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Scattering Resonances as Viscosity Limits

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Algebraic and Analytic Microlocal Analysis (AAMA 2013)

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Abstract

Using the method of complex scaling we show that scattering resonances of \( - \Delta + V \), \( V \in L^\infty _\mathrm{{c}} ( {\mathbb {R}}^n ) \), are limits of eigenvalues of \( - \Delta + V - i \varepsilon x^2 \) as \( \varepsilon \rightarrow 0+\). That justifies a method proposed in computational chemistry and reflects a general principle for resonances in other settings.

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References

  1. Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971)

    Article  Google Scholar 

  2. Balslev, E., Combes, J.M.: Spectral properties of many-body Schrödinger operators wth dilation analytic interactions. Commun. Math. Phys. 22, 280–294 (1971)

    Article  Google Scholar 

  3. Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    Article  MathSciNet  Google Scholar 

  4. Bindel, D., Zworski, M.: Theory and computation of resonances in 1D scattering. Online presentation, Including MATLAB codes. http://www.cims.nyu.edu/~dbindel/resonant1d

  5. Datchev, K., Vasy, A.: Gluing semiclassical resolvent estimates via propagation of singularities. IMRN 23, 5409–5443 (2012)

    Article  MathSciNet  Google Scholar 

  6. Datchev, K., Vasy, A.: Propagation through trapped sets and semiclassical resolvent estimates. Annales de l’Institut Fourier 62, 2379–2384 (2012)

    Article  MathSciNet  Google Scholar 

  7. Davies, E.B.: Pseudospectra, the harmonic oscillator and complex resonances. Proc. R. Soc. Lond. A 455, 585–599 (1999)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Dyatlov, S., Zworski, M.: Stochastic stability of Pollicott–Ruelle resonances. Preprint. arXiv:1407.8531

  10. Dyatlov, S., Zworski, M.: Mathematical theory of scattering resonances. Book in preparation. http://math.mit.edu/~dyatlov/res/res.pdf

  11. Helffer, B., Sjöstrand, J.: Resonances en limite semiclassique. Bull. Soc. Math. France 114(24–25) (1986)

    Google Scholar 

  12. Hitrik, M., Sjöstrand, J., Viola, J.: Resolvent Estimates for elliptic quadratic differential operators. Anal. PDE 6, 181–196 (2013)

    Article  MathSciNet  Google Scholar 

  13. Jagau, T.-C., Zuev, D., Bravaya, K.B., Epifanovsky, E., Krylov, A.I.: A fresh look at resonances and complex absorbing potentials: density matrix-based approach. J. Phys. Chem. Lett. 5, 310–315 (2014)

    Article  Google Scholar 

  14. Lebeau, G.: Fonctions harmoniques et spectre singulier. Ann. Sci. École Norm. Sup. 13, 269–291 (1980)

    Article  MathSciNet  Google Scholar 

  15. Martinez, A.: Prolongement des solution holomorphe des problèmes aux limits. Ann. Inst. Fourier 35, 93–116 (1985)

    Article  Google Scholar 

  16. Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering. Acta Math. 203, 149–233 (2009)

    Article  MathSciNet  Google Scholar 

  17. Nonnenmacher, S., Zworski, M.: Decay of correlations for normally hyperbolic trapping. Invent. Math. 200 (2015, to appear)

    Article  MathSciNet  Google Scholar 

  18. Reinhardt, W.P.: Complex scaling in atomic and molecular physics. In and out of external fields. AMS Proceedings Series: Proceedings of Symposia in Pure Mathematics, vol. 76, pp. 357–377 (2007)

    Google Scholar 

  19. Riss, U.V., Meyer, H.D.: Reflection-free complex absorbing potentials. J. Phys. B 28, 1475–1493 (1995)

    Article  Google Scholar 

  20. Seideman, T., Miller, W.H.: Calculation of the cumulative reaction probability via a discrete variable representation with absorbing boundary conditions. J. Chem. Phys. 96, 4412–4422 (1992)

    Article  Google Scholar 

  21. Simon, B.: The definition of molecular resonance curves by the method of exterior complex scaling. Phys. Lett. A 71, 211–214 (1979)

    Article  Google Scholar 

  22. Sjöstrand, J.: Singularités analytiques microlocales. Astérisque, vol. 95 (1982)

    Google Scholar 

  23. Sjöstrand, J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60, 1–57 (1990)

    Article  MathSciNet  Google Scholar 

  24. Sjöstrand, J.: Lectures on resonances, version préliminaire, printemps 2002

    Google Scholar 

  25. Sjöstrand, J., Zworski, M.: Complex scaling and the distribution of scattering poles. J. Am. Math. Soc. 4, 729–769 (1991)

    Article  MathSciNet  Google Scholar 

  26. Stefanov, P.: Approximating resonances with the complex absorbing potential method. Commun. Partial Differ. Equ. 30, 1843–1862 (2005)

    Article  MathSciNet  Google Scholar 

  27. Vasy, A.: Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, with an appendix by Semyon Dyatlov. Invent. Math. 194, 381–513 (2013). arXiv:1012.4391

    Article  MathSciNet  Google Scholar 

  28. Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. AMS, Providence (2012)

    Google Scholar 

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Acknowledgements

The author would like to thank Mike Christ, Semyon Dyatlov, Jeff Galkowski, John Strain and Joe Viola for helpful discussions. I am also grateful to the anonymous referee for the careful reading of the first version and for the valuable comments. This project was supported in part by the National Science Foundation grant DMS-1201417.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Maciej Zworski

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  1. Maciej Zworski
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Correspondence to Maciej Zworski .

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Editors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, USA

    Michael Hitrik

  2. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Dmitry Tamarkin

  3. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Boris Tsygan

  4. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Steve Zelditch

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Zworski, M. (2018). Scattering Resonances as Viscosity Limits. In: Hitrik, M., Tamarkin, D., Tsygan, B., Zelditch, S. (eds) Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proceedings in Mathematics & Statistics, vol 269. Springer, Cham. https://doi.org/10.1007/978-3-030-01588-6_14

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