Time-dependent step-like potential with a freezable bound state in the continuum


  • Izamar Gutiérrez Altamirano Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Ciudad Universitaria. Francisco J. Mújica s/n. Col. Felícitas del Río. 58040 Morelia, Michoacán, México
  • Alonso Contreras-Astorga CONACyT–Physics Department, Cinvestav, P.O. Box. 14-740, 07000 Mexico City, Mexico
  • Alfredo Raya Montaño Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Ciudad Universitaria. Francisco J. Mújica s/n. Col. Felícitas del Río. 58040 Morelia, Michoacán, México; Centro de Ciencias Exactas, Universidad del Bío-Bío, Avda. Andrés Bello 720, Casilla 447, 3800708, Chillán, Chile




bound states in the continuum, supersymmetric quantum mechanics, time-dependent quantum systems


In this work, we construct a time-dependent step-like potential supporting a normalizable state with energy embedded in the continuum. The potential is allowed to evolve until a stopping time ti, where it becomes static. The normalizable state also evolves but remains localized at every fixed time up to ti. After this time, the probability density of this state freezes becoming a Bound state In the Continuum. Closed expressions for the potential, the freezable bound state in the continuum, and scattering states are given.


Download data is not yet available.


J. von Neuman, E. Wigner. Uber merkwürdige diskrete Eigenwerte. Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen. Physikalische Zeitschrift 30:467–470, 1929. https://ui.adsabs.harvard.edu/abs/1929PhyZ...30..467V Provided by the SAO/NASA Astrophysics Data System.

B. Simon. On positive eigenvalues of one-body Schrödinger operators. Communications on Pure and Applied Mathematics 22:531–538, 1969. https://doi.org/10.1002/cpa.3160220405.

F. H. Stillinger, D. R. Herrick. Bound states in the continuum. Physical Review A 11:446–454, 1975. https://doi.org/10.1103/PhysRevA.11.446.

B. Gazdy. On the bound states in the continuum. Physics Letters A 61(2):89–90, 1977. https://doi.org/10.1016/0375-9601(77)90845-3.

M. Klaus. Asymptotic behavior of Jost functions near resonance points for Wigner–von Neumann type potentials. Journal of Mathematical Physics 32:163–174, 1991. https://doi.org/10.1063/1.529140.

I. M. Gel’fand, B. M. Levitan. On the determination of a differential equation from its spectral function. Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 15(4):309–360, 1951.

T. A. Weber, D. L. Pursey. Continuum bound states. Physical Review A 50:4478–4487, 1994. https://doi.org/10.1103/PhysRevA.50.4478.

A. A. Stahlhofen. Completely transparent potentials for the schrödinger equation. Physical Review A 51:934–943, 1995. https://doi.org/10.1103/PhysRevA.51.934.

D. Lohr, E. Hernandez, A. Jauregui, A. Mondragon. Bound states in the continuum and time evolution of the generalized eigenfunctions. Revista Mexicana de Física 64(5):464–471, 2018. https://doi.org/10.31349/RevMexFis.64.464.

J. Pappademos, U. Sukhatme, A. Pagnamenta. Bound states in the continuum from supersymmetric quantum mechanics. Physical Review A 48:3525–3531, 1993. https://doi.org/10.1103/PhysRevA.48.3525.

N. Fernández-García, E. Hernández, A. Jáuregui, A. Mondragón. Exceptional points of a Hamiltonian of von Neumann–Wigner type. Journal of Physics A: Mathematical and Theoretical 46(17):175302, 2013. https://doi.org/10.1088/1751-8113/46/17/175302.

L. López-Mejía, N. Fernández-García. Truncated radial oscillators with a bound state in the continuum via Darboux transformations. Journal of Physics: Conference Series 1540:012029, 2020. https://doi.org/10.1088/1742-6596/1540/1/012029.

A. Demić, V. Milanović, J. Radovanović. Bound states in the continuum generated by supersymmetric quantum mechanics and phase rigidity of the corresponding wavefunctions. Physics Letters A 379(42):2707–2714, 2015. https://doi.org/10.1016/j.physleta.2015.08.017.

C. W. Hsu, B. Zhen, A. D. Stone, et al. Bound states in the continuum. Nature Reviews Materials 1(9):16048, 2016. https://doi.org/10.1038/natrevmats.2016.48.

D. L. Hill, J. A. Wheeler. Nuclear constitution and the interpretation of fission phenomena. Physical Review 89:1102–1145, 1953. https://doi.org/10.1103/PhysRev.89.1102.

S. W. Doescher, M. H. Rice. Infinite square-well potential with a moving wall. American Journal of Physics 37:1246–1249, 1969. https://doi.org/10.1119/1.1975291.

K. Cooney. The infinite potential well with moving walls, 2017. arXiv:1703.05282.

M. V. Berry. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society Lond A 392:45–57, 1984. https://doi.org/10.1098/rspa.1984.0023.

J. R. Ray. Exact solutions to the time-dependent Schrödinger equation. Physical Review A 26:729–733, 1982. https://doi.org/10.1103/PhysRevA.26.729.

G. W. Bluman. On mapping linear partial differential equations to constant coefficient equations. SIAM Journal on Applied Mathematics 43(6):1259–1273, 1983. https://doi.org/10.1137/0143084.

A. Schulze-Halberg, B. Roy. Time dependent potentials associated with exceptional orthogonal polynomials. Journal of Mathematical Physics 55(12):123506, 2014. https://doi.org/10.1063/1.4903257.

K. Zelaya, O. Rosas-Ortiz. Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations. Physica Scripta 95(6):064004, 2020. https://doi.org/10.1088/1402-4896/ab5cbf.

K. Zelaya, V. Hussin. Point transformations: Exact solutions of the quantum time-dependent mass nonstationary oscillator. In M. B. Paranjape, R. MacKenzie, Z. Thomova, et al. (eds.), Quantum Theory and Symmetries, pp. 295–303. Springer International Publishing, Cham, 2021. https://doi.org/10.1007/978-3-030-55777-5_28.

A. Contreras-Astorga, V. Hussin. Infinite Square-Well, Trigonometric Pöschl-Teller and Other Potential Wells with a Moving Barrier, pp. 285–299. Springer International Publishing, Cham, 2019. https://doi.org/10.1007/978-3-030-20087-9_11.

V. B. Matveev, M. A. Salle. Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin, Heidelberg, 1992. ISBN 9783662009246, https://books.google.com.mx/books?id=pJDjvwEACAAJ.

F. Cooper, A. Khare, U. Sukhatme. Supersymmetry and quantum mechanics. Physics Reports 251(5-6):267–385, 1995. https://doi.org/10.1016/0370-1573(94)00080-M.

D. J. Fernández C., N. Fernández-García. Higher-order supersymmetric quantum mechanics. AIP Conference Proceedings 744:236–273, 2004. https://doi.org/10.1063/1.1853203.

A. Gangopadhyaya, J. V. Mallow, C. Rasinariu. Supersymmetric Quantum Mechanics: An Introduction (Second Edition). World Scientific Publishing Company, 2017. ISBN 9789813221062.

G. Junker. Supersymmetric Methods in Quantum, Statistical and Solid State Physics. IOP Expanding Physics. Institute of Physics Publishing, 2019. ISBN 9780750320245.

C. Cohen-Tannoudji, B. Diu, F. Laloë. Quantum mechanics. 1st ed. Wiley, New York, NY, 1977. Trans. of : Mécanique quantique. Paris : Hermann, 1973, https://cds.cern.ch/record/101367.

R. Shankar. Principles of quantum mechanics. Plenum, New York, NY, 1980. https://cds.cern.ch/record/102017.




How to Cite

Gutiérrez Altamirano, I., Contreras-Astorga, A. ., & Raya Montaño , A. (2022). Time-dependent step-like potential with a freezable bound state in the continuum. Acta Polytechnica, 62(1), 56–62. https://doi.org/10.14311/AP.2022.62.0056



Analytic and Algebraic Methods in Physics