International Journal of Theoretical Physics

, Volume 53, Issue 5, pp 1614–1627 | Cite as

The Infinite Square Well with a Point Interaction: A Discussion on the Different Parameterizations

  • M. GadellaEmail author
  • M. A. García-Ferrero
  • S. González-Martín
  • F. H. Maldonado-Villamizar


The construction of Dirac delta type potentials has been achieved with the use of the theory of self adjoint extensions of non-self adjoint formally Hermitian (symmetric) operators. The application of this formalism to investigate the possible self adjoint extensions of the one dimensional kinematic operator \(K=-\frac{d^{2}}{dx^{2}}\) on the infinite square well potential is quite illustrative and has been given elsewhere. This requires the definition and use of four independent real parameters, which relate the boundary values of the wave functions at the walls. By means of a different approach, that fixes matching conditions at the origin for the wave functions, it is possible to define a perturbation of the type (x)+′(x), thus depending on two parameters, on the infinite square well. The objective of this paper is to investigate whether these two approaches are compatible in the sense that perturbations like (x)+′(x) can be fixed and determined using the first approach.


Point potentials Parameterizations of self adjoint extensions 



Financial support is acknowledged to the Ministry of Economy and Innovation of Spain through the Grant MTM2009-10751.


  1. 1.
    Bonneau, G., Faraut, J., Valent, G.: Self adjoint extensions of operators and the teaching of quantum mechanics. Am. J. Phys. 69, 322–331 (2001) ADSCrossRefGoogle Scholar
  2. 2.
    Kurasov, P.: Distribution theory for discontinuous test functions and differential operators with generalized coefficients. J. Math. Anal. Appl. 201, 297–323 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Golovaty, Y.: Schrödinger operators with (αδ′+βδ)-like potentials: norm resolvent convergence and solvable models. Methods Funct. Anal. Topol. 18, 243–255 (2012) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Zolotaryuk, A.V.: Boundary conditions for the states with resonant tunnelling across the δ′-potential. Phys. Lett. A 374, 1636–1641 (2010) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Seba, P.: Some remarks on the δ′ interaction in one dimension. Rep. Math. Phys. 24, 111–120 (1986) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Toyama, F.M., Nogami, Y.: Transmission-reflection problem with a potential of the form of the derivative of the delta function. J. Phys. A, Math. Theor. 40, F685–F690 (2007) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Zolotaryuk, A.V., Zolotaryuk, Y.: Controlling a resonant transmission across the δ′-potential: the inverse problem. J. Phys. A, Math. Theor. 45, 375305 (2011). Corrigendum: J. Phys. A, Math. Theor. 45, 119501 (2012) CrossRefGoogle Scholar
  8. 8.
    Gadella, M., Negro, J., Nieto, L.M.: Bound states and scattering coefficients of the −(x)+′(x) potential. Phys. Lett. A 373, 1310–1313 (2009) ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bohm, A., Gadella, M., Wickramasekara, W.: Some little things about rigged Hilbert spaces and quantum mechanics and all that. In: Antoniou, I., Lummer, E. (eds.) Generalized Functions, Operator Theory and Dynamical Systems, pp. 202–250. CRC Press, Boca Raton (1999) Google Scholar
  10. 10.
    Albeverio, S., Gesztesy, F., Høeg-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, vol. 350.H. AMS, Providence (2005) zbMATHGoogle Scholar
  11. 11.
    Gadella, M., Glasser, M.L., Nieto, L.M.: The infinite square well with a singular perturbation. Int. J. Theor. Phys. 50, 2191–2200 (2011) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • M. Gadella
    • 1
    Email author
  • M. A. García-Ferrero
    • 1
  • S. González-Martín
    • 1
  • F. H. Maldonado-Villamizar
    • 2
  1. 1.Department of Theoretical, Atomic Physics and Optics, Facultad de CienciasUniversity of ValladolidValladolidSpain
  2. 2.Departamento de FísicaCentro de Investigación y Estudios Avanzados del IPNMéxico DFMexico

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