Inverse Regge poles problem on a warped ball

Jack Borthwick; Nabile Boussaïd; Thierry Daudé

doi:10.3934/ipi.2023031

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2024, Volume 18, Issue 1: 239-270. Doi: 10.3934/ipi.2023031

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Inverse Regge poles problem on a warped ball

1.
Université de Franche-Comté, Laboratoire de mathématiques de Besançon (CNRS UMR 6623), 16 route de Gray, 25030 Besançon cedex, France

^*Corresponding author: Jack Borthwick
^*Corresponding author: Jack Borthwick

Received: December 2022

Revised: May 2023

Early access: June 2023

Published: February 2024

The first author gratefully acknowledges that part of this work was supported by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-0003).

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Abstract

In this paper, we study a new type of inverse problem on warped product Riemannian manifolds with connected boundary that we name warped balls. Using the symmetry of the geometry, we first define the set of Regge poles as the poles of the meromorphic continuation of the Dirichlet-to-Neumann map with respect to the complex angular momentum appearing in the separation of variables procedure. These Regge poles can also be viewed as the set of eigenvalues and resonances of a one-dimensional Schrödinger equation on the half-line, obtained after separation of variables. Secondly, we find a precise asymptotic localisation of the Regge poles in the complex plane and prove that they uniquely determine the warping function of the warped balls.

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Mathematics Subject Classification: Primary: 35R30, 34A55, 81U24.

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Figure 1. Sketch of $ S_{\varepsilon,\delta} $ and a generic curve $ \gamma_n $

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[1]	V. de Alfaro and T. Regge, Potential Scattering, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1965, ix+205.
[2]	C. Bennewitz, A proof of the local Borg-Marchenko theorem, Comm. Math. Phys., 218 (2001), 131-132. doi: 10.1007/s002200100384.
[3]	B. M. Brown, I. Knowles and R. Weikard, On the inverse resonance problem, J. London Math. Soc., 68 (2003), 383-401. doi: 10.1112/S0024610703004654.
[4]	B. M. Brown and R. Weikard, The inverse resonance problem for perturbations of algebro-geometric potentials, Inverse Problems, 20 (2004), 481-494. doi: 10.1088/0266-5611/20/2/011.
[5]	M. L. Cartwright, The zeros of certain integral functions, The Quarterly Journal of Mathematics, os-1 (1930), 38-59. doi: 10.1093/qmath/os-1.1.38.
[6]	M. L. Cartwright, The zeros of certain integral functions. (Ⅱ), The Quarterly Journal of Mathematics, os-2 (1931), 113-129. doi: 10.1093/qmath/os-2.1.113.
[7]	T. Daudé, N. Kamran and F. Nicoleau, Stability in the inverse Steklov problem on warped product Riemannian manifolds, J. Geom. Anal., 31 (2021), 1821-1854. doi: 10.1007/s12220-019-00326-9.
[8]	K. Destounis, R. P. Macedo, E. Berti, V. Cardoso and J. L. Jaramillo, Pseudospectrum of Reissner-Nordström black holes: Quasinormal mode instability and universality, Phys. Rev. D, 104 (2021), Paper No. 084091, 18 pp. doi: 10.1103/PhysRevD.104.084091.
[9]	S. Dyatlov and M. Zworski, Mathematical Theory of Scattering Resonances, Vol. 200. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2019, xi+634. doi: 10.1090/gsm/200.
[10]	G. Gendron, Uniqueness results in the inverse spectral Steklov problem, Inverse Probl. Imaging, 14 (2020), 631-664. doi: 10.3934/ipi.2020029.
[11]	G. Gendron, Stability estimates for an inverse Steklov problem in a class of hollow spheres, Asymptot. Anal., 126 (2022), 323-377. doi: 10.3233/ASY-211684.
[12]	F. Gesztesy and B. Simon, On local Borg-Marchenko uniqueness results, Comm. Math. Phys., 211 (2000), 273-287. doi: 10.1007/s002200050812.
[13]	G. H. Hardy, On the zeroes of certain classes of integral Taylor series. Part Ⅱ. On the integral function formula and other similar functions, Proc. London Math. Soc., s2-2 (1905), 401-431. doi: 10.1112/plms/s2-2.1.401.
[14]	H. Isozaki and E. L. Korotyaev, Inverse spectral theory and the Minkowski problem for the surface of revolution, Dyn. Partial Differ. Equ., 14 (2017), 321-341. doi: 10.4310/DPDE.2017.v14.n4.a1.
[15]	H. Isozaki and E. Korotyaev, Inverse resonance scattering on rotationally symmetric manifolds, Asymptot. Anal., 125 (2021), 347-363. doi: 10.3233/ASY-201659.
[16]	E. Korotyaev, Inverse resonance scattering on the half line, Asymptot. Anal., 37 (2004), 215-226.
[17]	B. Y. Levin, Lectures on Entire Functions, Vol. 150. Translations of Mathematical Monographs. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko. American Mathematical Society, Providence, RI, 1996, xvi+248. doi: 10.1090/mmono/150.
[18]	B. M. Levitan, Inverse Sturm-Liouville Problems, Berlin, Boston: De Gruyter, 1987. doi: 10.1515/9783110941937.
[19]	M. Marletta, R. Shterenberg and R. Weikard, On the inverse resonance problem for Schrödinger operators, Comm. Math. Phys., 295 (2010), 465-484. doi: 10.1007/s00220-009-0928-8.
[20]	P. Petersen, Riemannian Geometry, Third. Vol. 171. Graduate Texts in Mathematics. Springer, Cham, 2016, xviii+499. doi: 10.1007/978-3-319-26654-1.
[21]	A. Ramm and B. Simon, A new approach to inverse spectral theory. Ⅲ. Short-range potentials, J. Anal. Math., 80 (2000), 319-334. doi: 10.1007/BF02791540.
[22]	T. Regge, Introduction to complex orbital momenta, Nuovo Cimento, 14 (1959), 951-976. doi: 10.1007/BF02728177.
[23]	W. Rudin, Real and Complex Analysis, Third Edition. McGraw-Hill Book Co., New York, 1987, xiv+416.
[24]	B. Simon, A new approach to inverse spectral theory. I. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057. doi: 10.2307/121061.
[25]	G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Reprint of the second (1944) edition. Cambridge University Press, Cambridge, 1995, Pp. viii+804.
[26]	M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal., 73 (1987), 277-296. doi: 10.1016/0022-1236(87)90069-3.