The semi-classical limit with a delta potential

Abstract

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.

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Notes

  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 \(|\xi | \le 1\).

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Correspondence to Andrea Posilicano.

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

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Cacciapuoti, C., Fermi, D. & Posilicano, A. The semi-classical limit with a delta potential. Annali di Matematica (2020). https://doi.org/10.1007/s10231-020-01002-4

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Keywords

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

Mathematics Subject Classification

  • 81Q20
  • 81Q10
  • 47A40