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Resonance Asymptotics

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Spectral Theory of Infinite-Area Hyperbolic Surfaces

Part of the book series: Progress in Mathematics ((PM,volume 318))

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Abstract

One implication of the trace formula of Corollary 11.5 is immediately clear: for a non-elementary geometrically finite hyperbolic surface the set \(\mathcal{R}_{X}\) must be infinite, to account for the singularities on the right-hand side of (11.20). The trace formula contains much more information on the distribution of resonances, and in this chapter we will discuss methods by which this information can be extracted.

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References

  1. Borthwick, D.: Upper and lower bounds on resonances for manifolds hyperbolic near infinity. Commun. PDE 33, 1507–1539 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borthwick, D.: Sharp upper bounds on resonances for perturbations of hyperbolic space. Asymptot. Anal. 69, 45–85 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Borthwick, D.: Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends. Anal. PDE 5, 513–552 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borthwick, D., Philipp, P.: Resonance asymptotics for asymptotically hyperbolic manifolds with warped-product ends. Asymptot. Anal. 90, 281–323 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Bruneau, V., Petkov, V.: Meromorphic continuation of the spectral shift function. Duke Math. J. 116, 389–430 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bunke, U., Olbrich, M.: Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. Ann. Math. 149, 627–689 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carron, G.: Déterminant relatif et la fonction Xi. Am. J. Math. 124, 307–352 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Christiansen, T.: Weyl asymptotics for the Laplacian on asymptotically Euclidean spaces. Am. J. Math. 121, 1–22 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Christiansen, T.: Weyl asymptotics for the Laplacian on manifolds with asymptotically cusp ends. J. Funct. Anal. 187, 211–226 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Christiansen, T., Zworski, M.: Spectral asymptotics for manifolds with cylindrical ends. Ann. Inst. Fourier (Grenoble) 45, 251–263 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dyatlov, S., Guillarmou, C.: Scattering phase asymptotics with fractal remainders. Commun. Math. Phys. 324, 425–444 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Froese, R.: Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions. Can. J. Math. 50, 538–546 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guillarmou, C.: Generalized Krein formula, determinants, and Selberg zeta function in even dimension. Am. J. Math. 131, 1359–1417 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guillopé, L., Zworski, M.: Scattering asymptotics for Riemann surfaces. Ann. Math. 145, 597–660 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guillopé, L., Zworski, M.: The wave trace for Riemann surfaces. Geom. Funct. Anal. 9, 1156–1168 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hörmander, L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Springer, Berlin (1994). Corrected reprint of the 1985 original

    Google Scholar 

  18. Melrose, R.B.: Weyl asymptotics for the phase in obstacle scattering. Commun. PDE 13, 1431–1439 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Melrose, R.B.: Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. In: Spectral and Scattering Theory (Sanda, 1992). Lecture Notes in Pure and Applied Mathematics, vol. 161, pp. 85–130. Dekker, New York (1994)

    Google Scholar 

  20. Müller, W.: Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109, 265–305 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nedelec, L.: Asymptotique du nombre de résonances de l’opérateur de Schrödinger avec potentiel linéaire et matriciel. Math. Res. Lett. 4, 309–320 (1997)

    Article  MathSciNet  Google Scholar 

  22. Olver, F.W.J.: Asymptotics and Special Functions. Academic, New York/London (1974)

    MATH  Google Scholar 

  23. Parnovski, L.B.: Spectral asymptotics of Laplace operators on surfaces with cusps. Math. Ann. 303, 281–296 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Parnovski, L.B.: Spectral asymptotics of the Laplace operator on manifolds with cylindrical ends. Int. J. Math. 6, 911–920 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Patterson, S.J., Perry, P.A.: The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J. 106, 321–390 (2001). Appendix A by Charles Epstein

    Google Scholar 

  26. Perry, P.A.: A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds. Int. Math. Res. Not. 2003 (34), 1837–1851 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Perry, P.A.: The spectral geometry of geometrically finite hyperbolic manifolds. Festschrift for the Sixtieth Birthday of Barry Simon. Proc. Sympos. Pure Math. 76, 289–327 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Robert, D.: Sur la formule de Weyl pour des ouverts non bornés. C. R. Acad. Sci. Paris Sér. I Math. 319, 29–34 (1994)

    Google Scholar 

  29. Selberg, A.: Göttingen lectures. In: Collected Works, vol. I, pp. 626–674. Springer, Berlin (1989)

    Google Scholar 

  30. Sjöstrand, J.: A trace formula and review of some estimates for resonances. In: Microlocal Analysis and Spectral Theory (Lucca, 1996). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 490, pp. 377–437. Kluwer, Dordrecht (1997)

    Google Scholar 

  31. Sjöstrand, J.: A trace formula for resonances and application to semi-classical Schrödinger operators. In: Séminaire sur les Équations aux Dérivées Partielles, 1996–1997. École Polytechnique, Palaiseau (1997). p. Exp. No. II

    Google Scholar 

  32. Sjöstrand, J., Zworski, M.: Lower bounds on the number of scattering poles. II. J. Funct. Anal. 123, 336–367 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  33. Stefanov, P.: Sharp upper bounds on the number of the scattering poles. J. Funct. Anal. 231, 111–142 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Vodev, G.: Asymptotics on the number of scattering poles for degenerate perturbations of the Laplacian. J. Funct. Anal. 138, 295–310 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zworski, M.: Distribution of poles for scattering on the real line. J. Funct. Anal. 73, 277–296 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zworski, M.: Sharp polynomial bounds on the number of scattering poles of radial potentials. J. Funct. Anal. 82, 370–403 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zworski, M.: Counting scattering poles. In: Spectral and Scattering Theory. Lecture Notes in Pure and Applied Mathematics, vol. 161, pp. 301–331. Dekker, New York (1994)

    Google Scholar 

  38. Zworski, M.: Resonances in physics and geometry. Not. Am. Math. Soc. 46, 319–328 (1999)

    MathSciNet  MATH  Google Scholar 

  39. Zworski, M.: Quantum resonances and partial differential equations. In: Proceedings of the International Congress of Mathematicians, vol. III (Beijing, 2002), pp. 243–252. Higher Education Press, Beijing (2002)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

    David Borthwick

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  1. David Borthwick
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Borthwick, D. (2016). Resonance Asymptotics. In: Spectral Theory of Infinite-Area Hyperbolic Surfaces. Progress in Mathematics, vol 318. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-33877-4_12

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