianchi type models","volume":"13","author":"Tkach","year":"1996","journal-title":"Class. Quant. Grav."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"R1750","DOI":"10.1103\/PhysRevD.53.R1750","article-title":"Superfield description of the FRW universe","volume":"53","author":"Rosales","year":"1996","journal-title":"Phys. Rev. D"},{"key":"ref_101","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. 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