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}, eid = {3}, pages = {1--18}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2008}, doi = {10.5802/jedp.47}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jedp.47/} }
TY - JOUR AU - Christiansen, T. J. AU - Hislop, P. D. TI - Resonances for Schrödinger operators with compactly supported potentials JO - Journées équations aux dérivées partielles PY - 2008 SP - 1 EP - 18 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.47/ DO - 10.5802/jedp.47 LA - en ID - JEDP_2008____A3_0 ER -
%0 Journal Article %A Christiansen, T. J. %A Hislop, P. D. %T Resonances for Schrödinger operators with compactly supported potentials %J Journées équations aux dérivées partielles %D 2008 %P 1-18 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.47/ %R 10.5802/jedp.47 %G en %F 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/articles/10.5802/jedp.47/
[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 | Zbl
[2] T. Christiansen, Some lower bounds on the number of resonances in Euclidean scattering, Math. Res. Lett. 6 (1999), no. 2, 203–211. | MR | Zbl
[3] T. Christiansen, Several complex variables and the distribution of resonances for potential scattering, Commun. Math. Phys 259 (2005), 711-728. | MR | Zbl
[4] T. Christiansen, Schrödinger operators with complex-valued potentials and no resonances, Duke Math Journal 133, no. 2 (2006), 313-323. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[9] A. Intissar, A polynomial bound on the number of the scattering poles for a potential in even dimensional spaces ${\mathbb{R}}^{n}$, Comm. in Partial Diff. Eqns. 11, No. 4 (1986), 367–396. | MR | Zbl
[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 | Zbl
[11] P. Lelong and L. Gruman, Entire functions of several complex variables, Springer Verlag, Berlin, 1986. | MR | Zbl
[12] G. P. Menzala, T. Schonbek, Scattering frequencies for the wave equation with a potential term, J. Funct. Anal. 55 (1984), 297–322. | MR | Zbl
[13] R. B. Melrose, Polynomial bounds on the number of scattering poles, J. Funct. Anal. 53 (1983), 287–303. | MR | Zbl
[14] R. B. Melrose, Geometric scattering theory, Cambridge University Press, 1995. | MR | Zbl
[15] R. G. Newton, Analytic properties of radial wave functions, J. Math. Phys. 1, no. 4, 319–347 (1960). | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[20] T. Ransford, Potential theory in the complex plane, Cambridge University Press, Cambridge, 1995. | MR | Zbl
[21] T. Regge, Analytic properties of the scattering matrix, Il Nuovo Cimento 8 (1958), no. 10, 671–679. | MR | Zbl
[22] A. Sá Barreto, Remarks on the distribution of resonances in odd dimensional Euclidean scattering, Asymptot. Anal. 27 (2001), no. 2, 161–170. | MR | Zbl
[23] A. Sá Barreto, Lower bounds for the number of resonances in even dimensional potential scattering, J. Funct. Anal. 169 (1999), 314–323. | MR | Zbl
[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 | Zbl
[25] A. Sá Barreto, M. Zworski, Existence of resonances in three dimensions, Comm. Math. Phys. 173 (1995), no. 2, 401–415. | MR | Zbl
[26] A. Sá Barreto, M. Zworski, Existence of resonances in potential scattering, Comm. Pure Appl. Math. 49 (1996), no. 12, 1271–1280. | MR | Zbl
[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 | Zbl
[28] B. Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178 (2000), no. 2, 396–420. | MR | Zbl
[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 | Zbl
[30] B. Simon, Operators with singular continuous spectrum: I. general operators, Ann. Math. 141 (1995), 131–145. | MR | Zbl
[31] J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1–57. | MR | Zbl
[32] J. Sjöstrand, M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4(1991), no. 4, 729–769. | MR | Zbl
[33] P. Stefanov, Sharp bounds on the number of the scattering poles, J. Func. Anal., 231 (1) (2006), 111–142. | MR | Zbl
[34] A. Vasy, Scattering poles for negative potentials, Comm. Partial Differential Equations 22 (1997), no. 1-2, 185–194 | MR | Zbl
[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 | Zbl
[36] G. Vodev, Sharp bounds on the number of scattering poles in even-dimensional spaces, Duke Math. J. 74 (1) (1994), 1–17. | MR | Zbl
[37] G. Vodev, Sharp bounds on the number of scattering poles in the two-dimensional case, Math. Nachr. 170 (1994), 287–297. | MR | Zbl
[38] G. Vodev, Resonances in Euclidean scattering, Cubo Matemática Educacional 3 No. 1, Enero 2001, 319–360. | Zbl
[39] G. N. Watson, Treatise on the theory of Bessel functions, Cambridge University Press, 1966. | Zbl
[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 | Zbl
[41] M. Zworski, Sharp polynomial poles on the number of scattering poles, Duke Math. J. 59 (1989), 311–323. | MR | Zbl
[42] M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), 277–296. | MR | Zbl
[43] M. Zworski, Sharp polynomial bounds on the number of scattering poles of radial potentials, J. Funct. Anal. 82 (1989), 370–403. | MR | Zbl
[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
[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 | Zbl
Cited by Sources: