Spectral theory of translation surfaces : A short introduction

Luc Hillairet[1]

  • [1] Université de Nantes Laboratoire de mathématiques Jean Leray UMR CNRS 6629 2 rue de la Houssinière BP 92208 44322 Nantes cedex 3 (France)

Séminaire de théorie spectrale et géométrie (2009-2010)

  • Volume: 28, page 51-62
  • ISSN: 1624-5458

Abstract

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We define translation surfaces and, on these, the Laplace operator that is associated with the Euclidean (singular) metric. This Laplace operator is not essentially self-adjoint and we recall how self-adjoint extensions are chosen. There are essentially two geometrical self-adjoint extensions and we show that they actually share the same spectrum

How to cite

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Hillairet, Luc. "Spectral theory of translation surfaces : A short introduction." Séminaire de théorie spectrale et géométrie 28 (2009-2010): 51-62. <http://eudml.org/doc/116465>.

@article{Hillairet2009-2010,
abstract = {We define translation surfaces and, on these, the Laplace operator that is associated with the Euclidean (singular) metric. This Laplace operator is not essentially self-adjoint and we recall how self-adjoint extensions are chosen. There are essentially two geometrical self-adjoint extensions and we show that they actually share the same spectrum},
affiliation = {Université de Nantes Laboratoire de mathématiques Jean Leray UMR CNRS 6629 2 rue de la Houssinière BP 92208 44322 Nantes cedex 3 (France)},
author = {Hillairet, Luc},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {translation surfaces; flat Laplace operator; isospectrality},
language = {eng},
pages = {51-62},
publisher = {Institut Fourier},
title = {Spectral theory of translation surfaces : A short introduction},
url = {http://eudml.org/doc/116465},
volume = {28},
year = {2009-2010},
}

TY - JOUR
AU - Hillairet, Luc
TI - Spectral theory of translation surfaces : A short introduction
JO - Séminaire de théorie spectrale et géométrie
PY - 2009-2010
PB - Institut Fourier
VL - 28
SP - 51
EP - 62
AB - We define translation surfaces and, on these, the Laplace operator that is associated with the Euclidean (singular) metric. This Laplace operator is not essentially self-adjoint and we recall how self-adjoint extensions are chosen. There are essentially two geometrical self-adjoint extensions and we show that they actually share the same spectrum
LA - eng
KW - translation surfaces; flat Laplace operator; isospectrality
UR - http://eudml.org/doc/116465
ER -

References

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  1. M. Sh. Birman, M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, (1987), D. Reidel Publishing Co., Dordrecht MR1192782
  2. J. Cheeger, M. Taylor, On the diffraction of waves by conical singularities. I, Comm. Pure Appl. Math. 35 (1982), 275-331 Zbl0526.58049MR649347
  3. G. Forni, Sobolev regularity of solutions of the cohomological equation Zbl0893.58037
  4. G. Grubb, Distributions and operators, 252 (2009), Springer, New York Zbl1171.47001MR2453959
  5. L. Hillairet, Spectral decomposition of square-tiled surfaces, Math. Z. 260 (2008), 393-408 Zbl1156.58012MR2429619
  6. L. Hillairet, A. Kokotov, S -matrix and Krein formula on Euclidean surfaces with conical singularities 
  7. D. Jakobson, M. Levitin, N. Nadirashvili, I. Polterovich, Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond, J. Comput. Appl. Math. 194 (2006), 141-155 Zbl1121.58027MR2230975
  8. A. Kokotov, Compact polyhedral surfaces of an arbitrary genus and determinants of Laplacians Zbl1129.58010
  9. A. Kokotov, D. Korotkin, Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-Singer formula, J. Differential Geom. 82 (2009), 35-100 Zbl1175.30041MR2504770
  10. V. Kostrykin, R. Schrader, Laplacians on metric graphs: eigenvalues, resolvents and semigroups, Quantum graphs and their applications 415 (2006), 201-225, Amer. Math. Soc., Providence, RI Zbl1122.34066MR2277618
  11. M. Reed, B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, (1975), Academic Press [Harcourt Brace Jovanovich Publishers], New York Zbl0242.46001MR493420
  12. M. Reed, B. Simon, Methods of modern mathematical physics. I, (1980), Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York Zbl0459.46001MR751959
  13. M. Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2) 32 (1986), 79-94 Zbl0611.53035MR850552
  14. A. Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I (2006), 437-583, Springer, Berlin Zbl1129.32012MR2261104

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