Spectral Analysis of the 2 + 1 Fermionic Trimer with Contact Interactions

  • Simon Becker
  • Alessandro MichelangeliEmail author
  • Andrea Ottolini


We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise the discrete spectrum and prove its finiteness, qualify the angular symmetry of the eigenfunctions, and prove the increasing monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence or absence of bound states in the physically relevant regimes of masses.


Particle systems with zero-range/contact interactions Ter-martirosyan-skornyakov hamiltonians Fermonic 2 + 1 system 

Mathematics Subject Classification (2010)

46N50 47A60 47B25 47N50 70F07 81Q10 


  1. 1.
    Bethe, H., Peierls, R.: Quantum theory of the diplon, proceedings of the royal society of London. Series A Math. Phys. Sci. 148, 146–156 (1935a)zbMATHGoogle Scholar
  2. 2.
    Bethe, H.A., Peierls, R.: The scattering of neutrons by protons, proceedings of the royal society of London. Series A Math. Phys. Sci. 149, 176–183 (1935b)zbMATHGoogle Scholar
  3. 3.
    Braaten, E., Hammer, H.-W.: Universality in few-body systems with large scattering length. Phys. Rep. 428, 259–390 (2006)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Castin, Y., Werner, F.: The unitary gas and its symmetry properties, in the BCS-BEC crossover and the unitary fermi gas. In: Zwerger, W. (ed.) Lecture Notes in Physics, vol. 836, pp 127–191. Springer, Berlin (2012)Google Scholar
  5. 5.
    Correggi, M., Dell’Antonio, G., Finco, D., Michelangeli, A., Teta, A.: Stability for a system of N fermions plus a different particle with zero-range interactions. Rev. Math. Phys. 24, 1250017, 32 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Correggi, M., Dell’Antonio, G., Finco, D., Michelangeli, A., Teta, A.: A class of hamiltonians for a three-particle fermionic system at unitarity. Math. Phys. Anal. Geom. 18, 32 (2015). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dell’Antonio, G.F., Figari, R., Teta, A.: Hamiltonians for systems of N particles interacting through point interactions. Ann. Inst. H. Poincaré Phys. Théor. 60, 253–290 (1994)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Endo, S., Naidon, P., Ueda, M.: Universal physics of 2 + 1 particles with non-zero angular momentum. Few-Body Syst. 51, 207–217 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    Finco, D., Teta, A.: Quadratic forms for the fermionic unitary gas model. Rep. Math. Phys. 69, 131–159 (2012)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Gallone, M., Michelangeli, A., Ottolini, A.: Kreı̆n-Višik-Birman self-adjoint extension theory revisited SISSA preprint 25/2017/MATE (2017)Google Scholar
  11. 11.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products, Elsevier/Academic Press, Amsterdam, eighth ed., 2015. Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition [MR2360010]Google Scholar
  12. 12.
    Kartavtsev, O.I., Malykh, A.V.: Universal descritpion of three two-species particles. In: Malykh, A.V. (ed.) Proc. of the 4th South Africa-JINR Symposium Few to Many Body Systems: Models, Methods, and Applications, pp. 23–29 (2016)Google Scholar
  13. 13.
    Kartavtsev, O.I., Malykh, A.V.: Low-energy three-body dynamics in binary quantum gases. J. Phys. B Atomic Mol. Phys. 40, 1429 (2007)ADSCrossRefGoogle Scholar
  14. 14.
    Kartavtsev, O.I., Malykh, A.V.: Universal description of three two-component fermions. EPL 115, 36005 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    Michelangeli, A., Ottolini, A.: On point interactions realised as Ter-Martirosyan–Skornyakov hamiltonians. Rep. Math. Phys. 79, 215–260 (2017)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Michelangeli, A., Ottolini, A.: Multiplicity of self-adjoint realisations of the (2 + 1)-fermionic model of Ter-Martirosyan–Skornyakov type. Rep. Math. Phys. 81, 1–38 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Michelangeli, A., Pfeiffer, P.: Stability of the (2 + 2)-fermionic system with zero-range interaction. J. Phys. A Math. Theor. 49, 105301 (2016)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Michelangeli, A., Schmidbauer, C.: Binding properties of the (2 + 1)-fermion system with zero-range interspecies interaction. Phys. Rev. A 87, 053601 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    Minlos, R.A.: On the point interaction of three particles. In: Applications of selfadjoint extensions in quantum physics (Dubna, 1987), vol. 324 of Lecture Notes in Phys., pp 138–145. Springer, Berlin (1989)Google Scholar
  20. 20.
    Minlos, R.A.: On pointlike interaction between N fermions and another particle. In: Dell’Antonio, A., Figari, R., Teta, A. (eds.) Proceedings of the Workshop on Singular Schrodinger̈ Operators, Trieste 29 September - 1 October 1994, pp ILAS/FM-16 (1995)Google Scholar
  21. 21.
    Minlos, R.A.: On point-like interaction between n fermions and another particle. Mosc. Math. J. 11, 113–127, 182 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Minlos, R.A.: On point-like interaction between three particles: two fermions and another particle. ISRN Mathematical Physics 2012, 230245 (2012)CrossRefGoogle Scholar
  23. 23.
    Minlos, R.A.: A system of three pointwise interacting quantum particles. Uspekhi Mat. Nauk 69, 145–172 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Minlos, R.A.: On point-like interaction of three particles: two fermions and another particle. II. Mosc. Math. J. 14, 617–637, 642–643 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Minlos, R.A., Faddeev, L.D.: Comment on the problem of three particles with point interactions. Sov. Phys. JETP 14, 1315–1316 (1962)MathSciNetGoogle Scholar
  26. 26.
    Minlos, R.A., Faddeev, L.D.: On the point interaction for a three-particle system in quantum mechanics. Sov. Phys. Dokl. 6, 1072–1074 (1962)ADSMathSciNetGoogle Scholar
  27. 27.
    Minlos, R.A., Shermatov, M.K.: Point interaction of three particles. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 97, 7–14 (1989)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Pethick, C.J., Smith, H.: Bose–Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  29. 29.
    Petrov, D.S.: The few-atom problem. In: Many-Body Physics With Ultracold Gases (Les Houches 2010) Lecture Notes of the Les Houches Summer School, vol. 94, pp 109–160. Oxford Univ. Press, Oxford (2013)Google Scholar
  30. 30.
    Schmüdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space, vol. 265 of Graduate Texts in Mathematics. Springer, Dordrecht (2012)CrossRefGoogle Scholar
  31. 31.
    Skornyakov, G.V., Ter-Martirosyan, K.A.: Three body problem for short range forces. I. Scattering of low energy neutrons by deuterons. Sov. Phys. JETP 4, 648–661 (1956)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Teta, A.: Quadratic forms for singular perturbations of the Laplacian. Publ. Res. Inst. Math. Sci. 26, 803–817 (1990)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Yoshitomi, K.: Finiteness of the discrete spectrum in a three-body system with point interaction. Math. Slovaca 67, 1031–1042 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.International School for Advanced Studies – SISSATriesteItaly
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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