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

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Abstract

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 aδ(x)+bδ′(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 aδ(x)+bδ′(x) can be fixed and determined using the first approach.

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Notes

  1. The operator B extends the operator A if \({\mathcal{D}}_{A}\subset{\mathcal{D}}_{B}\) and Aψ=Bψ for all \(\psi\in{\mathcal{D}}_{A}\). Then, we write A≺B.

  2. Technically, \({\mathcal{D}}^{*}\) is the space of absolutely continuous functions in L 2[−c,c] with first absolutely continuous derivative and such that \(\int_{-c}^{c} \{|f(x)|^{2}+|f''(x)|^{2}\}\,dx<\infty\).

  3. Or in more technical terms, the Sobolev space \(W^{2}_{2}({\mathbb{R}})\).

  4. This function may be even of class C ∞ on the whole real line. See an example in the Appendix of [9]

  5. Here, we have avoided some technicalities. As a matter of fact this domain is the Sobolev space \(W^{2}_{2}({\mathbb{R}}\setminus \{0\})\) [2].

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Acknowledgement

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

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Authors and Affiliations

  1. Department of Theoretical, Atomic Physics and Optics, Facultad de Ciencias, University of Valladolid, Paseo Belén 7, 47011, Valladolid, Spain

    M. Gadella, M. A. García-Ferrero & S. González-Martín

  2. Departamento de Física, Centro de Investigación y Estudios Avanzados del IPN, 07360, México DF, Mexico

    F. H. Maldonado-Villamizar

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  1. M. Gadella
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  2. M. A. García-Ferrero
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  3. S. González-Martín
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  4. F. H. Maldonado-Villamizar
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Correspondence to M. Gadella.

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Gadella, M., García-Ferrero, M.A., González-Martín, S. et al. The Infinite Square Well with a Point Interaction: A Discussion on the Different Parameterizations. Int J Theor Phys 53, 1614–1627 (2014). https://doi.org/10.1007/s10773-013-1959-7

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  • DOI: https://doi.org/10.1007/s10773-013-1959-7

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