DOI: 10.1215/s0012-7094-90-06001-6. Geometric bounds on the density of resonances for semiclassical problems

Geometric bounds on the density of resonances for semiclassical problems

10.1215/s0012-7094-90-06001-6

Crossref journal-article

Duke University Press

Duke Mathematical Journal (73)

Bibliography

SjÃ¶strand, J. (1990). Geometric bounds on the density of resonances for semiclassical problems. Duke Mathematical Journal, 60(1).

Authors 1

Johannes Sjöstrand (first)

References 40 Referenced 110

[ACo] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269–279. (10.1007/BF01877510)
[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. (10.1007/BF01877511)
[BLeR] C. Bardos, G. Lebeau, and J. Rauch, Scattering frequencies and Gevrey $3$ singularities, Invent. Math. 90 (1987), no. 1, 77–114. (10.1007/BF01389032)
[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. (10.1515/9781400881444)
[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. (10.1016/0022-247X(87)90077-1)
[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. (10.1080/03605308708820488)
[CF] A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), no. 11, 979–1005. (10.1080/03605307808820083)
[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. (10.1007/BF01212317)
[GS2] C. Gérard and J. Sjöstrand, Resonances en limite semiclassique et exposants de Lyapunov, Comm. Math. Phys. 116 (1988), no. 2, 193–213. (10.1007/BF01225255)
[Go] W. Goodhue, Scattering theory for hyperbolic systems with coefficients of Gevrey type, Trans. Amer. Math. Soc. 180 (1973), 337–346. (10.2307/1996669)
[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. (10.1016/B978-0-12-501658-2.50008-7)
[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. (10.1080/03605308608820428)
[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. (10.1016/0022-0396(67)90016-2)
[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. (10.1007/BF01207259)
[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. (10.1080/03605308408820368)
[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. (10.5802/jedp.285)
[N1] S. Nakamura, A note on the absence of resonances for Schrödinger operators, Lett. Math. Phys. 16 (1988), no. 3, 217–223. (10.1007/BF00398958)
[N2] S. Nakamura, Shape resonances for distortion analytic Schrödinger operators, Preprint. (10.1080/03605308908820659)
[PSt] V. Petkov Stojanov, Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles, Preprint, 1987. (10.5802/jedp.332)
[Sh] M. Shub, Global stability of dynamical systems, Springer Verlag, New York, 1987. (10.1007/978-1-4757-1947-5)
[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. (10.1017/S0305004100059119)
[Z] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Preprint. (10.1215/S0012-7094-89-05913-9)

Dates

Type	When
Created	20 years, 9 months ago (Nov. 11, 2004, 9:28 a.m.)
Deposited	1 year, 7 months ago (Jan. 30, 2024, 12:20 p.m.)
Indexed	2 weeks, 2 days ago (Aug. 21, 2025, 1:26 p.m.)
Issued	35 years, 7 months ago (Feb. 1, 1990)
Published	35 years, 7 months ago (Feb. 1, 1990)
Published Print	35 years, 7 months ago (Feb. 1, 1990)

Funders 0

None

BibTeX

@article{Sj_strand_1990, title={Geometric bounds on the density of resonances for semiclassical problems}, volume={60}, ISSN={0012-7094}, url={http://dx.doi.org/10.1215/s0012-7094-90-06001-6}, DOI={10.1215/s0012-7094-90-06001-6}, number={1}, journal={Duke Mathematical Journal}, publisher={Duke University Press}, author={Sjöstrand, Johannes}, year={1990}, month=feb }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        21
      ]
    ],
    "date-time": "2025-08-21T17:26:58Z",
    "timestamp": 1755797218170
  },
  "reference-count": 40,
  "publisher": "Duke University Press",
  "issue": "1",
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1990,
        2,
        1
      ]
    ]
  },
  "DOI": "10.1215/s0012-7094-90-06001-6",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2004,
        11,
        11
      ]
    ],
    "date-time": "2004-11-11T14:28:40Z",
    "timestamp": 1100183320000
  },
  "source": "Crossref",
  "is-referenced-by-count": 110,
  "title": "Geometric bounds on the density of resonances for\n              semiclassical problems",
  "prefix": "10.1215",
  "volume": "60",
  "author": [
    {
      "given": "Johannes",
      "family": "Sj\u00f6strand",
      "sequence": "first",
      "affiliation": []
    }
  ],
  "member": "73",
  "reference": [
    {
      "key": "1",
      "doi-asserted-by": "publisher",
      "unstructured": "[ACo] J. Aguilar and J. M. Combes, <i>A class of analytic perturbations for one-body Schr\u00f6dinger Hamiltonians</i>, Comm. Math. Phys. <b>22</b> (1971), 269\u2013279.",
      "DOI": "10.1007/BF01877510"
    },
    {
      "key": "2",
      "doi-asserted-by": "publisher",
      "unstructured": "[BaCo] E. Balslev and J. M. Combes, <i>Spectral properties of many-body Schr\u00f6dinger operators with dilatation analytic interactions</i>, Comm. Math. Phys. <b>22</b> (1971), 280\u2013294.",
      "DOI": "10.1007/BF01877511"
    },
    {
      "key": "3",
      "doi-asserted-by": "publisher",
      "unstructured": "[BLeR] C. Bardos, G. Lebeau, and J. Rauch, <i>Scattering frequencies and Gevrey $3$ singularities</i>, Invent. Math. <b>90</b> (1987), no. 1, 77\u2013114.",
      "DOI": "10.1007/BF01389032"
    },
    {
      "key": "4",
      "doi-asserted-by": "crossref",
      "unstructured": "[BoGu] L. Boutet de Monvel and V. Guillemin, <i>The spectral theory of Toeplitz operators</i>, Ann. of Math. Studies, vol. 99, Princeton Univ. Press, Princeton, NJ, 1981.",
      "DOI": "10.1515/9781400881444"
    },
    {
      "key": "5",
      "doi-asserted-by": "publisher",
      "unstructured": "[BrCoDu1] P. Briet, J. M. Combes, and P. Duclos, <i>On the location of resonances for Schr\u00f6dinger operators in the semiclassical limit. I. Resonances free domains</i>, J. Math. Anal. Appl. <b>126</b> (1987), no. 1, 90\u201399.",
      "DOI": "10.1016/0022-247X(87)90077-1"
    },
    {
      "key": "6",
      "doi-asserted-by": "publisher",
      "unstructured": "[BrCoDu2] P. Briet, J. M. Combes, and P. Duclos, <i>On the location of resonances for Schr\u00f6dinger operators in the semiclassical limit. II. Barrier top resonances</i>, Comm. Partial Differential Equations <b>12</b> (1987), no. 2, 201\u2013222.",
      "DOI": "10.1080/03605308708820488"
    },
    {
      "key": "7",
      "doi-asserted-by": "publisher",
      "unstructured": "[CF] A. Cordoba and C. Fefferman, <i>Wave packets and Fourier integral operators</i>, Comm. Partial Differential Equations <b>3</b> (1978), no. 11, 979\u20131005.",
      "DOI": "10.1080/03605307808820083"
    },
    {
      "key": "8",
      "unstructured": "[DSc] N. Dunford and J. T. Schwartz, <i>Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space</i>, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley &amp; Sons New York-London, 1963."
    },
    {
      "key": "9",
      "unstructured": "[F] K. J. Falconer, <i>The geometry of fractal sets</i>, Cambridge tracts in mathematics, vol. 95, Cambridge University press, Cambridge, 1985."
    },
    {
      "key": "10",
      "unstructured": "[G] C. G\u00e9rard, <i>Asymptotique des p\u00f4les de la matrice de scattering pour deux obstacles strictement convexes</i>, M\u00e9m. Soc. Math. France (N.S.) (1988), no. 31, 146."
    },
    {
      "key": "11",
      "doi-asserted-by": "publisher",
      "unstructured": "[GS1] C. G\u00e9rard and J. Sj\u00f6strand, <i>Semiclassical resonances generated by a closed trajectory of hyperbolic type</i>, Comm. Math. Phys. <b>108</b> (1987), no. 3, 391\u2013421.",
      "DOI": "10.1007/BF01212317"
    },
    {
      "key": "12",
      "doi-asserted-by": "publisher",
      "unstructured": "[GS2] C. G\u00e9rard and J. Sj\u00f6strand, <i>Resonances en limite semiclassique et exposants de Lyapunov</i>, Comm. Math. Phys. <b>116</b> (1988), no. 2, 193\u2013213.",
      "DOI": "10.1007/BF01225255"
    },
    {
      "key": "13",
      "doi-asserted-by": "publisher",
      "unstructured": "[Go] W. Goodhue, <i>Scattering theory for hyperbolic systems with coefficients of Gevrey type</i>, Trans. Amer. Math. Soc. <b>180</b> (1973), 337\u2013346.",
      "DOI": "10.2307/1996669"
    },
    {
      "key": "14",
      "unstructured": "[HS] B. Helffer and J. Sj\u00f6strand, <i>R\u00e9sonances en limite semiclassique</i>, M\u00e9m. Soc. Math. France (N.S.) (1986), no. 24-25, iv+228."
    },
    {
      "key": "15",
      "unstructured": "[HMa] B. Helffer and A. Martinez, <i>Comparaison entre les diverses notions de r\u00e9sonances</i>, Helv. Phys. Acta <b>60</b> (1987), no. 8, 992\u20131003."
    },
    {
      "key": "16",
      "unstructured": "[H\u00f6] L. H\u00f6rmander, <i>The analysis of linear partial differential operators III</i>, Grundlehren der math. wiss., vol. 274, Springer Verlag, Berlin, 1985."
    },
    {
      "key": "17",
      "unstructured": "[I1]<sup>1</sup> M. Ikawa, <i>On the poles of the scattering matrix for two strictly convex obstacles</i>, J. Math. Kyoto Univ. <b>23</b> (1983), no. 1, 127\u2013194."
    },
    {
      "key": "18",
      "unstructured": "[I1]<sup>2</sup> M. Ikawa, <i>On the poles of the scattering matrix for two strictly convex obstacles: An addendum</i>, J. Math. Kyoto Univ. <b>23</b> (1983), no. 4, 795\u2013802."
    },
    {
      "key": "19",
      "unstructured": "[I1]<sup>3</sup> M. Ikawa, <i>On the distribution of the poles of the scattering matrix for two strictly convex obstacles</i>, Hokkaido Math. J. <b>12</b> (1983), no. 3, 343\u2013359."
    },
    {
      "key": "20",
      "unstructured": "[I2]<sup>1</sup> M. Ikawa, <i>Trapping obstacles with a sequence of poles of the scattering matrix converging to the real axis</i>, Preprint."
    },
    {
      "key": "21",
      "doi-asserted-by": "crossref",
      "unstructured": "[I2]<sup>2</sup> Mitsuru Ikawa, <i>On the scattering matrix for two convex obstacles</i>, Hyperbolic equations and related topics (Katata/Kyoto, 1984), Academic Press, Boston, MA, 1986, pp. 63\u201384.",
      "DOI": "10.1016/B978-0-12-501658-2.50008-7"
    },
    {
      "key": "22",
      "unstructured": "[I3] M. Ikawa, <i>Decay of solutions of the wave equations in the exterior of several convex bodies</i>, Preprint."
    },
    {
      "key": "23",
      "unstructured": "[I4] M. Ikawa, 1988, Proc. of Japan Academy."
    },
    {
      "key": "24",
      "doi-asserted-by": "publisher",
      "unstructured": "[In] A. Intissar, <i>A polynomial bound on the number of the scattering poles for a potential in even-dimensional spaces $\\bf R\\sp n$</i>, Comm. Partial Differential Equations <b>11</b> (1986), no. 4, 367\u2013396.",
      "DOI": "10.1080/03605308608820428"
    },
    {
      "key": "25",
      "doi-asserted-by": "publisher",
      "unstructured": "[Ke] A. Kelley, <i>The stable, center-stable, center, center-unstable and unstable manifolds</i>, W. A. Benjamin Inc., New York, Amsterdam, 1967, Appendix C in Transversal mappings and flows by R. Abraham and J. Robbin.",
      "DOI": "10.1016/0022-0396(67)90016-2"
    },
    {
      "key": "26",
      "doi-asserted-by": "publisher",
      "unstructured": "[K] M. Klein, <i>On the absence of resonances for Schr\u00f6dinger operators with nontrapping potentials in the classical limit</i>, Comm. Math. Phys. <b>106</b> (1986), no. 3, 485\u2013494.",
      "DOI": "10.1007/BF01207259"
    },
    {
      "key": "27",
      "unstructured": "[LPh] P. Lax and R. Phillips, <i>Scattering theory</i>, Pure and Appl. Math., vol. 26, Academic Press, New York, 1967."
    },
    {
      "key": "28",
      "doi-asserted-by": "publisher",
      "unstructured": "[Le] G. Lebeau, <i>R\u00e9gularit\u00e9 Gevrey $3$ pour la diffraction</i>, Comm. Partial Differential Equations <b>9</b> (1984), no. 15, 1437\u20131494.",
      "DOI": "10.1080/03605308408820368"
    },
    {
      "key": "29",
      "doi-asserted-by": "crossref",
      "unstructured": "[M] R. Melrose, <i>Polynomials bound on the distribution of poles in scattering by an obstacle</i>, Proceedings of the Journ\u00e9es \u201cEquations aux d\u00e9riv\u00e9es partielles\u201d \u00e0 St Jean de Montes, Soci\u00e9t\u00e9 Math\u00e9matique de France, 4\u20138 Juin 1984.",
      "DOI": "10.5802/jedp.285"
    },
    {
      "key": "30",
      "doi-asserted-by": "publisher",
      "unstructured": "[N1] S. Nakamura, <i>A note on the absence of resonances for Schr\u00f6dinger operators</i>, Lett. Math. Phys. <b>16</b> (1988), no. 3, 217\u2013223.",
      "DOI": "10.1007/BF00398958"
    },
    {
      "key": "31",
      "doi-asserted-by": "publisher",
      "unstructured": "[N2] S. Nakamura, <i>Shape resonances for distortion analytic Schr\u00f6dinger operators</i>, Preprint.",
      "DOI": "10.1080/03605308908820659"
    },
    {
      "key": "32",
      "doi-asserted-by": "crossref",
      "unstructured": "[PSt] V. Petkov Stojanov, <i>Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles</i>, Preprint, 1987.",
      "DOI": "10.5802/jedp.332"
    },
    {
      "key": "33",
      "doi-asserted-by": "crossref",
      "unstructured": "[Sh] M. Shub, <i>Global stability of dynamical systems</i>, Springer Verlag, New York, 1987.",
      "DOI": "10.1007/978-1-4757-1947-5"
    },
    {
      "key": "34",
      "unstructured": "[Si] Ya. G. Sinai, <i>Development of Krylov's ideas</i>, Princeton Univ. Press, 1979, Addendum to Works on the foundations of statistical physics, by N. S. Krylov."
    },
    {
      "key": "35",
      "unstructured": "[S1] J. Sj\u00f6strand, <i>Singularit\u00e9s analytiques microlocales</i>, Ast\u00e9risque, 95, Ast\u00e9risque, vol. 95, Soc. Math. France, Paris, 1982, pp. 1\u2013166."
    },
    {
      "key": "36",
      "unstructured": "[S2] J. Sj\u00f6strand, <i>Propagation of singularities for operators with multiple involutive characteristics</i>, Ann. Inst. Fourier (Grenoble) <b>26</b> (1976), no. 1, v, 141\u2013155."
    },
    {
      "key": "37",
      "unstructured": "[S3] J. Sj\u00f6strand, <i>Semiclassical resonances generated by nondegenerate critical points</i>, Pseudodifferential operators (Oberwolfach, 1986), Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987, pp. 402\u2013429."
    },
    {
      "key": "38",
      "unstructured": "[S4] J. Sj\u00f6strand, <i>Estimates on the number of resonances for semiclassical Schr\u00f6dinger operators</i>, Partial Differential Equations (Rio de Janeiro, 1986), Lecture Notes in Math., vol. 1324, Springer, Berlin, 1988, pp. 286\u2013292."
    },
    {
      "key": "39",
      "doi-asserted-by": "publisher",
      "unstructured": "[T] C. Tricot, <i>Two definitions of fractional dimension</i>, Math. Proc. Cambridge Philos. Soc. <b>91</b> (1982), no. 1, 57\u201374.",
      "DOI": "10.1017/S0305004100059119"
    },
    {
      "key": "40",
      "doi-asserted-by": "publisher",
      "unstructured": "[Z] M. Zworski, <i>Sharp polynomial bounds on the number of scattering poles</i>, Preprint.",
      "DOI": "10.1215/S0012-7094-89-05913-9"
    }
  ],
  "container-title": "Duke Mathematical Journal",
  "original-title": [],
  "link": [
    {
      "URL": "https://projecteuclid.org/journalArticle/Download?urlid=10.1215/S0012-7094-90-06001-6",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2024,
        1,
        30
      ]
    ],
    "date-time": "2024-01-30T17:20:07Z",
    "timestamp": 1706635207000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://projecteuclid.org/journals/duke-mathematical-journal/volume-60/issue-1/Geometric-bounds-on-the-density-of-resonances-for/10.1215/S0012-7094-90-06001-6.full"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1990,
        2,
        1
      ]
    ]
  },
  "references-count": 40,
  "journal-issue": {
    "issue": "1",
    "published-online": {
      "date-parts": [
        [
          1990,
          2,
          1
        ]
      ]
    }
  },
  "URL": "http://dx.doi.org/10.1215/S0012-7094-90-06001-6",
  "relation": {},
  "ISSN": [
    "0012-7094"
  ],
  "subject": [],
  "container-title-short": "Duke Math. J.",
  "published": {
    "date-parts": [
      [
        1990,
        2,
        1
      ]
    ]
  }
}