Point Interaction

Part of the UNITEXT for Physics book series (UNITEXTPH)


We introduce the self-adjoint Hamiltonian of a particle in dimension one subject to a point or delta interaction. We characterize the spectrum, compute proper, and generalized eigenfunctions and prove the eigenfunction expansion theorem. These results are applied to obtain the long time behavior of the unitary group generated by the Hamiltonian and to study the scattering problem in full detail. Finally, we consider the case of two point interactions and we study eigenvalues and eigenvectors in the semiclassical regime, showing the typical behavior occurring in the double well problem.


  1. 1.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, 3th edn. Pergamon Press, Oxford (1977)MATHGoogle Scholar
  2. 2.
    Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2th edn. with an Appendix by P. Exner. AMS Chelsea Publ, Providence (2004)Google Scholar
  3. 3.
    Agmon, S.: Spectral Properties of Schrödinger Operators and Scattering Theory. Ann. Scoula Norm. Sup. Pisa Cl. Sci. 4(2), 151–218 (1975)Google Scholar
  4. 4.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III: Scattering Theory. Academic Press, New York (1979)MATHGoogle Scholar
  5. 5.
    Simon, B.: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton University Press, Princeton (1971)MATHGoogle Scholar
  6. 6.
    Thaller, B.: Visual Quantum Mechanics. Springer, New York (2000)MATHGoogle Scholar
  7. 7.
    Gottfried, K., Yan, T.: Quantum Mechanics: Fundamentals, 2th edn. Springer, New York (2004)Google Scholar
  8. 8.
    Merzbacher, E.: Quantum Mechanics, 3th edn. Wiley, New York (1998)Google Scholar
  9. 9.
    Alonso, A., Simon, B.: The Birman-Krein-Vishik theory of selfadjoint extensions of semibounded operators. J. Oper. Theory 4, 251–270 (1980)MathSciNetMATHGoogle Scholar
  10. 10.
    Flamand, G.: Mathematical theory of non-relativistic two- and three-particle systems with point interactions. In Cargese Lectures in Theoretical Physics: Application of Mathematics to Problems in Theoretical Physics, pp. 247–287. Gordon and Breach Science Publ., New York (1967)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica Guido CastelnuovoUniversità degli Studi di Roma “La Sapienza”RomeItaly

Personalised recommendations