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Exponential estimates of solutions of pseudodifferential equations on the lattice \({(h \mathbb{Z})^{n}}\) : applications to the lattice Schrödinger and Dirac operators

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Abstract

The main aim of the paper is the application of the calculus of pseudodifferential operators on the lattice \({(h \mathbb{Z})^{n}=\{ x\in \mathbb{R}^{n}:x=hy,y\in\mathbb{Z}^{n}\}}\) depending on a small parameter h > 0 to exponential estimates of solutions of difference equations on \({( h\mathbb{Z})^{n}}\). As an application of general results we obtain the local exponential decreasing of eigenfunctions of Schrödinger and Dirac operators on the lattice \({(h\mathbb{Z})^{n}}\) for h → 0. These results can be considered as an analog of the well known tunnel effect for Schrödinger operators on \({\mathbb{R}^{n}}\).

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References

  1. Agmon S.: Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Princeton University Press, Princeton (1982)

    MATH  Google Scholar 

  2. Basu C., Roy C.L., Macía E., Domínguez-Adame F., Sánchez A.: Localization of relativistic electrons in a one-dimensional disordered system. J. Phys. A 27, 3285–3291 (1994)

    Article  Google Scholar 

  3. Bjorken S.D., Drell J.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1965)

    Google Scholar 

  4. Brüning J., Dobrokhotov S.Yu., Semenov E.S.: Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem. Regular Chaotic Dyn V. 11, 167–180 (2006)

    Article  MATH  Google Scholar 

  5. Dimassi M., Sjöstrand J.: Spectral Asymptotics in the Semi-Classical Limit. London Math. Soc, Lecture Note Se. 268. Cambridge Univesity Press, Cambridge (1999)

    Book  Google Scholar 

  6. Froese R., Herbst I.: Exponential bound and absence of positive eigenvalue for N-body Schrödinger operators. Comm. Math. Phys. 87, 429–447 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Froese R., Herbst I., Hoffman-Ostenhof M., Hoffman-Ostenhof T.: L 2− exponential lower bound of the solutions of the Schrödinger equation. Comm. Math. Phys. 87, 265–286 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Helffer B., Sjöstrand J.: Effect tunnel pour l’é quation de Schrödinger avec champ magnétique. Ann. della Scuola Normale Superiore di Pisa. Classe di Scienze 4 serie, t.14 4, 625–657 (1987)

    Google Scholar 

  9. Helffer B., Sjöstrand J.: Multipple wells in the semiclassical limit. Math. Nachr 124, 263–313 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Helffer, B.: Semi-classical analysis for the Schrö dinger operator and applications. In: Lecture Notes in Mathematics, vol. 1336. Springer, Berlin

  11. Klein M., Rosenberger E.: Harmonic approximation of difference operators. J. Funct. Anal. 257, 3409–3453 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Martinez, A.: Microlocal exponential estimates and application to tunneling. In: Rodino, L. (Ed.) Microlocal Analysis and Spectral Theory, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 490, pp. 349–376 (1996)

  13. Martinez A.: An Introduction to Semiclassical and Microlocal Analysis. Springer, New York (2002)

    MATH  Google Scholar 

  14. Maslov, V.P.: Global exponential asymptotic behavior of solutions of the tunnel equations and the problem of large deviations. In: International Conference on Analytical Methods in the Number Theory and Analysis, Moskow, 1981 (In Russian). Trudy Mat. Inst. Steklov., vol. 163, pp. 150–180 (1984)

  15. Maslov V.P., Fedoriuk M.V.: Semi-Classical Approximation in Quantum Mechanics. Kluwer, Dordericht (2002)

    Google Scholar 

  16. Mattis D.C.: The few-body problem on a lattice. Rev. Modern. Phys 58, 361–379 (1986)

    Article  MathSciNet  Google Scholar 

  17. Matte O.: Correlation asymptotics for non-translation invariant lattice spin systems. Math. Nachr. 281(5), 721–759 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mogilner, A.: Hamiltonians in solid state physics as multiparticle discrete Schrödinger operators: problems and results. Advances in Soviet Mathematics, vol. 5, AMS

  19. Nakamura S.: Agmon-type exponential decay estimates for pseudodifferential operators. J. Math. Sci. Univ. Tokyo 5, 693–712 (1998)

    MATH  MathSciNet  Google Scholar 

  20. de Oliveira C.R., Prado R.A.: Dynamical delocalization for the 1D Bernoulli discrete Dirac operator. J. Phys. A Math. Gen. 38, L115–L119 (2005)

    Article  MATH  Google Scholar 

  21. de Oliveira C.R., Prado R.A.: Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator. J. Math. Phys. 46(072105), 17 (2005)

    Google Scholar 

  22. Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and its Applications in the Operator Theory. In: Ser. Operator Theory: Advances and Applications, vol. 150. Birkhäuser (2004)

  23. Rabinovich V.S., Roch S.: Pseudodifference operators on weighted spaces, and applications to discrete Schrodinger operators. Acta Appl. Math. 84, 55–96 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rabinovich V.S., Roch S.: The essential spectrum of Schrödinger operators on lattice. J. Phys. A Math. Theor. 39, 8377–8394 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rabinovich V.S., Roch S.: Essential spectra of difference operators on Z n-periodic graphs. J. Phys. A Math. Theor. 40, 10109–10128V (2007) ISSN 1751-8113

    Article  MATH  MathSciNet  Google Scholar 

  26. Rabinovich V., Roch S.: Essential spectrum and exponential decay estimates of elliptic systems of partial differential equations, Applications to Schrödinger and Dirac operators. Georg. Math. J. 15(2), 1–19 (2008)

    MathSciNet  Google Scholar 

  27. Rabinovich, V.S., Roch, S.: Essential spectra of pseudodifferential operators and exponential decay of their solutions. Applications to Schrödinger Operators. In: Operator Algebras, Operator Theory and Applications, Ser. Operator Theory: Advances and Applications, vol. 181, pp. 335–384. (2008) ISBN: 978-3-7643-8683-2

  28. Rabinovich, V.S.: Pseudodifferential operators with analytic aymbols and some of its applications. Linear Topological Spaces and Complex Analysis 2, Metu-Tübitak, Ankara, pp. 79–98 (1995)

  29. Rabinovich V.: Pseudodifferential operators with analytic symbols and estimates for eigenfunctions of Schrödinger operators. Z. f. Anal. Anwend. (J. Anal. Appl.) 21(2), 351–370 (2002)

    MATH  MathSciNet  Google Scholar 

  30. Reed M., Simon B.: Methods of Modern Mathematical Physics III. Scattering Theory, Academic Press, London (1979)

    MATH  Google Scholar 

  31. Roy C.L.: Some special features of relativistic tunnelling through multi-barrier systems with function barriers. Phys. Lett. A 189, 345–350 (1994)

    Article  Google Scholar 

  32. Roy C.L., Basu C.: Relativistic study of electrical conduction in disordered systems. Phys. Rev. B 45, 14293–14301 (1992)

    Article  Google Scholar 

  33. Simon B.: Semiclassical analysis of low lying eigenvalues. II. Tunneling. Ann. Math. 120, 89–118 (1984)

    Article  Google Scholar 

  34. Stahlhofen A.A.: Supertransparent potentials for the Dirac equation. J. Phys. A 27, 8279–8290.1 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  35. Taylor M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  36. Thaller B.: The Dirac equation. Springer, Berlin (1991)

    MATH  Google Scholar 

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  1. National Polytechnic Institute, ESIME Zacatenco, Av. IPN, 07738, Mexico, D.F., Mexico

    V. Rabinovich

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  1. V. Rabinovich
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Correspondence to V. Rabinovich.

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The work was supported by the CONACYT project 81615, Mexico.

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Rabinovich, V. Exponential estimates of solutions of pseudodifferential equations on the lattice \({(h \mathbb{Z})^{n}}\) : applications to the lattice Schrödinger and Dirac operators. J. Pseudo-Differ. Oper. Appl. 1, 233–253 (2010). https://doi.org/10.1007/s11868-010-0005-2

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