On the construction of position-dependent mass models with quadratic spectra. (English) Zbl 07914651
Kielanowski, Piotr (ed.) et al., Geometric methods in physics XXXIX. Workshop, Białystok, Poland, June 19–25, 2022. Cham: Birkhäuser. Trends Math., 57-74 (2023).
Summary: The construction of position-dependent mass Hamiltonian hierarchies is considered in the factorization context. It is shown that the superpotentials, the deformed potentials, and their bound state wave functions can be expressed in terms of a transformation function \(u\) that satisfy a second-order linear equation. The connection of this equation with a family of Ermakov equations allows the generation of different types of position-dependent mass models with quadratic spectra.
For the entire collection see [Zbl 1531.53004].
For the entire collection see [Zbl 1531.53004].
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 |
References:
| [1] | Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied mathematics series. Dover, Washington D.C. (1970) |
| [2] | Andrianov, A., Borisov, N., Ioffe, M.: The factorization method and quantum systems with equivalent energy spectra. Physics Letters A 105(1), 19-22 (1984). https://doi.org/10.1016/0375-9601(84)90553-X |
| [3] | Andrianov, A.A., Borisov, N.V., Ioffe, M.V., Éides, M.I.: Supersymmetric mechanics: A new look at the equivalence of quantum systems. Theoretical and Mathematical Physics 61(1), 965-972 (1984). https://doi.org/10.1007/BF01038543 |
| [4] | Bagchi, B., Banerjee, A., Quesne, C., Tkachuk, V.: Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass. Journal of Physics A: Mathematical and General 38(13), 2929 (2005). https://doi.org/10.1088/0305-4470/38/13/008 · Zbl 1084.81072 |
| [5] | Bagchi, B., Quesne, C.: sl(2, \(c)\) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues. Physics Letters A 273(5), 285-292 (2000). https://doi.org/10.1016/S0375-9601(00)00512-0 · Zbl 1050.81546 |
| [6] | Bagchi, B., Quesne, C.: Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework. Physics Letters A 300(1), 18-26 (2002). https://doi.org/10.1016/S0375-9601(02)00689-8 · Zbl 0997.81036 |
| [7] | Bagchi, B.K.: Supersymmetry in quantum and classical mechanics, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 116. Chapman & Hall/CRC, Boca Raton, FL (2001) · Zbl 0970.81021 |
| [8] | Blanco-Garcia, Z., Rosas-Ortiz, O., Zelaya, K.: Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex-valued potentials with real energy spectrum. Mathematical Methods in the Applied Sciences 42(15), 4925-4938 (2019). https://doi.org/10.1002/mma.5069 · Zbl 1426.81039 |
| [9] | Cariñena, J.F., Rañada, M.F., Santander, M.: Curvature-dependent formalism, Schrödinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces. Journal of Physics A: Mathematical and Theoretical 45(26), 265303 (2012). https://doi.org/10.1088/1751-8113/45/26/265303 · Zbl 1250.81029 |
| [10] | Chumakov, S.M., Wolf, K.B.: Supersymmetry in Helmholtz optics. Physics Letters A 193(1), 51-53 (1994). https://doi.org/10.1016/0375-9601(94)00616-4 · Zbl 0959.78500 |
| [11] | Cooper, F., Khare, A., Sukhatme, U.: Supersymmetry and quantum mechanics. Physics Reports 251(5), 267-385 (1995). https://doi.org/10.1016/0370-1573(94)00080-M |
| [12] | Cruz y Cruz, S., Gress, Z.: Group approach to the paraxial propagation of Hermite-Gaussian modes in a parabolic medium. Annals of Physics 383, 257-277 (2017). https://doi.org/10.1016/j.aop.2017.05.020 · Zbl 1373.81239 |
| [13] | Cruz y Cruz, S., Gress, Z., Jiménez-Macías, P., Rosas-Ortiz, O.: Laguerre-Gaussian Wave Propagation in Parabolic Media. In: P. Kielanowski, A. Odzijewicz, E. Previato (eds.) Geometric Methods in Physics XXXVIII, pp. 117-128. Springer International Publishing, Birkhäuser (2020). https://doi.org/10.1007/978-3-030-53305-2_8 · Zbl 1479.35839 |
| [14] | Cruz y Cruz, S., Negro, J., Nieto, L.: Classical and quantum position-dependent mass harmonic oscillators. Physics Letters A 369(5), 400-406 (2007). https://doi.org/10.1016/j.physleta.2007.05.040 · Zbl 1209.81095 |
| [15] | Cruz y Cruz, S., Razo, R., Rosas-Ortiz, O., Zelaya, K.: Coherent states for exactly solvable time-dependent oscillators generated by Darboux transformations. Physica Scripta 95(4), 044009 (2020). https://doi.org/10.1088/1402-4896/ab6525 |
| [16] | Cruz y Cruz, S., Rosas-Ortiz, O.: Position-dependent mass oscillators and coherent states. Journal of Physics A: Mathematical and Theoretical 42(18), 185205 (2009). https://doi.org/10.1088/1751-8113/42/18/185205 · Zbl 1162.81388 |
| [17] | Cruz y Cruz, S., Rosas-Ortiz, O.: Dynamical equations, invariants and spectrum generating algebras of mechanical systems with position-dependent mass. SIGMA Symmetry Integrability Geom. Methods Appl. 9, Paper 004, 21 (2013). https://doi.org/10.3842/SIGMA.2013.004 · Zbl 1291.70005 |
| [18] | Cruz y Cruz, S., Santiago-Cruz, C.: Position dependent mass Scarf Hamiltonians generated via the Riccati equation. Mathematical Methods in the Applied Sciences 42(15), 4909-4924 (2019). https://doi.org/10.1002/mma.5068 · Zbl 1426.81037 |
| [19] | Cruz y Cruz, S., Rosas-Ortiz, O.: su(1, 1) Coherent States for Position-Dependent Mass Singular Oscillators. International Journal of Theoretical Physics 50(7), 2201-2210 (2011). https://doi.org/10.1007/s10773-011-0728-8 · Zbl 1226.81106 |
| [20] | De, R., Dutt, R., Sukhatme, U.: Mapping of shape invariant potentials under point canonical transformations. Journal of Physics A: Mathematical and General 25(13), L843 (1992). https://doi.org/10.1088/0305-4470/25/13/013 |
| [21] | Ermakov, V.P.: Second-order differential equations: conditions of complete integrability. Appl. Anal. Discrete Math. 2(2), 123-145 (2008). https://doi.org/10.2298/AADM0802123E. Translated from the 1880 Russian original by A. O. Harin and edited by P. G. L. Leach · Zbl 1199.34004 |
| [22] | Fernández-García, N., Rosas-Ortiz, O.: Gamow-Siegert functions and Darboux-deformed short range potentials. Annals of Physics 323(6), 1397-1414 (2008). https://doi.org/10.1016/j.aop.2007.11.002 · Zbl 1139.81338 |
| [23] | Gress, Z., Cruz y Cruz, S.: Hermite coherent states for quadratic refractive index optical media. In: Integrability, supersymmetry and coherent states, CRM Ser. Math. Phys., pp. 323-339. Springer, Cham (2019) · Zbl 1425.81054 |
| [24] | Infeld, L., Hull, T.E.: The factorization method. Rev. Modern Physics 23, 21-68 (1951). https://doi.org/10.1103/revmodphys.23.21 · Zbl 0043.38602 |
| [25] | Khordad, R.: Hydrogenic donor impurity in a cubic quantum dot: effect of position-dependent effective mass. The European Physical Journal B 85(4), 114 (2012). https://doi.org/10.1140/epjb/e2012-20435-6 |
| [26] | Kuru, Ş., Negro, J.: Dynamical algebras for Pöschl-Teller Hamiltonian hierarchies. Annals of Physics 324(12), 2548-2560 (2009). https://doi.org/10.1016/j.aop.2009.08.004 · Zbl 1179.81086 |
| [27] | Mielnik, B.: Factorization method and new potentials with the oscillator spectrum. J. Math. Phys. 25(12), 3387-3389 (1984). https://doi.org/10.1063/1.526108 · Zbl 0558.35064 |
| [28] | Mielnik, B., Nieto, L., Rosas-Ortiz, O.: The finite difference algorithm for higher order supersymmetry. Physics Letters A 269(2), 70-78 (2000). https://doi.org/10.1016/S0375-9601(00)00226-7 · Zbl 1115.81350 |
| [29] | Mielnik, B., Rosas-Ortiz, O.: Factorization: little or great algorithm? Journal of Physics A: Mathematical and General 37(43), 10007-10035 (2004). https://doi.org/10.1088/0305-4470/37/43/001 · Zbl 1064.81025 |
| [30] | Mustafa, O.: PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian. Physics Letters A 384(13), 126265 (2020). https://doi.org/10.1016/j.physleta.2020.126265 · Zbl 1434.81031 |
| [31] | Mustafa, O., Mazharimousavi, S.H.: Ordering Ambiguity Revisited via Position Dependent Mass Pseudo-Momentum Operators. International Journal of Theoretical Physics 46(7), 1786-1796 (2007). https://doi.org/10.1007/s10773-006-9311-0 · Zbl 1128.81009 |
| [32] | Olivar-Romero, F., Rosas-Ortiz, O.: Factorization of the Quantum Fractional Oscillator. Journal of Physics: Conference Series 698(1), 012025 (2016). https://doi.org/10.1088/1742-6596/698/1/012025 |
| [33] | Plastino, A.R., Rigo, A., Casas, M., Garcias, F., Plastino, A.: Supersymmetric approach to quantum systems with position-dependent effective mass. Phys. Rev. A 60, 4318-4325 (1999). https://doi.org/10.1103/PhysRevA.60.4318 |
| [34] | Quesne, C.: First-order intertwining operators and position-dependent mass Schrödinger equations in \(d\) dimensions. Annals of Physics 321(5), 1221-1239 (2006). https://doi.org/10.1016/j.aop.2005.11.013 · Zbl 1092.81016 |
| [35] | Quesne, C.: Point canonical transformation versus deformed shape invariance for position-dependent mass Schrödinger equations. SIGMA Symmetry Integrability Geom. Methods Appl. 5, Paper 046, 17 (2009). https://doi.org/10.3842/SIGMA.2009.046 · Zbl 1160.81377 |
| [36] | von Roos, O.: Position-dependent effective masses in semiconductor theory. Phys. Rev. B 27, 7547-7552 (1983). https://doi.org/10.1103/PhysRevB.27.7547 |
| [37] | Rosas-Ortiz, J.O.: Exactly solvable hydrogen-like potentials and the factorization method. Journal of Physics A: Mathematical and General 31(50), 10163 (1998). https://doi.org/10.1088/0305-4470/31/50/012 · Zbl 0934.34074 |
| [38] | Rosas-Ortiz, O.: On the factorization method in quantum mechanics. In: A.B. Castañeda, F.J. Herranz, J. Negro Vadillo, L.M. Nieto, C. Pereña (eds.) Symmetries in quantum mechanics and quantum optics, pp. 285-299. Servicio de Publicaciones de la Universidad de Burgos, Spain (1999) |
| [39] | Rosas-Ortiz, O.: Position-Dependent Mass Systems: Classical and Quantum Pictures. In: P. Kielanowski, A. Odzijewicz, E. Previato (eds.) Geometric Methods in Physics XXXVIII, pp. 351-361. Springer International Publishing, Birkhäuser (2020). https://doi.org/10.1007/978-3-030-53305-2_24 · Zbl 1477.70035 |
| [40] | Rosas-Ortiz, O., Castaños, O., Schuch, D.: New supersymmetry-generated complex potentials with real spectra. Journal of Physics A: Mathematical and Theoretical 48(44), 445302 (2015). https://doi.org/10.1088/1751-8113/48/44/445302 · Zbl 1327.81216 |
| [41] | Rosas-Ortiz, O., Cruz y Cruz, S.: Superpositions of bright and dark solitons supporting the creation of balanced gain-and-loss optical potentials. Mathematical Methods in the Applied Sciences 45(7), 3381-3392 (2022). https://doi.org/10.1002/mma.6666 · Zbl 1538.81029 |
| [42] | Rosas-Ortiz, O., Zelaya, K.: Bi-orthogonal approach to non-Hermitian Hamiltonians with the oscillator spectrum: Generalized coherent states for nonlinear algebras. Annals of Physics 388, 26-53 (2018). https://doi.org/10.1016/j.aop.2017.10.020 · Zbl 1382.81118 |
| [43] | Sukumar, C.V.: Supersymmetry, factorisation of the Schrodinger equation and a Hamiltonian hierarchy. Journal of Physics A: Mathematical and General 18(2), L57 (1985). https://doi.org/10.1088/0305-4470/18/2/001 · Zbl 0586.35020 |
| [44] | Zelaya, K., Cruz y Cruz, S., Rosas-Ortiz, O.: On the Construction of Non-Hermitian Hamiltonians with All-Real Spectra through Supersymmetric Algorithms. In: P. Kielanowski, A. Odzijewicz, E. Previato (eds.) Geometric Methods in Physics XXXVIII, pp. 283-292. Springer International Publishing, Birkhäuser (2020). https://doi.org/10.1007/978-3-030-53305-2_18 · Zbl 1472.81109 |
| [45] | Zelaya, K., Rosas-Ortiz, O.: Exactly Solvable Time-Dependent Oscillator-Like Potentials Generated by Darboux Transformations. Journal of Physics: Conference Series 839(1), 012018 (2017). https://doi.org/10.1088/1742-6596/839/1/012018 |
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