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Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs

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Abstract

We prove a Feynman path integral formula for the unitary group exp(—itLυ,θ), t ≥ 0, associated with a discrete magnetic Schrödinger operator Lυ,θ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate \(\left| {\exp \left({- it{L_{v,\theta}}} \right)\left({x,y} \right)} \right| \le \exp \left({- t{L_{- \deg ,0}}} \right)\left({x,y} \right),\) which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential deg is the weighted degree function of the graph.

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References

  1. A. Avila and S. Jitomirskaya, The Ten Martini Problem, Ann. of Math. (2) 170 (2009), 303–342.

    Article  MathSciNet  Google Scholar 

  2. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators,Birkhäuser Boston, Boston, MA, 1990.

    Book  Google Scholar 

  3. Z.-Q. Chen, P. Kim and T. Kumagai, Discrete approximation of symmetric jump processes on metric measure spaces, Probab. Theory Related Fields 155 (2013), 703–749.

    Article  MathSciNet  Google Scholar 

  4. M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 2011.

    MATH  Google Scholar 

  5. S. Golénia, Unboundedness of adjacency matrices of locally finite graphs, Lett. Math. Phys. 93 (2010), 127–140.

    Article  MathSciNet  Google Scholar 

  6. B. Güneysu, On generalized Schrödinger semigroups, J. Funct. Anal. 262 (2012), 4639–4674.

    Article  MathSciNet  Google Scholar 

  7. B. Güneysu, Semiclassical limits of quantum partition functions on infinite graphs, J. Math. Phys. 56 (2015), 022102.

    Article  MathSciNet  Google Scholar 

  8. B. Güneysu, M. Keller and M. Schmidt, A Feynman-Kac-Itô formula for magnetic Schrödinger operators on graphs, Probab. Theory Related Fields 165 (2016), 365–399.

    Article  MathSciNet  Google Scholar 

  9. B. Güneysu, O. Milatovic and F Truc, Generalized Schrödinger semigroups on infinite graphs, Potential Anal. 41 (2014), 517–541.

    Article  MathSciNet  Google Scholar 

  10. W. Hackenbroch and A. Thalmaier, Stochastische Analysis, B. G. Teubner, Stuttgart, 1994.

    Book  Google Scholar 

  11. S. Haeseler, M. Keller, D. Lenz and R. K. Wojciechowski, Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions, J. Spectr. Theory 2 (2012), 397–432.

    Article  MathSciNet  Google Scholar 

  12. P. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. A 68 (1955), 874–878.

    Article  Google Scholar 

  13. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White Noise, Kluwer Academic, Dordrecht, 1993.

    Book  Google Scholar 

  14. E. Korotyaev and J. S. Møller, Weighted estimates for the discrete Laplacian on the cubic lattice, Ark. Mat. 57 (2017).

  15. E. Korotyaev and N. Saburova, Magnetic Schrödinger operators on periodic discrete graphs, J. Funct. Anal. 272 (2017), 1625–1660.

    Article  MathSciNet  Google Scholar 

  16. O. Milatovic, Essential self-adjointness of discrete magnetic Schrödinger operators on locally finite graphs, Integral Equations Operator Theory 71 (2011), 13–27.

    Article  MathSciNet  Google Scholar 

  17. O. Milatovic, A Sears-type self-adjointness result for discrete magnetic Schrödinger operators, J. Math. Anal. Appl. 396 (2012) 801–809.

    Article  MathSciNet  Google Scholar 

  18. M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1980.

    MATH  Google Scholar 

  19. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier analysis, Self-Adjointness, Academic Press, New York, 1980.

    MATH  Google Scholar 

  20. B. Simon, Functional Integration and Quantum Physics, American Mathematical Society, Chelsea Publishing, Providence, RI, 2005.

  21. T. Sunada, A discrete analogue of periodic magnetic Schrödinger operators, in Geometry of the Spectrum, American Mathematical Society, Providence, RI, 1994, pp. 283–299.

    Chapter  Google Scholar 

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Acknowledgements

The authors would like to thank Burkhard Eden, Evgeny Korotyaev, Ognjen Milatovic and Matthias Staudacher for very helpful discussions.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, 12489, Berlin, Germany

    Batu Güneysu

  2. Institut für Mathematik, Universität Potsdam, 14476, Potsdam, Germany

    Matthias Keller

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  1. Batu Güneysu
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  2. Matthias Keller
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Correspondence to Batu Güneysu.

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Güneysu, B., Keller, M. Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs. JAMA 141, 751–770 (2020). https://doi.org/10.1007/s11854-020-0110-y

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  • DOI: https://doi.org/10.1007/s11854-020-0110-y

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