We consider the quantum evolution $e^{- i \frac{t}{ℏ} H_{β}} ψ_{ξ}^{ℏ}$ of a Gaussian coherent state $ψ_{ξ}^{ℏ} \in L^{2} (ℝ)$ localized close to the classical state $ξ \equiv (q, p) \in ℝ^{2}$ , where $H_{β}$ denotes a self-adjoint realization of the formal Hamiltonian $- \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} + β δ_{0}^{'}$ , with $δ_{0}^{'}$ the derivative of Dirac’s delta distribution at $x = 0$ and $β$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (with respect to the $L^{2} (ℝ)$ -norm, uniformly for any $t \in ℝ$ away from the collision time) by $e^{\frac{i}{ℏ} A_{t}} e^{i t L_{B}} ϕ_{x}^{ℏ}$ , where $A_{t} = \frac{p^{2} t}{2 m}$ , $ϕ_{x}^{ℏ} (ξ) : = ψ_{ξ}^{ℏ} (x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $𝒞_{c}^{\infty} (ℳ_{0})$ , $ℳ_{0} : = {(q, p) \in ℝ^{2} | q \neq 0}$ , of ( $- i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi and A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453–489], in the present case the approximation gives a smaller error: it is of order $ℏ^{7 / 2 - λ}$ , $0 < λ < 1 / 2$ , whereas it turns out to be of order $ℏ^{3 / 2 - λ}$ , $0 < λ < 3 / 2$ , for the delta potential. We also provide similar approximation results for both the wave and scattering operators.

Keywords:

AMSC: 81Q20, 81Q10, 47A40

References

1. S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edn. (AMS Chelsea Publishing, Providence, RI, 2005), with an appendix by Pavel Exner. Google Scholar
2. S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators (Cambridge University Press, Cambridge, 2000). Crossref, Google Scholar
3. C. Cacciapuoti and D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453–489. Crossref, Google Scholar
4. C. Cacciapuoti, D. Fermi and A. Posilicano, The semiclassical limit on a star-graph with Kirchhoff conditions, Anal. Math. Phys. 11 (2021) Paper No. 45, 43 pp. Crossref, Google Scholar
5. M. Combescure and D. Robert, Coherent States and Applications in Mathematical Physics (Springer, The Netherlands, 2012). Crossref, Google Scholar
6. T. A. Filatova and A. I. Shafarevich, Semiclassical spectral series of the Schrödinger operator with a delta potential on a straight line and on a sphere, Theor. Math. Phys. 164 (2010) 1064–1080. Crossref, Google Scholar
7. Y. Golovaty, Two-parametric $δ^{'}$ -interactions: Approximation by Schrödinger operators with localized rank-two perturbations, J. Phys. A: Math. Theor. 51(25) (2018) 255202. Crossref, ADS, Google Scholar
8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edn. (Elsevier/Academic Press, Amsterdam, 2007). Google Scholar
9. G. A. Hagedorn, Semiclassical quantum mechanics I. The $ℏ \to 0$ limit for coherent states, Comm. Math. Phys. 71 (1980) 77–93. Crossref, ISI, ADS, Google Scholar
10. T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1976). Crossref, Google Scholar
11. A. Kostenko and M. Malamud, 1-D Schrödinger operators with local point interactions: A review, in Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday, eds. H. HoldenB. SimonG. Teschl, Proceedings of Symposia in Pure Mathematics, Vol. 87 (American Mathematical Society, Providence, RI, 2013), pp. 235–262. Crossref, Google Scholar
12. A. Mantile, A. Posilicano and M. Sini, Limiting absorption principle, generalized eigenfunctions and scattering matrix for Laplace operators with boundary conditions on hypersurfaces, J. Spectr. Theory 8 (2018) 1443–1486. Crossref, ISI, Google Scholar
13. T. Ohsawa, Approximation of semiclassical expectation values by symplectic Gaussian wave packet dynamics, Lett. Math. Phys. 111(121) (2021) 26 pp. Google Scholar
14. F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010). Google Scholar
15. A. Posilicano, A Kreĭn-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal. 183 (2001) 109–147. Crossref, Google Scholar
16. T. S. Ratiu, T. A. Filatova and A. I. Shafarevich, Noncompact Lagrangian manifolds corresponding to the spectral series of the Schrödinger operator with delta-potential on a surface of revolution, Dokl. Math. 86 (2012) 694–696. Crossref, Google Scholar
17. T. S. Ratiu, A. A. Suleimanova and A. I. Shafarevich, Spectral series of the Schrödinger operator with delta-potential on a three-dimensional spherically symmetric manifold, Russ. J. Math. Phys. 20 (2013) 326–335. Crossref, Google Scholar
18. D. Robert, Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results, Helv. Phys. Acta 71 (1998) 44–116. Google Scholar
19. A. I. Shafarevich and O. A. Shchegortsova, Semiclassical asymptotics of the solution to the Cauchy problem for the Schrödinger equation with a Delta potential localized on a codimension 1 surface, Proc. Steklov Inst. Math. 310 (2020) 304–313. Crossref, Google Scholar
20. W. Thirring, Classical Dynamical Systems. A Course in Mathematical Physics, Vol. I (Springer-Verlag, 1978). Crossref, Google Scholar

Vol. 34, No. 06

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History

Received 20 April 2021

Revised 11 February 2022

Accepted 13 February 2022

Published: 31 March 2022

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The semi-classical limit with a delta-prime potential

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