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Semiclassical limits of the Schrödinger kernels on the \(h\)-Heisenberg group

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Abstract

We consider the Heisenberg group with the parameter \(h\) in the group multiplication, which tends to the Euclidean space when we take the limit as \(h\) tends to \(0\). We construct the Schrödinger kernels of the sub-Laplacian and the full Laplacian on the \(h\)-Heisenberg group and compute the limits of the Schrödinger kernels when \(h\) tends to \(0\).

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References

  1. Dasgupta, A., Wong, M.W.: Weyl transforms and the Heat Equation for the Sub-Laplacian on the Heisenberg Group. In: New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications, vol. 189, pp. 33–42. Birkhäuser, Basel (2009)

  2. Duan, X.: The heat kernel and Green function of the sub-Laplacian on the Heisenberg group. In: Pseudo-Differential Operators, Generalized Functions and Asymptotics, Operator Theory: Advances and Applications, vol. 231, pp. 55–75. Birkhäuser, Basel (2013)

  3. Greiner, P.C., Holcman, D., Kannai, Y.: Wave kernels related to second-order operators. Duke Math. J. 114(2), 329–386 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kagawa, T., Wong, M.W.: Semiclassical limits of heat kernels of Laplacians on the \(h\)-Heisenberg group. Math. Model. Nat. Phenom. 8, 132–142 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kagawa, T., Wong, M.W.: Weyl transforms and solutions to Schrödinger equations for time-dependent hermite operaters. J. Pseudo-Differ. Oper. Appl. 3(1), 31–48 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Molahajloo, S., Wong, M.W.: The Schrödinger kernel of the twisted Laplacian and cyclic models. Arch. Math (Basel) 95, 593–599 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Pauri, S., Ratnakumar, P.K., Thangavelu, S.: Analyticity of the Schrödinger propagator on the Heisenberg group. Monatsh. Math. 168(2), 279–303 (2012)

    Article  MathSciNet  Google Scholar 

  8. Saïd, S.B., Thangavelu, S.: Uniqueness of Solutions to the Schrödinger equation on the Heisenberg group, ArXiv:1006.5310 (2010)

  9. Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Birkhäuser, Basel (1998)

  10. Wong, M.W.: Weyl Transforms. Springer, New York (1998)

    MATH  Google Scholar 

  11. Wong, M.W.: Weyl transforms, the heat kernel and Green function of degenerate elliptic operator. Ann. Glob. Anal. Geom. 28, 271–283 (2005)

    Article  MATH  Google Scholar 

  12. Wong, M.W.: Weyl transforms and convolution operators on the Heisenberg group. In: Pseudo-Differential Operators and Related Topics, Operator Theory : Advances and Applications, vol. 164, pp. 115–120. Birkhäuser, Basel (2005)

  13. Wong, M.W.: The heat equation for the Hermite operator on the Heisenberg group. Hokkaido Math. J. 34, 383–404 (2005)

    Article  Google Scholar 

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Acknowledgments

The author would like to express my gratitude to Professor K. Furutani and Professor K. Yoshino for valuable advice on this work. The author is also grateful to reviewer for many suggestions.

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  1. Department of Mathematics, Tokyo University of Science, 2641 Yamazaki Noda, Chiba, 278-8510, Japan

    Toshinao Kagawa

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  1. Toshinao Kagawa
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Correspondence to Toshinao Kagawa.

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Kagawa, T. Semiclassical limits of the Schrödinger kernels on the \(h\)-Heisenberg group. J. Pseudo-Differ. Oper. Appl. 5, 1–25 (2014). https://doi.org/10.1007/s11868-013-0089-6

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  • DOI: https://doi.org/10.1007/s11868-013-0089-6

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