Tunisian Journal of Mathematics

Semiclassical approximation of the magnetic Schrödinger operator on a strip: dynamics and spectrum

Mouez Dimassi

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In the semiclassical regime (i.e., ϵ0), we study the effect of a slowly varying potential V(ϵt,ϵz) on the magnetic Schrödinger operator P=Dx2+(Dz+μx)2 on a strip [a,a]×z. The potential V(t,z) is assumed to be smooth. We derive the semiclassical dynamics and we describe the asymptotic structure of the spectrum and the resonances of the operator P+V(ϵt,ϵz) for ϵ small enough. All our results depend on the eigenvalues corresponding to Dx2+(μx+k)2 on L2([a,a]) with Dirichlet boundary condition.

Article information

Tunisian J. Math., Volume 2, Number 1 (2020), 197-215.

Received: 10 October 2018
Revised: 17 November 2018
Accepted: 2 December 2018
First available in Project Euclid: 2 April 2019

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Zentralblatt MATH identifier

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47N50: Applications in the physical sciences 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 81Q15: Perturbation theories for operators and differential equations

semiclassical analysis periodic Schrödinger operator Bohr–Sommerfeld quantization spectral shift function asymptotic expansions limiting absorption theorem


Dimassi, Mouez. Semiclassical approximation of the magnetic Schrödinger operator on a strip: dynamics and spectrum. Tunisian J. Math. 2 (2020), no. 1, 197--215. doi:10.2140/tunis.2020.2.197. https://projecteuclid.org/euclid.tunis/1554170465

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  • V. Bonnaillie-Noël, F. Hérau, and N. Raymond, “Magnetic WKB constructions”, Arch. Ration. Mech. Anal. 221:2 (2016), 817–891.
  • J.-F. Bony, V. Bruneau, P. Briet, and G. Raikov, “Resonances and SSF singularities for magnetic Schrödinger operators”, Cubo 11:5 (2009), 23–38.
  • P. Briet, G. Raikov, and E. Soccorsi, “Spectral properties of a magnetic quantum Hamiltonian on a strip”, Asymptot. Anal. 58:3 (2008), 127–155.
  • P. Briet, P. D. Hislop, G. Raikov, and E. Soccorsi, “Mourre estimates for a 2D magnetic quantum Hamiltonian on strip-like domains”, pp. 33–46 in Spectral and scattering theory for quantum magnetic systems, edited by P. Briet et al., Contemp. Math. 500, Amer. Math. Soc., Providence, RI, 2009.
  • J. Brüning, S. Y. Dobrokhotov, and K. V. Pankrashkin, “The spectral asymptotics of the two-dimensional Schrödinger operator with a strong magnetic field, II”, Russ. J. Math. Phys. 9:4 (2002), 400–416.
  • S. De Bièvre and J. V. Pulé, “Propagating edge states for a magnetic Hamiltonian”, Math. Phys. Electron. J. 5 (1999), Paper 3.
  • M. Dimassi, “Développements asymptotiques des perturbations lentes de l'opérateur de Schrödinger périodique”, Comm. Partial Differential Equations 18:5-6 (1993), 771–803.
  • M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999.
  • S. Fournais and B. Helffer, Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications 77, Birkhäuser, Boston, 2010.
  • C. Gérard and I. Łaba, Multiparticle quantum scattering in constant magnetic fields, Mathematical Surveys and Monographs 90, Amer. Math. Soc., Providence, RI, 2002.
  • V. A. Geĭler and M. M. Senatorov, “The structure of the spectrum of the Schrödinger operator with a magnetic field in a strip, and finite-gap potentials”, Mat. Sb. 188:5 (1997), 21–32. In Russian; translated in Sb. Math. 188:05 (1997), 657–669.
  • B. Helffer and A. Martinez, “Comparaison entre les diverses notions de résonances”, Helv. Phys. Acta 60:8 (1987), 992–1003.
  • P. D. Hislop and I. M. Sigal, Introduction to spectral theory: with applications to Schrödinger operators, Applied Mathematical Sciences 113, Springer, 1996.
  • L. Hörmander, “Fourier integral operators, I”, Acta Math. 127:1-2 (1971), 79–183.
  • V. Ivrii, “Microlocal Analysis, Sharp spectral Asymptotics and Applications”, research monograph, 2018, http://www.math.toronto.edu/ivrii/monsterbook.pdf.
  • T. Kato, Perturbation theory for linear operators, Grundlehren der Math. Wissenschaften 132, Springer, 1966.
  • J. B. Keller, “Corrected Bohr–Sommerfeld quantum conditions for nonseparable systems”, Ann. Physics 4 (1958), 180–188.
  • V. A. Marchenko, Sturm–Liouville operators and applications, Operator Theory: Advances and Applications 22, Birkhäuser, Basel, 1986.
  • A. Martinez, “Résonances dans l'approximation de Born–Oppenheimer, I”, J. Differential Equations 91:2 (1991), 204–234.
  • A. Martinez, “Résonances dans l'approximation de Born–Oppenheimer, II: Largeur des résonances”, Comm. Math. Phys. 135:3 (1991), 517–530.
  • V. P. Maslov and M. V. Fedoriuk, Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics 7, Reidel, Dordrecht, Netherlands, 1981.
  • M. Reed and B. Simon, Methods of modern mathematical physics, IV: Analysis of operators, Academic, New York, 1978.
  • D. Spehner, R. Narevich, and E. Akkermans, “Semiclassical spectrum of integrable systems in a magnetic field”, Journal of Physics A: Mathematical and General 31:30 (1998), 6531–6545.
  • O. Viehweger, W. Pook, M. Janßen, and J. Hajdu, “Note on the quantum Hall Hamiltonian in cylinder geometry”, Z. Phys. B 78:1 (1990), 11–16.