## Duke Mathematical Journal

### Geometric bounds on the density of resonances for semiclassical problems

Johannes Sjöstrand

#### Article information

Source
Duke Math. J., Volume 60, Number 1 (1990), 1-57.

Dates
First available in Project Euclid: 20 February 2004

https://projecteuclid.org/euclid.dmj/1077297139

Digital Object Identifier
doi:10.1215/S0012-7094-90-06001-6

Mathematical Reviews number (MathSciNet)
MR1047116

Zentralblatt MATH identifier
0702.35188

#### Citation

Sjöstrand, Johannes. Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60 (1990), no. 1, 1--57. doi:10.1215/S0012-7094-90-06001-6. https://projecteuclid.org/euclid.dmj/1077297139

#### References

• [ACo] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269–279.
• [BaCo] E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatation analytic interactions, Comm. Math. Phys. 22 (1971), 280–294.
• [BLeR] C. Bardos, G. Lebeau, and J. Rauch, Scattering frequencies and Gevrey $3$ singularities, Invent. Math. 90 (1987), no. 1, 77–114.
• [BoGu] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Ann. of Math. Studies, vol. 99, Princeton Univ. Press, Princeton, NJ, 1981.
• [BrCoDu1] P. Briet, J. M. Combes, and P. Duclos, On the location of resonances for Schrödinger operators in the semiclassical limit. I. Resonances free domains, J. Math. Anal. Appl. 126 (1987), no. 1, 90–99.
• [BrCoDu2] P. Briet, J. M. Combes, and P. Duclos, On the location of resonances for Schrödinger operators in the semiclassical limit. II. Barrier top resonances, Comm. Partial Differential Equations 12 (1987), no. 2, 201–222.
• [CF] A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), no. 11, 979–1005.
• [DSc] N. Dunford and J. T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963.
• [F] K. J. Falconer, The geometry of fractal sets, Cambridge tracts in mathematics, vol. 95, Cambridge University press, Cambridge, 1985.
• [G] C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes, Mém. Soc. Math. France (N.S.) (1988), no. 31, 146.
• [GS1] C. Gérard and J. Sjöstrand, Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys. 108 (1987), no. 3, 391–421.
• [GS2] C. Gérard and J. Sjöstrand, Resonances en limite semiclassique et exposants de Lyapunov, Comm. Math. Phys. 116 (1988), no. 2, 193–213.
• [Go] W. Goodhue, Scattering theory for hyperbolic systems with coefficients of Gevrey type, Trans. Amer. Math. Soc. 180 (1973), 337–346.
• [HS] B. Helffer and J. Sjöstrand, Résonances en limite semiclassique, Mém. Soc. Math. France (N.S.) (1986), no. 24-25, iv+228.
• [HMa] B. Helffer and A. Martinez, Comparaison entre les diverses notions de résonances, Helv. Phys. Acta 60 (1987), no. 8, 992–1003.
• [Hö] L. Hörmander, The analysis of linear partial differential operators III, Grundlehren der math. wiss., vol. 274, Springer Verlag, Berlin, 1985.
• [I1]1 M. Ikawa, On the poles of the scattering matrix for two strictly convex obstacles, J. Math. Kyoto Univ. 23 (1983), no. 1, 127–194.
• [I1]2 M. Ikawa, On the poles of the scattering matrix for two strictly convex obstacles: An addendum, J. Math. Kyoto Univ. 23 (1983), no. 4, 795–802.
• [I1]3 M. Ikawa, On the distribution of the poles of the scattering matrix for two strictly convex obstacles, Hokkaido Math. J. 12 (1983), no. 3, 343–359.
• [I2]1 M. Ikawa, Trapping obstacles with a sequence of poles of the scattering matrix converging to the real axis, Preprint.
• [I2]2 Mitsuru Ikawa, On the scattering matrix for two convex obstacles, Hyperbolic equations and related topics (Katata/Kyoto, 1984), Academic Press, Boston, MA, 1986, pp. 63–84.
• [I3] M. Ikawa, Decay of solutions of the wave equations in the exterior of several convex bodies, Preprint.
• [I4] M. Ikawa, 1988, Proc. of Japan Academy.
• [In] A. Intissar, A polynomial bound on the number of the scattering poles for a potential in even-dimensional spaces $\bf R\sp n$, Comm. Partial Differential Equations 11 (1986), no. 4, 367–396.
• [Ke] A. Kelley, The stable, center-stable, center, center-unstable and unstable manifolds, W. A. Benjamin Inc., New York, Amsterdam, 1967, Appendix C in Transversal mappings and flows by R. Abraham and J. Robbin.
• [K] M. Klein, On the absence of resonances for Schrödinger operators with nontrapping potentials in the classical limit, Comm. Math. Phys. 106 (1986), no. 3, 485–494.
• [LPh] P. Lax and R. Phillips, Scattering theory, Pure and Appl. Math., vol. 26, Academic Press, New York, 1967.
• [Le] G. Lebeau, Régularité Gevrey $3$ pour la diffraction, Comm. Partial Differential Equations 9 (1984), no. 15, 1437–1494.
• [M] R. Melrose, Polynomials bound on the distribution of poles in scattering by an obstacle, Proceedings of the Journées “Equations aux dérivées partielles” à St Jean de Montes, Société Mathématique de France, 4–8 Juin 1984.
• [N1] S. Nakamura, A note on the absence of resonances for Schrödinger operators, Lett. Math. Phys. 16 (1988), no. 3, 217–223.
• [N2] S. Nakamura, Shape resonances for distortion analytic Schrödinger operators, Preprint.
• [PSt] V. Petkov Stojanov, Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles, Preprint, 1987.
• [Sh] M. Shub, Global stability of dynamical systems, Springer Verlag, New York, 1987.
• [Si] Ya. G. Sinai, Development of Krylov's ideas, Princeton Univ. Press, 1979, Addendum to Works on the foundations of statistical physics, by N. S. Krylov.
• [S1] J. Sjöstrand, Singularités analytiques microlocales, Astérisque, 95, Astérisque, vol. 95, Soc. Math. France, Paris, 1982, pp. 1–166.
• [S2] J. Sjöstrand, Propagation of singularities for operators with multiple involutive characteristics, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 1, v, 141–155.
• [S3] J. Sjöstrand, Semiclassical resonances generated by nondegenerate critical points, Pseudodifferential operators (Oberwolfach, 1986), Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987, pp. 402–429.
• [S4] J. Sjöstrand, Estimates on the number of resonances for semiclassical Schrödinger operators, Partial Differential Equations (Rio de Janeiro, 1986), Lecture Notes in Math., vol. 1324, Springer, Berlin, 1988, pp. 286–292.
• [T] C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57–74.
• [Z] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Preprint.