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Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

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Abstract

We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers–Fokker–Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.

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References

  1. Boutet de Monvel, L.: Hypoelliptic operators with double characteristics and related pseudo-differential operators. Commun. Pure Appl. Math. 27, 585–639 (1974)

    Google Scholar 

  2. Caliceti, E., Graffi, S., Hitrik, M., Sjöstrand, J.: Quadratic PT-symmetric operators with real spectrum and similarity to self-adjoint operators. J. Phys. A Math. Theor. (to appear)

  3. Davies, E.B.: Semi-classical states for non-self-adjoint Schrödinger operators. Commun. Math. Phys. 200(1), 35–41 (1999)

    Article  MATH  Google Scholar 

  4. Davies, E.B., Kuijlaars, A.B.J.: Spectral asymptotics of the non-self-adjoint harmonic oscillator. J. Lond. Math. Soc. 70(2), 420–426 (2004)

    Google Scholar 

  5. Dencker, N., Sjöstrand, J., Zworski, M.: Pseudospectra of semiclassical (pseudo-) differential operators. Commun. Pure Appl. Math. 57(3), 384–415 (2004)

    Article  MATH  Google Scholar 

  6. Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society. Lecture Note Series, 268, pp. xii+227. Cambridge University Press, Cambridge (1999). ISBN: 0-521-66544-2

  7. Gohberg, I.C., Goldberg, S., Kaashoek, M.A.: Classes of linear operators, vol. I. In: Operator Theory: Advances and Applications, vol. 49. Birkhäuser Verlag, Basel (1990)

  8. Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker–Planck operators and Witten Laplacians. Lecture Notes in Mathematics, vol. 1862. Springer, Berlin (2005)

    Google Scholar 

  9. Hérau, F., Sjöstrand, J., Stolk, C.C.: Semiclassical analysis for the Kramers–Fokker–Planck equation. Commun. Partial Differ. Equ. 30(4–6), 689–760 (2005)

    Article  MATH  Google Scholar 

  10. Hislop, P.D., Sigal, I.M.: Introduction to Spectral Theory: With applications to Schrödinger operators. Applied Mathematical Sciences, vol. 113. Springer, New York (1996)

  11. Hitrik, M., Pravda-Starov, K.: Spectra and semigroup smoothing for non-elliptic quadratic operators. Math. Ann. 344(4), 801–846 (2009)

    Google Scholar 

  12. Hitrik, M., Pravda-Starov, K.: Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics. Commun. Partial Differ. Equ. 35(6), 988–1028 (2010)

    Google Scholar 

  13. Hitrik, M., Pravda-Starov, K.: Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics. Ann. Inst. Fourier (to appear)

  14. Hitrik, M., Sjöstrand, J., Viola, J.: Resolvent estimates for elliptic quadratic differential operators. Anal. PDE (to appear)

  15. Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Classics in Mathematics. Springer, Berlin (2007) (Pseudo-differential operators, Reprint of the 1994 edition)

  16. Hörmander, L.: On the Legendre and Laplace transformations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(3–4), 517–568 (1998) (1997, Dedicated to Ennio De Giorgi)

  17. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990) (Corrected reprint of the 1985 original)

  18. Kaidi, Nourredine, Kerdelhué, Philippe: Forme normale de Birkhoff et résonances. Asymptot. Anal. 23(1), 1–21 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)

    Article  MathSciNet  MATH  Google Scholar 

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

  21. Miller, Peter D.: Applied Asymptotic Analysis. Graduate Studies in Mathematics, vol. 75. American Mathematical Society, Providence, RI (2006)

    Google Scholar 

  22. Ottobre, M., Pavliotis, G., Pravda-Starov, K.: Exponential return to equilibrium for hypoelliptic quadratic systems. J. Funct. Anal. 262(9), 4000–4039 (2012)

    Google Scholar 

  23. Pravda-Starov, K.: Contraction semigroups of elliptic quadratic differential operators. Math. Z. 259(2), 363–391 (2008)

    Google Scholar 

  24. Pravda-Starov, K.: On the pseudospectrum of elliptic quadratic differential operators. Duke Math. J. 145(2), 249–279 (2008)

    Google Scholar 

  25. Pravda-Starov, K.: Subelliptic estimates for quadratic differential operators. Am. J. Math. 133(1), 39–89 (2011)

    Google Scholar 

  26. Risken, H.: The Fokker-Planck equation. Springer Series in Synergetics, vol. 18, 2nd edn. Springer, Berlin (1989) (Methods of solution and applications)

  27. Sjöstrand, J.: Function spaces associated to global I-Lagrangian manifolds. In: Structure of Solutions of Differential Equations (Katata/Kyoto 1995), pp. 369–423. World Scientific Publishing, River Edge, NJ (1996)

  28. Sjöstrand, J.: Lectures on resonances (2002). http://www.math.polytechnique.fr/~sjoestrand/CoursgbgWeb.pdf

  29. Sjöstrand, Johannes: Parametrices for pseudodifferential operators with multiple characteristics. Ark. Mat. 12, 85–130 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. Princeton University Press, Princeton (2005)

  31. Viola, J.: Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators. Int. Math. Res. Not. (2012). doi:10.1093/imrn/rns188

  32. Zworski, M.: A remark on a paper of E. B Davies: “Semi-classical states for non-self-adjoint Schrödinger operators” [Commun. Math. Phys. 200(1), 35–41 (1999); MR 1671904 (99m:34197)]. Proc. Am. Math. Soc. 129(10), 2955–2957 (2001, electronic)

  33. Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence, RI (2012)

    Google Scholar 

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Acknowledgments

The author would like to thank Michael Hitrik for helpful discussions at the outset of this work. He would also like to thank Frédéric Hérau for helpful suggestions and Fredrik Andersson for a valuable discussion contributing to the proof of Proposition 4.1. Finally, the author would like to thank the referee for a careful reading and many helpful corrections and suggestions.

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  1. Laboratoire de Mathématiques Jean Leray, Université de Nantes, BP 92208, 2 rue de la Houssinière, 44322 , Nantes Cedex 3, France

    Joe Viola

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Viola, J. Spectral projections and resolvent bounds for partially elliptic quadratic differential operators. J. Pseudo-Differ. Oper. Appl. 4, 145–221 (2013). https://doi.org/10.1007/s11868-013-0066-0

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