Analysis & PDE

  • Anal. PDE
  • Volume 5, Number 3 (2012), 513-552.

Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

David Borthwick

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Abstract

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary conditions.

Article information

Source
Anal. PDE, Volume 5, Number 3 (2012), 513-552.

Dates
Received: 31 July 2010
Accepted: 26 February 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731229

Digital Object Identifier
doi:10.2140/apde.2012.5.513

Mathematical Reviews number (MathSciNet)
MR2994506

Zentralblatt MATH identifier
1264.35157

Subjects
Primary: 35P25: Scattering theory [See also 47A40] 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]

Keywords
resonances hyperbolic surfaces scattering theory

Citation

Borthwick, David. Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends. Anal. PDE 5 (2012), no. 3, 513--552. doi:10.2140/apde.2012.5.513. https://projecteuclid.org/euclid.apde/1513731229


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References

  • R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954.
  • D. Borthwick, Spectral theory of infinite-area hyperbolic surfaces, Progress in Mathematics 256, Birkhäuser, Boston, 2007.
  • D. Borthwick, “Upper and lower bounds on resonances for manifolds hyperbolic near infinity”, Comm. Partial Differential Equations 33:7-9 (2008), 1507–1539.
  • D. Borthwick, “Sharp upper bounds on resonances for perturbations of hyperbolic space”, Asymptot. Anal. 69:1-2 (2010), 45–85.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, I, McGraw-Hill, New York, 1953.
  • R. Froese, “Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions”, Canad. J. Math. 50:3 (1998), 538–546.
  • C. Guillarmou, “Absence of resonance near the critical line on asymptotically hyperbolic spaces”, Asymptot. Anal. 42:1-2 (2005), 105–121.
  • L. Guillopé and M. Zworski, “Upper bounds on the number of resonances for non-compact Riemann surfaces”, J. Funct. Anal. 129:2 (1995), 364–389.
  • L. Guillopé and M. Zworski, “Scattering asymptotics for Riemann surfaces”, Ann. of Math. $(2)$ 145:3 (1997), 597–660.
  • H. P. McKean, “Selberg's trace formula as applied to a compact Riemann surface”, Comm. Pure Appl. Math. 25 (1972), 225–246.
  • W. Müller, “Spectral geometry and scattering theory for certain complete surfaces of finite volume”, Invent. Math. 109:2 (1992), 265–305.
  • F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974. http://www.emis.de/cgi-bin/MATH-item?0303.41035Zbl 0303.41035
  • L. B. Parnovski, “Spectral asymptotics of Laplace operators on surfaces with cusps”, Math. Ann. 303:2 (1995), 281–296.
  • P. Stefanov, “Sharp upper bounds on the number of the scattering poles”, J. Funct. Anal. 231:1 (2006), 111–142.
  • A. B. Venkov, Spectral theory of automorphic functions and its applications, Mathematics and its Applications (Soviet Series) 51, Kluwer Academic, Dordrecht, 1990.
  • G. Vodev, “Sharp bounds on the number of scattering poles in even-dimensional spaces”, Duke Math. J. 74:1 (1994), 1–17.