Position dependent mass Scarf Hamiltonians generated via the Riccati equation. (English) Zbl 1426.81037
Summary: The construction of position dependent mass Scarf Hamiltonians of the trigonometric as well as the hyperbolic types is addressed by means of the factorization method and the Riccati equation. These Hamiltonians are shown to be independent of the ordering parameter of the kinetic term. Additionally, new families of Hamiltonians with the Scarf spectrum are also determined by supersymmetry. Some examples for masses with and without singularities are considered to illustrate our results.
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81Q60 | Supersymmetry and quantum mechanics |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 81Q80 | Special quantum systems, such as solvable systems |
References:
| [1] | InfeldL, HullTE. The factorization method. Rev Mod Phys. 1951;23:21‐68. · Zbl 0043.38602 |
| [2] | AndrianovAA, BorisovNV, IoffeMV. The factorization method and quantum‐systems with equivalent energy‐spectra Phys. Lett A. 1984;105:19. |
| [3] | MielnikB, Rosas‐OrtizO. Factorization: little or great algorithm. J Phys A: Math Gen. 2004;37:10007‐10035. · Zbl 1064.81025 |
| [4] | KuruŞ, NegroJ. Dynamical algebras for Pöschl‐Teller Hamiltonians hierarchies. Ann Phys. 2009;324:2548‐2560. · Zbl 1179.81086 |
| [5] | MielnikB. Factorization method and new potentials with the oscillator spectrum. J Math Phys. 1984;25:387‐389. |
| [6] | AndrianovAA, BorisovNV, IoffeMV, EidesMI. Supersymmetric mechanics: a new look at the equivalence of quantum systems. Theor Math Phys. 1984;61:965‐972. |
| [7] | SukumarCV. Supersymmetry, factorisation of the Schrödinger equation and a Hamiltonian hierarchy. J Phys A: Math Gen. 1985;18:L57‐L61. · Zbl 0586.35020 |
| [8] | DeR, DuttR, SukhatmeU. Mapping of shape invariant potentials under point canonical transformations. J Phys A: Math Gen. 1992;25:L843‐L850. |
| [9] | CooperF, KhareA, SukhatmeU. Supersymmetry in quantum mechanics. Phys Rep. 1995;251:267‐385. |
| [10] | BagchiBK. Supersymmetry in Quantum and Classical Mechanics. London Boca Raton FL: Chapman and Hall CRC Press; 2000. |
| [11] | WannierGH. The structure of electronic excitation levels in insulating crystals. Phys Rev. 1937;52:191‐197. · Zbl 0017.23601 |
| [12] | vonRoosO. Position‐dependent effective masses semiconductor theory. Phys Rev B. 1983;27:7547. |
| [13] | KhordadR. Hydrogenic donor impurity in a cubic quantum dot: effect of position‐dependent effective mass. Europhys J B. 2012;85:114. |
| [14] | PanahiH, GolshahiS, DoostdarM. Influence of position dependent effective mass on donor binding energy in square and V‐shaped quantum wells in the presence of a magnetic field. Phys B. 2013;418:47‐51. |
| [15] | CariñenaJF, RañadaMF, SantanderM. Curvature‐dependent formalism, Schrödinger equation and energy levels for the harmonic oscillator on three‐dimensional spherical and hyperbolic spaces. J Phys A: Math Theor. 2012;45(265303). · Zbl 1250.81029 |
| [16] | CariñenaJF, RañadaMF, SantanderM. The quantum free particle on spherical and hyperbolic spaces: a curvature dependent approach. II. J Math Phys. 2012;53(102109). · Zbl 1278.81092 |
| [17] | RichstoneDO, PotterMD. Galactic mass loss: a mild evolutionary correction to the angular size test. Astrophys J. 1982;254:451‐455. |
| [18] | MilanovićV, IkonićZ. Generation of isospectral combinations of the potential and the effective‐mass variations by supersymmetric quantum mechanics. J Phys A: Math Gen. 1999;32:7001‐7015. · Zbl 0966.81022 |
| [19] | PlastinoAR, RigoA, CasasM, GraciasF, PlastinoA. Supersymmetric approach to quantum systems with position‐dependent effective mass. Phys Rev A. 1999;60:4318. |
| [20] | GönülB, GönülB, TutcuD, ÖzerO. Supersymmetric approach to exactly solvable systems with position‐dependent effective mass. Mod Phys Lett A. 2002;17:2057‐2066. · Zbl 1083.81525 |
| [21] | KoçR, KocaM. A systematic study on the exact solution of the position dependent mass Schrödinger equation. J Phys A Math Gen. 2003;36:8105‐8112. · Zbl 1048.81019 |
| [22] | RoyB, RoyP. Effective mass Schrödinger equation and nonlinear algebras. Phys Lett A. 2005;340:70. · Zbl 1145.81362 |
| [23] | QuesneC. First‐order intertwining operators and position‐dependent mass Schrödinger equations in d dimensions. Ann Phys. 2006;321:1221. · Zbl 1092.81016 |
| [24] | Cruz y CruzS, NegroJ, NietoLM. Classical and quantum position‐dependent mass harmonic oscillators. Phys Lett A. 2007;369:400‐406. · Zbl 1209.81095 |
| [25] | BagchiB, TanakaT. A generalized non‐Hermitian oscillator Hamiltonian, N‐fold supersymmetry and position‐dependent mass models. Phys Lett A. 2008;372:5390‐5393. · Zbl 1223.81089 |
| [26] | QuesneC. Point canonical transformation versus deformed shape invariance for position‐dependent mass Schrödinger equations. SIGMA. 2009;5:046. · Zbl 1160.81377 |
| [27] | Cruz y CruzS, Rosas‐OrtizO. Position‐dependent mass oscillators and coherent states. J Phys A: Math Theor. 2009;42(185205). · Zbl 1162.81388 |
| [28] | Cruz y CruzS. Factorization Method and the Position‐dependent Mass Problem, Trends in Mathematics (Germany Springer Basel) 229; 2013. · Zbl 1264.81174 |
| [29] | Santiago‐CruzC. Isospectral trigonometric Pöschl‐Teller potentials with position dependent mass generated by supersymmetry. J Phys Conf Ser. 2016;698:012028. |
| [30] | YahiaouiS‐H, BentaibaM. Isospectral Hamiltonian for position‐dependent mass for an arbitrary quantum system and coherent states. J Math Phys. 2017;58:063507. · Zbl 1366.81202 |
| [31] | AmirN, IqbalS. Ladder operators and associated algebra for position‐dependent effective mass systems. EPL. 2015;111:20005. |
| [32] | SuzkoAA, TralleI. Reconstruction of quantum well potentials via the intertwining operator technique. Acta Phys Pol B. 2008;39:545. · Zbl 1371.81296 |
| [33] | Rosas‐OrtizO, CastañosO, DieterS. New supersymmetry‐generated complex potentials with real spectra. J Phys A: Math Theor. 2015;48:445302. · Zbl 1327.81216 |
| [34] | Lévy‐LeblondJM. Position‐dependent effective mass and Galilean invariance. Phys Rev A. 1995;52:1845‐1849. |
| [35] | GangulyA, KuruŞ, NegroJ, NietoLM. A study of the bound states for square potential wells with position‐dependent mass. Phys Lett A. 2006;360:228‐233. · Zbl 1236.81181 |
| [36] | MustafaO, MazharimousaviSH. Ordering ambiguity revisited via position‐dependent mass pseudo‐momentum operators. Int J Theor Phys. 2007;46:1786‐1796. · Zbl 1128.81009 |
| [37] | AbramowitzM, StegunI. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Washington DC: Dover; 1970. |
| [38] | Cruz y CruzS, KuruŞ, NegroJ. Classical motion and coherent states for Pöschl‐Teller potentials. Phys Lett A. 2008;372:1391‐1405. · Zbl 1217.81077 |
| [39] | Cruz y CruzS, NegroJ, NietoLM. On the position dependent mass harmonic oscillatos. J Phys Conf Ser. 2008;128(012053). |
| [40] | PerelomovA. Generalized Coherent States and Their Applications. Berlin: Springer‐Verlag; 1986. · Zbl 0605.22013 |
| [41] | Rosas‐OrtizO, Cruz y CruzS, EnriquezM. SU(1,1) and SU(2) approaches to the radial oscillator: generalized coherent states and squeezing of variances. Ann Phys. 2016;373:346‐373. · Zbl 1380.81161 |
| [42] | Cruz y CruzS, GressZ. Group approach to the paraxial propagation of Hermite‐Gaussian modes in a parabolic medium. Ann Phys. 2017;383:257‐277. · Zbl 1373.81239 |
| [43] | PerstchF. Exact solution of the Schrodinger equation for a potential well with a barrier and other potentials. J Phys A: Math Gen. 1990;23:4145. · Zbl 0719.35077 |
| [44] | ZhangX‐C, LiuQ‐W, JiaC‐S, WangL‐Z. Bound states of the Dirac equation with vector and scalar Scarf‐type potentials. Phys Lett A. 2005;340:59. · Zbl 1145.81363 |
| [45] | BagchiB, QuesneC. sl \((2, \mathbb{C} )\) as a complex Lie algebra and the associated non‐Hermitian Hamiltonians with real eigenvalues. Phys Lett A. 2000:285. · Zbl 1050.81546 |
| [46] | BagchiB, QuesneC. Non‐hermitian Hamiltonians with real and complex eigenvalues in a Lie‐algebraic framework. Phys Lett A. 2002;300:18. · Zbl 0997.81036 |
| [47] | BagchiB, QuesneC. An update on the \(\mathcal{P} T\)‐symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally extended supersymmetric partners. J Phys A: Math Theor. 2010;43:305301. · Zbl 1194.81083 |
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