Skip to main content
SpringerLink
Log in

The Schrödinger particle on the half-line with an attractive \(\delta \)-interaction: bound states and resonances

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this paper, we provide a detailed description of the eigenvalue \( E_{D}(x_0)\le 0\) (respectively, \( E_{N}(x_0)\le 0\)) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution \(-\lambda \delta (x-x_0)\) for any fixed value of the magnitude of the coupling constant. We also investigate the \(\lambda \)-dependence of both eigenvalues for any fixed value of \(x_0\). Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of the resolvent. These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverio’s et al. monograph, perturbed by an attractive \(\delta \)-distribution supported on the spherical shell of radius \(r_0\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holde, in Solvable Models in Quantum Mechanics, AMS (Chelsea Series), 2nd edn (2004)

  2. S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators (Cambridge UP, Cambridge, 1999)

    MATH  Google Scholar 

  3. S. Albeverio, L. Dabrowski, P. Kurasov, Lett. Math. Phys. 45, 33–47 (1998)

    Article  MathSciNet  Google Scholar 

  4. P. Kurasov, J. Math. Anal. Appl. 201, 297–323 (1996)

    Article  MathSciNet  Google Scholar 

  5. A.V. Zolotaryuk, J. Phys. A Math. Theor. 43, 105302 (2010)

    Article  ADS  Google Scholar 

  6. A.V. Zolotaryuk, Phys. Rev. A 87(5), 052121 (2013)

    Article  ADS  Google Scholar 

  7. A.V. Zolotaryuk, G.P. Tsironis, Y. Zolotaryuk, Front. Phys. 7, 87 (2019)

    Article  Google Scholar 

  8. Y.D. Golovaty, Methods Funct. Anal. Top. 18, 243–255 (2012)

    Google Scholar 

  9. F. Erman, H. Uncu, Phys. Lett. A 384(11), 126227 (2020)

    Article  MathSciNet  Google Scholar 

  10. M. Calçada, J.T. Lunardi, L.A. Manzoni, W. Monteiro, Front. Phys. 2, 2014 (2014)

    Google Scholar 

  11. M.A. Lee, J.T. Lunardi, L.A. Manzoni, E.A. Nyquist, Front. Phys. 4, 10 (2016)

    Article  Google Scholar 

  12. S. Fassari, F. Rinaldi, Rep. Math. Phys. 69(3), 353–370 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  13. M. Belloni, R.W. Robinett, Phys. Rep. 540, 25–122 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  14. F. Erman, M. Gadella, S. Tunali, H. Uncu, Eur. Phys. J. Plus 132, 352 (2017)

    Article  Google Scholar 

  15. A.V. Zolotaryuk, Phys. Lett. A 374(15–16), 1636–1641 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  16. A.V. Zolotaryuk, Y. Zolotaryuk, J. Phys. A Math. Theor. 44(37), 375305 (2011)

    Article  Google Scholar 

  17. P. Exner, H. Neidhardt, V.A. Zagrebnov, Commun. Math. Phys. 224, 593612 (2001)

    Article  Google Scholar 

  18. P. Seba, Czech. J. Phys. 36, 667–673 (1986)

    Article  ADS  Google Scholar 

  19. S. Albeverio, S. Fassari, F. Rinaldi, J. Phys. A Math. Theor. 46, 385305 (2013)

    Article  ADS  Google Scholar 

  20. S. Albeverio, S. Fassari, F. Rinaldi, J. Phys. A Math. Theor. 49, 025302 (2016)

    Article  ADS  Google Scholar 

  21. S. Fassari, M. Gadella, M.L. Glasser, L.M. Nieto, Annal. Phys. 389, 48–62 (2018)

    Article  ADS  Google Scholar 

  22. M. Gadella, M.L. Glasser, L.M. Nieto, Int. J. Theor. Phys. 50, 2144–2152 (2011)

    Article  Google Scholar 

  23. M. Gadella, M.L. Glasser, L.M. Nieto, Int. J. Theor. Phys. 50, 2191–2200 (2011)

    Article  Google Scholar 

  24. V.L. Kulinskii, D.Y. Panchenko, Phys. B Condens. Matt. 472, 78–83 (2015)

    Article  ADS  Google Scholar 

  25. V.L. Kulinskii, D.Y. Panchenko, Ann. Phys. 404, 47–56 (2019)

    Article  ADS  Google Scholar 

  26. S. Albeverio, P. Kurasov, Lett. Math. Phys. 41, 79–92 (1997)

    Article  MathSciNet  Google Scholar 

  27. M.H. Al-Hashimi, A.M. Shalabi, U.J. Wiese, Phys. Rev. D 89(12), 125023 (2014)

    Article  ADS  Google Scholar 

  28. S. Albeverio, S. Fassari, F. Rinaldi, J. Phys. A Math. Theor. 48(18), 185301 (2015)

    Article  ADS  Google Scholar 

  29. F. Erman, M. Gadella, H. Uncu, Phys. Rev. D 95, 045004 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  30. H. Uncu, D. Tarhan, E. Demiralp, Ö.E. Müstecaphoğlu, Phys. Rev A 76, 013618 (2007)

    Article  ADS  Google Scholar 

  31. F. Ferrari, V.G. Rostiashvili, T.A. Vilgis, Phys. Rev E 71, 061802 (2005)

    Article  ADS  Google Scholar 

  32. I. Alvarado-Rodríguez, P. Halevi, J.J. Sánchez-Mondragón, Phys. Rev. E 59, 3624 (1999)

    Article  ADS  Google Scholar 

  33. J.R. Zurita-Sánchez, P. Halevi, Phys. Rev. E 61, 5802 (2000)

    Article  ADS  Google Scholar 

  34. M.C. Lin, R.F. Jao, Phys. Rev. E 74, 046613 (2006)

    Article  ADS  Google Scholar 

  35. W. Häusler, B. Kramer, Phys. Rev. B 47, 16353–16357 (1993)

    Article  ADS  Google Scholar 

  36. S. Kvaal, M. Hjorth-Jensen, H. Møll Nilsen, Phys. Rev. B 76, 085421 (2007)

    Article  ADS  Google Scholar 

  37. S. Kvaal, Phys. Rev. B 80, 045321 (2009)

    Article  ADS  Google Scholar 

  38. G. Barton, J. Phys. A Math. Gen. 37, 1011–1049 (2004)

    Article  ADS  Google Scholar 

  39. J.M. Muñoz-Castañeda, J. Mateos Guilarte, A.M. Mosquera, Phys. Rev. D 87, 105020 (2013)

    Article  ADS  Google Scholar 

  40. M. Asorey, D. García-Alvarez, J.M. Muñoz-Castañeda, J. Phys. A Math. Gen. 39, 6127–6136 (2006)

    Article  ADS  Google Scholar 

  41. M. Asorey, J.M. Muñoz-Castañeda, Nucl. Phys. B 874, 852–876 (2013)

    Article  ADS  Google Scholar 

  42. J.M. Muñoz-Castañeda, K. Kirsten, M. Bordag, Lett. Math. Phys. 105, 523–549 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  43. J. Mateos Guilarte, J.M. Muñoz-Castañeda, Int. J. Theor. Phys. 50, 2227–22241 (2011)

    Article  Google Scholar 

  44. M. Bordag, J.M. Muñoz-Castañeda, J. Phys. A Math. Theor. 45, 374012 (2012)

    Article  Google Scholar 

  45. M. Bordag, Phys. Rev. D 89, 125015 (2014)

    Article  ADS  Google Scholar 

  46. G. Fucci, K. Kirstens, J. Phys. A Math. Theor. 44, 295403 (2011)

    Article  Google Scholar 

  47. G. Fucci, Nucl. Phys. B 891, 676–679 (2015)

    Article  ADS  Google Scholar 

  48. M. Gadella, J. Mateos Guilarte, J.M. Muñoz-Castañeda, L.M. Nieto, J. Phys. A Math. Theor. 49, 015204 (2016)

    Article  ADS  Google Scholar 

  49. J.P. Antoine, F. Gesztesy, J. Shabani, J. Phys. A Math. Gen. 20, 3687–3712 (1987)

    Article  ADS  Google Scholar 

  50. J. Shabani, Nouvo Cimento 101, 429–437 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  51. L. Dabrowski, J. Shabani, J. Math. Phys. 29, 2241–2244 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  52. J.M. Muñoz-Castañeda, L.M. Nieto, C. Romaniega, Ann. Phys. 400, 246–261 (2019)

    Article  ADS  Google Scholar 

  53. J. Derezinski, Operators on \(L^2({\mathbb{R}}^d)\) (Department of Mathematical Methods in Physics, Wasaw University, Lecture Notes, 2007)

  54. P. Seba, Lett. Math. Phys. 10, 21–27 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  55. H. Englisch, P. Seba, Rep. Math. Phys. 23(3), 341–348 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  56. A. Bohm, Quantum Mechanics: Foundations and Applications (Springer, New York, 1994)

    MATH  Google Scholar 

  57. L. Fonda, G.C. Ghirardi, A. Rimini, Rep. Prog. Phys. 41, 587–631 (1978)

    Article  ADS  Google Scholar 

  58. H.M. Nussenzveig, Causality and Dispersion Relations (Academic Press, New York, 1972)

    Google Scholar 

  59. M. Reed, B. Simon, Analysis of Operators (Academic Press, New York, 1978)

    MATH  Google Scholar 

  60. M. Klaus, Helv. Phys. Acta 52, 223–229 (1979)

    MathSciNet  Google Scholar 

  61. M. Klaus, J. Phys. A Math. Gen. 13, L295–L298 (1980)

    Article  ADS  Google Scholar 

  62. S. Fassari, Helv. Phys. Acta 63, 849–883 (1990)

    MathSciNet  Google Scholar 

  63. S. Fassari, F. Rinaldi, Rendiconti di Matematica, Serie VII, 31, 35–52 (2011)

  64. S. Fassari, F. Rinaldi, Rep. Math. Phys. 75(2), 257–265 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  65. S. Fassari, F. Rinaldi, Rep. Math. Phys. 64(3), 367–393 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  66. M. Reed, B. Simon, Functional Analysis (Academic Press, New York, 1972)

    MATH  Google Scholar 

  67. S. Albeverio, S. Fassari, F. Rinaldi, Nanosyst. Phys. Chem. Math. 7(5), 803–815 (2016)

    Article  Google Scholar 

  68. S. Albeverio, S. Fassari, F. Rinaldi, Nanosyst. Phys. Chem. Math. 8(2), 153–159 (2017)

    Article  Google Scholar 

  69. S. Fassari, I. Popov, F. Rinaldi, Phys. Scr. 95, 075209 (2020)

    Article  ADS  Google Scholar 

  70. R. de la Madrid, Nucl. Phys. A 962, 24–45 (2017)

    Article  ADS  Google Scholar 

  71. R.J. Lange, J. Math. Phys. 56, 122105 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  72. S. Albeverio, R. Høegh-Krohn, T.T. Wu, Phys. Lett. A 83(3), 105–109 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  73. H. Narnhofer, Acta Physica Austriaca 40, 308–322 (1974)

    Google Scholar 

  74. R. Arutyunyan, Y. Obukhov, P. Vabishchevich, Math. Mod. Anal. 24, 263–275 (2019)

    Article  Google Scholar 

  75. M. Reed, B. Simon, in Fourier Analysis (Academic Press, New York, 1975)

    MATH  Google Scholar 

Download references

Acknowledgements

S. Fassari would like to thank Prof. Igor Yu. Popov and the entire staff at the Department of Mathematics, ITMO University, St. Petersburg for their warm hospitality throughout his stay. Stimulating discussions with A. Colli, M.Sc.(Damor Pharmaceuticals SpA, Naples, Italy) are kindly acknowledged by S. Fassari. S. Fassari’s contribution to this work has been made possible by the financial support granted by the Government of the Russian Federation through the ITMO University Fellowship and Professorship Programme. This work was partially financially supported by the Government of the Russian Federation (grant 08-08) and by the Russian Science Foundation (grant 16-11-10330). M. Gadella and L.M. Nieto gratefully acknowledge partial financial support from Junta de Castilla y León and FEDER (Projects VA137G18 and BU229P18).

Author information

Authors and Affiliations

  1. Department of Higher Mathematics, ITMO University, S. Petersburg, Russian Federation

    S. Fassari

  2. Dipartimento di Scienze Ingegneristiche, Univ. degli Studi Guglielmo Marconi, Via Plinio 44, 00193, Rome, Italy

    S. Fassari & F. Rinaldi

  3. CERFIM, PO Box 1132, 6601, Locarno, Switzerland

    S. Fassari & F. Rinaldi

  4. Departamento de Física Teórica, Atómica y Óptica, and IMUVA, U. de Valladolid, 47011, Valladolid, Spain

    M. Gadella & L. M. Nieto

Authors
  1. S. Fassari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Gadella
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. M. Nieto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. F. Rinaldi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. M. Nieto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fassari, S., Gadella, M., Nieto, L.M. et al. The Schrödinger particle on the half-line with an attractive \(\delta \)-interaction: bound states and resonances. Eur. Phys. J. Plus 136, 673 (2021). https://doi.org/10.1140/epjp/s13360-021-01636-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01636-0

Search

Navigation