Abstract
We apply Cartwright’s theory in integral function theory to describe the angular distribution of scattering resonances in mathematical physics. A quantitative description on the counting function along rays in complex plane is obtained.
Research Article | Open Access
On the Angular Density of Three Dimensional Scattering Resonances
Academic Editor: F. Sugino, A. L. Salas-Brito, D. Dürr, N. E. Bjerrum-Bohr
Received01 Jan 2014
Accepted04 Feb 2014
Published09 Mar 2014
References
J. Sjöstrand and M. Zworski, “Lower bounds on the number of scattering poles. II,” Journal of Functional Analysis, vol. 123, no. 2, pp. 336–367, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetG. Vodev, “Sharp bounds on the number of scattering poles for perturbations of the Laplacian,” Communications in Mathematical Physics, vol. 146, no. 1, pp. 205–216, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. Zworski, “Poisson formulae for resonances,” in Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, École Polytechnique, Palaiseau, France, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNetA. Sá Barreto, “Remarks on the distribution of resonances in odd dimensional Euclidean scattering,” Asymptotic Analysis, vol. 27, no. 2, pp. 161–170, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. L. Cartwright, “On the directions of Borel of functions which are regular and of finite order in an angle,” Proceedings of the London Mathematical Society 2, vol. 38, no. 1, pp. 503–541, 1933.
View at: Publisher Site | Google Scholar | MathSciNetM. L. Cartwright, “On the directions of Borel of analytic functions,” Proceedings of the London Mathematical Society 2, vol. 38, no. 1, pp. 417–457, 1933.
View at: Publisher Site | Google Scholar | MathSciNetM. L. Cartwright, “On functions which are regular and of finite order in an angle,” Proceedings of the London Mathematical Society 2, vol. 38, no. 1, pp. 158–179, 1933.
View at: Publisher Site | Google Scholar | MathSciNetM. L. Cartwright, Integral Functions, Cambridge University Press, Cambridge, UK, 1956.
View at: MathSciNetB. Ja. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence, RI, USA, 1972.
View at: MathSciNetB. Ya. Levin, Lectures on Entire Functions, vol. 150, American Mathematical Society, Providence, RI, USA, 1996.
View at: MathSciNetM. Zworski, “Distribution of poles for scattering on the real line,” Journal of Functional Analysis, vol. 73, no. 2, pp. 277–296, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetT. J. Christiansen, “Schrödinger operators and the distribution of resonances in sectors,” Analysis & PDE, vol. 5, no. 5, pp. 961–982, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetR. Froese, “Asymptotic distribution of resonances in one dimension,” Journal of Differential Equations, vol. 137, no. 2, pp. 251–272, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetD. Drasin and A. Weitsman, “On the Julia directions and Borel directions of entire functions,” Proceedings of the London Mathematical Society, vol. 32, no. 2, pp. 199–212, 1976.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Copyright
Copyright © 2014 Lung-Hui Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.