Inverse spectral problem for analytic domains, II: $\mathbb Z_2$-symmetric domains

Abstract

This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a $\mathbb{Z}_2$ symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class.

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      ZBLNUMBER = {0869.35003},
      MRREVIEWER = {Luigi Rodino},
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    @incollection{V, MRKEY = {2182064},
      AUTHOR = {Vasy, Andr{á}s},
      TITLE = {Propagation of singularities for the wave equation on manifolds with corners},
      BOOKTITLE = {Séminaire: Équations aux Dérivées Partielles. 2004--2005},
      PAGES = {Exp. No. XX, 18},
      PUBLISHER = {École Polytech.},
      ADDRESS = {Palaiseau},
      YEAR = {2005},
      MRCLASS = {58J47 (35A21 35L05 58J40)},
      MRNUMBER = {2006j:58045},
      invmarginpar = {is this reference correct?},
      MRREVIEWER = {Sandro Coriasco},
      }
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    @article {V2, MRKEY = {MR2456883},
      AUTHOR = {Vasy, Andr{á}s},
      TITLE = {Propagation of singularities for the wave equation on manifolds with corners},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {168},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {749--812},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58J47 (35L05)},
      MRNUMBER = {2456883},
      DOI = {10.4007/annals.2008.168.749},
      }
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    @article{Z1, MRKEY = {1713144},
      AUTHOR = {Zelditch, Steve},
      TITLE = {Spectral determination of analytic bi-axisymmetric plane domains},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {6},
      YEAR = {1999},
      NUMBER = {3-4},
      PAGES = {457--464},
      ISSN = {1073-2780},
      MRCLASS = {58J53 (35P05 35R30 37D50 37N20 58J50)},
      MRNUMBER = {2000h:58060},
      ZBLNUMBER = {0960.58017},
      MRREVIEWER = {Dorothee Schueth},
      }
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    @article{Z2, MRKEY = {1779616},
      AUTHOR = {Zelditch, Steve},
      TITLE = {Spectral determination of analytic bi-axisymmetric plane domains},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {10},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {628--677},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {58J50 (35J05 35P99 35R30 58J37)},
      MRNUMBER = {2001k:58064},
      ZBLNUMBER = {0961.58012},
      MRREVIEWER = {Georgi Popov},
      DOI = {10.1007/PL00001633},
      }
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    @article{Z3, MRKEY = {1664907},
      AUTHOR = {Zelditch, Steve},
      TITLE = {The inverse spectral problem for surfaces of revolution},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {49},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {207--264},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {58G25 (53C22 58D27)},
      MRNUMBER = {99k:58188},
      ZBLNUMBER = {0938.58027},
      MRREVIEWER = {Edoh Amiran},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214461019},
      }
  • [Z4] S. Zelditch, "Inverse resonance problem for $\Bbb Z_2$-symmetric analytic obstacles in the plane," in Geometric Methods in Inverse Problems and PDE Control, Croke, C. B., Lasiecka, I., Uhlmann, G., and Vogelius, M. S., Eds., New York: Springer-Verlag, 2004, pp. 289-321.
    @incollection{Z4, MRKER = {2169909},
      AUTHOR = {Zelditch, Steve},
      EDITOR = {Croke, C. B. and Lasiecka, I. and Uhlmann, G. and Vogelius, M. S.},
      TITLE = {Inverse resonance problem for {$\Bbb Z\sb 2$}-symmetric analytic obstacles in the plane},
      BOOKTITLE = {Geometric Methods in Inverse Problems and PDE Control},
      SERIES = {IMA Vol. Math. Appl.},
      NUMBER = {137},
      PAGES = {289--321},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2004},
      MRCLASS = {58J50 (35P20 35R30)},
      MRNUMBER = {2007h:58055a},
      MRREVIEWER = {Jonathan Huntley},
      }
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    @article{Zfi, MRKEY = {2073139},
      AUTHOR = {Zelditch, Steve},
      TITLE = {Inverse spectral problem for analytic domains. {I}: {B}alian-{B}loch trace formula},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {248},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {357--407},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {58J53 (35P20 35R30)},
      MRNUMBER = {2005f:58063},
      ZBLNUMBER = {1086.58016},
      MRREVIEWER = {David Borthwick},
      DOI = {10.1007/s00220-004-1074-y},
      }
  • [Zsi] S. Zelditch, "Inverse spectral problem for analytic domains, II: Domains with symmetry," , preprint , 2001.
    @techreport{Zsi,
      author = {Zelditch, Steve},
      TITLE = {Inverse spectral problem for analytic domains, {II}: {D}omains with symmetry},
      ARXIV = {math.SP/0111078v1},
      TYPE = {preprint},
      YEAR = {2001},
      }
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    @techreport{H,
      author = {Hezari, H.},
      title = {Inverse spectral problem for Schrödinger operators},
      url = {http://tinyurl.com/cab9ks},
      type = {preprint},
      year = {2008},
      }
  • [HZ2] H. Hezari and S. Zelditch, "Inverse spectral problem for analytic $\mathbb Z_2^n$-symmetric domains in $\mathbb R^n$," , preprint.
    @techreport{HZ2,
      author = {H. Hezari and S. Zelditch},
      title = {Inverse spectral problem for analytic $\mathbb Z_2^n$-symmetric domains in $\mathbb R^n$},
      type = {preprint},
      arxiv = {0902.1373},
      }

Authors

Steve Zelditch

Department of Mathematics
Johns Hopkins University
Baltimore, MD 21218
United States