Skip to main content
Log in

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We prove the absolute continuity of the spectrum of the Schrödinger operator in \(L^2 ({\mathbb{R}}^n )\), \(n \geqslant 3\), with periodic (with a common period lattice \(\Lambda\)) scalar \(V\) and vector \(A \in C^1 ({\mathbb{R}}^n ,{\mathbb{R}}^n )\) potentials for which either \(\user1{A} \in H_{\user2{loc}}^\user1{q} \user2{(}\mathbb{R}^\user1{n} \user2{;}\mathbb{R}^\user1{n} \user2{)}\), \(2q > n - 2\), or the Fourier series of the vector potential \(A\) converges absolutely, \(V \in L_w^{p(n)} (K)\), where \(K\) is an elementary cell of the lattice \(\Lambda\), \(p(n) = n/2\) for \(n = 3, 4, 5, 6\), and \(p(n) = n - 3\) for \(n \geqslant 7\), and the value of \(\user2{lim}_{\user1{t} \to \user2{ + }\infty } \left\| {\theta _\user1{t} V} \right\|_{\user1{L}_\user1{w}^{\user1{p}\user2{(}\user1{n}\user2{)}} \user2{(}\user1{K}\user2{)}} \) is sufficiently small, where \(\theta _\user1{t} \user2{(}\user1{x}\user2{) = 0 if }\left| {V\user2{(}\user1{x}\user2{)}} \right| \leqslant \user1{t}\) and \(\theta _t (x) = 1\) otherwise, \(x \in K\), and \(t > 0\).

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

REFERENCES

  1. L. E. Thomas, “Time dependent approach to scattering from impurities in a crystal,” Comm. Math. Phys., 33 (1973), 335–343.

    Google Scholar 

  2. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 4. Analysis of Operators, Academic Press, New York, 1979.

    Google Scholar 

  3. M. Sh. Birman and T. A. Suslina, “Absolute continuity of a two-dimensional periodic magnetic Hamiltonian with a discontinuous vector potential,” Algebra i Analiz [St. Petersburg Math. J.], 10 (1998), no. 4, 1–36.

    Google Scholar 

  4. A. Sobolev, Absolute Continuity of a Periodic Magnetic Schrödinger Operator, Res. Report no. 97/06, Univ. of Sussex, 1997, Preprint ESI no. 495 (1997), Wien.

  5. M. Sh. Birman and T. A. Suslina, “Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity,” Algebra i Analiz [St. Petersburg Math. J.], 11 (1999), no. 2, 1–40.

    Google Scholar 

  6. P. Kuchment and S. Levendorskii, “On absolute continuity of spectra of periodic elliptic operators,” Oper. Theory Adv. Appl., 108 (1999), 291–297.

    Google Scholar 

  7. Z. Shen, On Absolute Continuity of Periodic Schrödinger Operators, Preprint ESI no. 597, The Erwin Schrödinger Internat. Institute for Math. Phys., Wien, 1998, Preprint no. 99-189 (1999).

    Google Scholar 

  8. A. Morame, “Absence of singular spectrum for a perturbation of a two-dimensional Laplace–Beltrami operator with periodic electro-magnetic potential,” J. Phys. A:Math. Gen., 31 (1998), 7593–7601.

    Google Scholar 

  9. A. Morame, The Absolute Continuity of the Spectrum of Maxwell Operator in Periodic Media, Preprint no. 99-308, Texas Math. Physics Archive, 1999.

  10. L. I. Danilov, “Estimates of the resolvent and of the spectrum of the Dirac operator with a periodic potential,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 103 (1995), no. 1, 3–22.

    Google Scholar 

  11. L. I. Danilov, “Absolute continuity of the spectrum of a periodic Dirac operator,”Differentsial'nye Uravneniya [Differential Equations], 36 (2000), no. 2, 233–240.

    Google Scholar 

  12. L. I. Danilov, “On the spectrum of a two-dimensional periodic Dirac operator,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 118 (1999), no. 1, 3–14.

    Google Scholar 

  13. L. I. Danilov, The Spectrum of the Dirac Operator with Periodic Potential. III [in Russian] (Manuscript deposited at VINITI on July 10, 1992, deposition no. 2252-B92), Phys.-Tech. Inst., The Ural Division of the Russian Academy of Sciences, Izhevsk, 1992.

    Google Scholar 

  14. M. Sh. Birman and T. A. Suslina, The Periodic Dirac Operator is Absolutely Continuous, Preprint ESI no. 603, Erwin Schrödinger Internat. Institute Math. Phys., Wien, 1998.

    Google Scholar 

  15. L. I. Danilov, “On the spectrum of a periodic Dirac operator,Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 124 (2000), no. 1, 3–17.

    Google Scholar 

  16. L. I. Danilov, On the Absolute Continuity of the Spectra of Periodic Schrödinger and Dirac Operators. I [in Russian], (Manuscript deposited at VINITI on June 15, 2000, deposition no. 1683-B00), Phys.-Tech. Inst., The Ural Division of the Russian Academy of Sciences, Izhevsk, 2000.

    Google Scholar 

  17. P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel, 1993.

    Google Scholar 

  18. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 2. Fourier Analysis and Self-Adjointness, Academic Press, New York, 1975.

    Google Scholar 

  19. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton (USA), 1970.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Danilov, L.I. Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator. Mathematical Notes 73, 46–57 (2003). https://doi.org/10.1023/A:1022169916738

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022169916738

Navigation