The semi-classical limit with a delta potential


We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian H is given, as sum of quadratic forms, by \( H= -\frac{{\hbar ^{2}}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,\alpha \delta _{0}\), with \(\alpha \in \mathbb R\) and \(\delta _{0}\) the Dirac delta-distribution at \(x=0\). We show that the quantum evolution can be approximated, uniformly for any time away from the collision time and with an error of order \({\hbar ^{3/2-\lambda }}\), \(0\!<\!\lambda \!<\!3/2\), by the quasi-classical evolution generated by a self-adjoint extension of the restriction to \(\mathcal C^{\infty }_{c}({\mathscr {M}}_{0})\), \({\mathscr {M}}_{0}:=\{(q,p)\!\in \!\mathbb R^{2}\,|\,q\!\not =\!0\}\), of (\(-i\) times) the generator of the free classical dynamics; such a self-adjoint extension does not correspond to the classical dynamics describing the complete reflection due to the infinite barrier. Similar approximation results are also provided for the wave and scattering operators.

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


  1. 1.

    In particular note that, for all \(b \in \mathbb R\), there holds

    $$\begin{aligned} \sup _{\xi \in \mathbb R}\,{b^2\, (|\xi | - 1)^{2} \over (1 \!+\! b^2) (1 \!+\! b^2 \xi ^2)} = {b^2 \over 1 \!+\! b^2}\; \mathrm{max}\left\{ \left. { (|\xi | \!-\! 1)^{2} \over 1 \!+\! b^2 \xi ^2}\right| _{\xi = 0}, \left. {(|\xi | \!-\! 1)^{2} \over 1 \!+\! b^2 \xi ^2}\right| _{\xi = \infty } \right\} = {\mathrm{max}\{b^2, 1\} \over 1 + b^2} \le 1\,. \end{aligned}$$
  2. 2.

    In particular, note that \( \sup _{\xi \in \mathbb R} [{\xi ^{2}/ (1 \!+\! \xi ^2)^2}] = 1/4\) .

  3. 3.

    Note that \({\xi ^2 \over 1 + \xi ^2} \ge {\xi ^2 \over 2}\) for \(|\xi | \le 1\).


  1. 1.

    Abraham, R., Marsden, J.E.: Foundations of Mechanics. Second Edition, Revised and Enlarged. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass (1978)

  2. 2.

    Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Second Edition. With an Appendix by Pavel Exner, 2nd edn. AMS Chelsea Publishing, Providence (2005)

    Google Scholar 

  3. 3.

    Chernyshev, V.L., Shafarevich, A.I.: Semiclassical asymptotics and statistical properties of Gaussian packets for the nonstationary Schrödinger equation on a geometric graph. Russ. J. Math. Phys. 15, 25–34 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Combescure, M., Robert, D.: Coherent States and Applications in Mathematical Physics. Springer, Netherlands (2012)

    Google Scholar 

  5. 5.

    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin (2000)

    Google Scholar 

  6. 6.

    Gaveau, B., Schulman, L.: Explicit time-dependent Schrödinger propagators. J. Phys. A 19, 1833–1846 (1986)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007)

    Google Scholar 

  8. 8.

    Hagedorn, G.A.: Semiclassical quantum mechanics I. The \(\hbar \rightarrow 0\) limit for coherent states. Commun. Math. Phys. 71, 77–93 (1980)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)

    Google Scholar 

  10. 10.

    Keller, J.B.: A geometrical theory of diffraction. In: Calculus of Variations and Its Applications. Proceedings of Symposia in Applied Mathematics, Vol. 8., pp. 27–52. McGraw-Hill Book Co. (1958)

  11. 11.

    Keller, J.B.: Geometrical theory of diffraction. J. Opt. Soc. Am. 52, 116–130 (1962)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Mantile, A., Posilicano, A., Sini, M.: Limiting absorption principle, generalized eigenfunctions and scattering matrix for Laplace operators with boundary conditions on hypersurfaces. J. Spectr. Theory 8, 1443–1486 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Maslov, V.P.: The Complex WKB Method for Nonlinear Equations I. Linear Theory. Birkhäuser Verlag, Basel (1994)

    Google Scholar 

  14. 14.

    Maslov, V.P., Fedoriuk, M.V.: Semiclassical Approximation in Quantum Mechanics. D. Reidel Publishing Co., Dordrecht-Boston (1981)

    Google Scholar 

  15. 15.

    Posilicano, A.: A Kreĭn-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183, 109–147 (2001)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Povzner, A.: A global existence theorem for a non-linear system and the defect index of a linear operator. (Russian) Sibirsk. Mat. Z.5 (1964), 377-386. English translation in American Mathematical Society Translations. Series 2, Vol. 51: 11 papers on differential equations, functional analysis and measure theory. American Mathematical Society, Providence, R.I. (1966)

  17. 17.

    Robert, D.: Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results. Helv. Phys. Acta 71, 44–116 (1998)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Schrödinger, E.: Der stetige Übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14, 664–666 (1926)

    Article  Google Scholar 

  19. 19.

    Teta, A.: A Mathematical Primer on Quantum Mechanics. Unitext for Physics. Springer, Cham (2018)

  20. 20.

    Thirring, W.: Classical Dynamical Systems. A Course in Mathematical Physics, vol. I. Springer, Berlin (1978)

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Andrea Posilicano.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors acknowledge the support of the National Group of Mathematical Physics (GNFM-INdAM)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cacciapuoti, C., Fermi, D. & Posilicano, A. The semi-classical limit with a delta potential. Annali di Matematica (2020).

Download citation


  • Semiclassical dynamics
  • Delta-interactions
  • Coherent states
  • Scattering theory

Mathematics Subject Classification

  • 81Q20
  • 81Q10
  • 47A40