Abstract
We show how a proper use of the Lippmann–Schwinger equation simplifies the calculations to obtain scattering states for one dimensional systems perturbed by N Dirac delta equations. Here, we consider two situations. In the former, attractive Dirac deltas perturbed the free one dimensional Schrödinger Hamiltonian. We obtain explicit expressions for scattering and Gamow states. For completeness, we show that the method to obtain bound states use comparable formulas, although not based on the Lippmann–Schwinger equation. Then, the attractive N deltas perturbed the one dimensional Salpeter equation. We also obtain explicit expressions for the scattering wave functions. Here, we need regularisation techniques that we implement via heat kernel regularisation.
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Notes
- 1.
As a matter of fact, this also follows because Φ(E) appears in the denominator of the resolvent of the total Hamiltonian H.
References
S. Albeverio, F. Gesztesy, R. Høeg-Krohn, H. Holden, Solvable Models in Quantum Mechanics (AMS Chelsea Series, Providence RI, 2004)
Y.N. Demkov, V.N. Ostrovskii, Zero-range Potentials and Their Applications in Atomic Physics (Plenum, New York, 1988)
M. Belloni, R.W. Robinett, The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics. Phys. Rep. 540, 25–122 (2014)
S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators Solvable Schrödinger-type Operators (Cambridge University Press, Cambridge, 2000)
M.H. Al-Hashimi, A.M. Shalaby, U.-J.Wiese, Asymptotic freedom, dimensional transmutation, and an infrared conformal fixed point for the δ-function potential in one-dimensional relativistic quantum mechanics. Phys. Rev. D 89, 125023 (2014)
F. Erman, M. Gadella, H. Uncu, One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials. Phys. Rev. D 95, 045004 (2017)
M. Calçada, J.T. Lunardi, L.A. Manzoni, W. Monteiro, Distributional approach to point interactions in one-dimensional quantum mechanics. Front. Phys. 2, 23 (2014)
F. Erman, M. Gadella, S. Tunalı, H. Uncu, A singular one-dimensional bound state problem and its degeneracies. Eur. Phys. J. Plus 132, 352 (2017)
F. Erman, M. Gadella, H. Uncu, On scattering from the one dimensional multiple Dirac delta potentials. Eur. J. Phys. 39, 035403 (2018)
R. de L. Kronig, W.G. Penney, Quantum mechanics of electrons in crystal lattices. Proc. R. Soc. A 130, 499 (1931)
C. Kittel, Introduction to Solid State Physics 8th edn. (Wiley, New York, 2005)
I.R. Lapidus, Resonance scattering from a double δ-function potential. Am. J. Phys. 50, 663–664 (1982)
P. Senn, Threshold anomalies in one dimensional scattering. Am. J. Phys. 56, 916–921 (1988)
P.R. Berman, Transmission resonances and Bloch states for a periodic array of delta function potentials. Am. J. Phys. 81, 190–201 (2013)
S.H. Patil, Quadrupolar, triple δ-function potential in one dimension. Eur. J. Phys. 629–640 (2009)
V.E. Barlette, M.M. Leite, S.K. Adhikari, Integral equations of scattering in one dimension. Am. J. Phys. 69, 1010–1013 (2001)
D. Lessie, J. Spadaro, One dimensional multiple scattering in quantum mechanics. Am. J. Phys. 54, 909–913 (1986)
J.J. Alvarez, M. Gadella, L.M. Nieto, A study of resonances in a one dimensional model with singular Hamiltonian and mass jump. Int. J. Theor. Phys. 50, 2161–2169 (2011)
J.J. Alvarez, M. Gadella, L.P. Lara, F.H. Maldonado-Villamizar, Unstable quantum oscillator with point interactions: Maverick resonances, antibound states and other surprises. Phys. Lett. A 377, 2510–2519 (2013)
A. Bohm, in The Rigged Hilbert Space and Quantum Mechanics. Springer Lecture Notes in Physics, vol. 78 (Springer, New York, 1978)
J.E. Roberts, Rigged Hilbert spaces in quantum mechanics. Commun. Math. Phys. 3, 98–119 (1966)
J.P. Antoine, Dirac formalism and symmetry problems in quantum mechanics. I. General formalism. J. Math. Phys. 10, 53–69 (1969)
O. Melsheimer, Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. J. Math. Phys. 15, 902–916 (1974)
M. Gadella, F. Gómez, On the mathematical basis of the Dirac formulation of quantum mechanics. Int. J. Theor. Phys. 42, 2225–2254 (2003)
A. Bohm, Quantum Mechanics. Foundations and Applications (Springer, Berlin, New York, 2002)
M.C. Fischer, B. Gutiérrez-Medina, M.G. Raizen, Observation of the quantum Zeno and anti-Zeno effects in an unstable system. Phys. Rev. Lett. 87, 40402 (2001)
C. Rothe, S.L. Hintschich, A.P. Monkman, Violation of the exponential-decay law at long times. Phys. Rev. Lett. 96, 163601 (2006)
A. Bohm, Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics. J. Math. Phys. 22 (12), 2813–2823 (1981)
A. Bohm, M. Gadella, in Dirac Kets, Gamow Vectors and Gelfand Triplets. Springer Lecture Notes in Physics, vol. 348 (Springer, Berlin, 1989)
O. Civitarese, M. Gadella, Physical and mathematical aspects of Gamow states. Phys. Rep. 396, 41–113 (2004)
O. Civitarese, M. Gadella, Gamow states as solutions of a modified Lippmann–Schwinger equation. Int. J. Mod. Phys. E 25, 1650075 (2016)
M. Reed, B. Simon, Analysis of Operators (Academic, New York, 1978), p. 55
M. Gadella, F. Gómez, The Lippmann–Schwinger equations in the rigged Hilbert space. J. Phys. A: Math. Gen. 35, 8505–8511 (2002)
T. Berggren, Expectation value of an operator in a resonant state. Phys. Lett. B 373, 1–4 (1996)
O. Civitarese, M. Gadella, R. Id Betan, On the mean value of the energy for resonance states. Nucl. Phys. A 660, 255–266 (1999)
Acknowledgements
We dedicate this paper to Professor Véronique Hussin for her contributions to science and her friendship. The present work has been fully financed by TUBITAK from Turkey under the “2221 - Visiting Scientist Fellowship Programme”. We are very grateful to TUBITAK for this support. We also acknowledge Osman Teoman Turgut for clarifying discussions and his interest in the present research. This work was also sponsored by the Ministerio de Economía y Competitividad of Spain (Project No. MTM2014-57129-C2-1-P with EU-FEDER support) and the Junta de Castilla y León (Projects VA057U16, VA137G18 and BU229P18).
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Erman, F., Gadella, M., Uncu, H. (2019). The Lippmann–Schwinger Formula and One Dimensional Models with Dirac Delta Interactions. In: Kuru, Ş., Negro, J., Nieto, L. (eds) Integrability, Supersymmetry and Coherent States. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-20087-9_13
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