Resonances for Schrödinger operators with compactly supported potentials
Journées équations aux dérivées partielles (2008), article no. 3, 18 p.

We describe the generic behavior of the resonance counting function for a Schrödinger operator with a bounded, compactly-supported real or complex valued potential in $d\ge 1$ dimensions. This note contains a sketch of the proof of our main results [5, 6] that generically the order of growth of the resonance counting function is the maximal value $d$ in the odd dimensional case, and that it is the maximal value $d$ on each nonphysical sheet of the logarithmic Riemann surface in the even dimensional case. We include a review of previous results concerning the resonance counting functions for Schrödinger operators with compactly-supported potentials.

@article{JEDP_2008____A3_0,
author = {Christiansen, T. J. and Hislop, P. D.},
title = {Resonances for Schr\"odinger operators with compactly supported potentials},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2008},
doi = {10.5802/jedp.47},
language = {en},
url = {http://www.numdam.org/item/JEDP_2008____A3_0}
}

Christiansen, T. J.; Hislop, P. D. Resonances for Schrödinger operators with compactly supported potentials. Journées équations aux dérivées partielles (2008), article  no. 3, 18 p. doi : 10.5802/jedp.47. http://www.numdam.org/item/JEDP_2008____A3_0/

[1] R. Bañuelos, A. Sá Barreto, On the heat trace of Schrödinger operators, Comm. Partial Differential Equations 20 (1995), no. 11-12, 2153–2164. | MR 1361734 | Zbl 0843.35016

[2] T. Christiansen, Some lower bounds on the number of resonances in Euclidean scattering, Math. Res. Lett. 6 (1999), no. 2, 203–211. | MR 1689210 | Zbl 0947.35102

[3] T. Christiansen, Several complex variables and the distribution of resonances for potential scattering, Commun. Math. Phys 259 (2005), 711-728. | MR 2174422 | Zbl 1088.81093

[4] T. Christiansen, Schrödinger operators with complex-valued potentials and no resonances, Duke Math Journal 133, no. 2 (2006), 313-323. | MR 2225694 | Zbl 1107.35094

[5] T. Christiansen and P. D. Hislop, The resonance counting function for Schrödinger operators with generic potentials, Math. Research Letters, 12 (6) (2005), 821-826. | MR 2189242 | Zbl 1155.35319

[6] T. Christiansen and P. D. Hislop, Maximal order of growth for the resonance counting function for generic potentials in even dimensions, submitted, arXiv:0811.4761v1.

[7] R. Froese, Asymptotic distribution of resonances in one dimension, J. Differential Equations 137 (1997), no. 2, 251–272. | MR 1456597 | Zbl 0955.35057

[8] R. Froese, Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions, Canad. J. Math. 50 (1998), no. 3, 538–546. | MR 1629819 | Zbl 0918.47005

[9] A. Intissar, A polynomial bound on the number of the scattering poles for a potential in even dimensional spaces ${ℝ}^{n}$, Comm. in Partial Diff. Eqns. 11, No. 4 (1986), 367–396. | MR 829322 | Zbl 0607.35069

[10] P. D. Lax and R. S. Phillips, Decaying modes for the wave equation in the exterior of an obstacle, Comm. Pure Appl. Math. 22 (1969), 737–787. | MR 254432 | Zbl 0181.38201

[11] P. Lelong and L. Gruman, Entire functions of several complex variables, Springer Verlag, Berlin, 1986. | MR 837659 | Zbl 0583.32001

[12] G. P. Menzala, T. Schonbek, Scattering frequencies for the wave equation with a potential term, J. Funct. Anal. 55 (1984), 297–322. | MR 734801 | Zbl 0536.35060

[13] R. B. Melrose, Polynomial bounds on the number of scattering poles, J. Funct. Anal. 53 (1983), 287–303. | MR 724031 | Zbl 0535.35067

[14] R. B. Melrose, Geometric scattering theory, Cambridge University Press, 1995. | MR 1350074 | Zbl 0849.58071

[15] R. G. Newton, Analytic properties of radial wave functions, J. Math. Phys. 1, no. 4, 319–347 (1960). | MR 115692 | Zbl 0090.19303

[16] H. M. Nussenzveig, The poles of the $S$-matrix of a rectangular potential well or barrier, Nuclear Phys. 11 (1959), 499–521.

[17] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, San Deigo, 1974. | MR 435697 | Zbl 0303.41035

[18] F. W. J. Olver, The asymptotic solution of linear differential equations of the second order for large values of a parameter, Phil. Trans. Royal Soc. London Ser. A 247, 307–327 (1954). | MR 67249 | Zbl 0070.30801

[19] F. W. J. Olver, The asymptotic expansion of Bessel functions of large order, Phil. Trans. Royal Soc. London ser. A 247, 328–368 (1954). | MR 67250 | Zbl 0070.30801

[20] T. Ransford, Potential theory in the complex plane, Cambridge University Press, Cambridge, 1995. | MR 1334766 | Zbl 0828.31001

[21] T. Regge, Analytic properties of the scattering matrix, Il Nuovo Cimento 8 (1958), no. 10, 671–679. | MR 95702 | Zbl 0080.41903

[22] A. Sá Barreto, Remarks on the distribution of resonances in odd dimensional Euclidean scattering, Asymptot. Anal. 27 (2001), no. 2, 161–170. | MR 1852004 | Zbl 1116.35344

[23] A. Sá Barreto, Lower bounds for the number of resonances in even dimensional potential scattering, J. Funct. Anal. 169 (1999), 314–323. | MR 1726757 | Zbl 0939.35133

[24] A. Sá Barreto, S.-H. Tang, Existence of resonances in even dimensional potential scattering, Commun. Part. Diff. Eqns. 25 (2000), no. 5-6, 1143–1151. | MR 1759805 | Zbl 0947.35101

[25] A. Sá Barreto, M. Zworski, Existence of resonances in three dimensions, Comm. Math. Phys. 173 (1995), no. 2, 401–415. | MR 1355631 | Zbl 0835.35099

[26] A. Sá Barreto, M. Zworski, Existence of resonances in potential scattering, Comm. Pure Appl. Math. 49 (1996), no. 12, 1271–1280. | MR 1414586 | Zbl 0877.35087

[27] N. Shenk, D. Thoe, Resonant states and poles of the scattering matrix for perturbations of $-\Delta$, J. Math. Anal. Appl. 37 (1972), 467–491. | MR 308616 | Zbl 0229.35072

[28] B. Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178 (2000), no. 2, 396–420. | MR 1802901 | Zbl 0977.34075

[29] B. Simon, Trace Ideals and their Applications, London Mathematical Society Lecture Note Series 35, Cambridge University Press, 1979; second edition, American Mathematical Society, Providence RI, 2005. | MR 2154153 | Zbl 0423.47001

[30] B. Simon, Operators with singular continuous spectrum: I. general operators, Ann. Math. 141 (1995), 131–145. | MR 1314033 | Zbl 0851.47003

[31] J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1–57. | MR 1047116 | Zbl 0702.35188

[32] J. Sjöstrand, M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4(1991), no. 4, 729–769. | MR 1115789 | Zbl 0752.35046

[33] P. Stefanov, Sharp bounds on the number of the scattering poles, J. Func. Anal., 231 (1) (2006), 111–142. | MR 2190165 | Zbl 1099.35074

[34] A. Vasy, Scattering poles for negative potentials, Comm. Partial Differential Equations 22 (1997), no. 1-2, 185–194 | MR 1434143 | Zbl 0884.35109

[35] G. Vodev, Sharp polynomial bounds on the number of scattering poles for perturbations of the Laplacian, Commun. Math. Phys. 146 (1992), 39–49. | MR 1163673 | Zbl 0754.35105

[36] G. Vodev, Sharp bounds on the number of scattering poles in even-dimensional spaces, Duke Math. J. 74 (1) (1994), 1–17. | MR 1271461 | Zbl 0813.35075

[37] G. Vodev, Sharp bounds on the number of scattering poles in the two-dimensional case, Math. Nachr. 170 (1994), 287–297. | MR 1302380 | Zbl 0829.35091

[38] G. Vodev, Resonances in Euclidean scattering, Cubo Matemática Educacional 3 No. 1, Enero 2001, 319–360. | Zbl 1075.35024

[39] G. N. Watson, Treatise on the theory of Bessel functions, Cambridge University Press, 1966. | Zbl 0174.36202

[40] D. Yafaev, Mathematical scattering theory. General theory, translated from the Russian by J. R. Schulenberger, Translations of Mathematical Monographs, 105, American Mathematical Society, Providence, RI, 1992 | MR 1180965 | Zbl 0761.47001

[41] M. Zworski, Sharp polynomial poles on the number of scattering poles, Duke Math. J. 59 (1989), 311–323. | MR 1016891 | Zbl 0705.35099

[42] M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), 277–296. | MR 899652 | Zbl 0662.34033

[43] M. Zworski, Sharp polynomial bounds on the number of scattering poles of radial potentials, J. Funct. Anal. 82 (1989), 370–403. | MR 987299 | Zbl 0681.47002

[44] M. Zworski, Poisson formulae for resonances, Séminaire sur les Equations aux Dérivées Partielles, 1996–1997, Exp. No. XIII, 14 pp., Ecole Polytech., Palaiseau, 1997 Seminaire Ecole Polytechnique. | Numdam | MR 1482819 | Zbl pre02124115

[45] M. Zworski, Counting scattering poles, In: Spectral and scattering theory (Sanda, 1992), 301–331, Lectures in Pure and Appl. Math. 161, New York: Dekker, 1994. | MR 1291649 | Zbl 0823.35139