Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-t4qhp Total loading time: 0.527 Render date: 2022-08-09T05:00:14.089Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Bibliography

Published online by Cambridge University Press:  05 March 2013

Bernard Helffer
Affiliation:
Université de Paris-Sud
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arendt, W., Batty, C., Hieber, M., and Neubrander, F. 2001. Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser.CrossRefGoogle Scholar
Abramowitz, M. and Stegun, I. A. 1964. Handbook of Mathematical Functions, Applied Mathematics Series, Vol. 55. National Bureau of Standards.Google Scholar
Agmon, S. 1982. Lecture on Exponential Decay of Solutions of Second Order Elliptic Equations, Mathematical Notes, Vol. 29. Princeton University Press.Google Scholar
Almog, Y., Helffer, B., and Pan, X. 2010. Superconductivity near the normal state under the action of electric currents and induced magnetic field in ℝ2. Commun. Math. Phys., 300 (1), 147–184.CrossRefGoogle Scholar
Almog, Y., Helffer, B., and Pan, X. 2011. Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field. To appear in Trans. Am. Math. Soc.
Avron, J., Herbst, I., and Simon, B. 1978. Schrödinger operators with magnetic fields I. Duke Math. J., 45, 847–883.CrossRefGoogle Scholar
Akhiezer, N. I. and Glazman, I. M. 1981. Theory of Linear Operators in Hilbert Space. Pitman.Google Scholar
Almog, Y. 2008. The stability of the normal state of superconductors in the presence of electric currents. SIAM J. Math. Anal., 40(2), 824–850.CrossRefGoogle Scholar
Aslayan, A. and Davies, E. B. 2000. Spectral instability for some Schrödinger operators. Numer. Math., 85, 525–552.CrossRefGoogle Scholar
Bernoff, A. and Sternberg, P. 1998. Onset of superconductivity in decreasing fields for general domains. J. Math. Phys., 39, 1272–1284.CrossRefGoogle Scholar
Berger, M., Gauduchon, P., and Mazet, E. 1971. Spectre d’une Variété Riemannienne, Lecture Notes in Mathematics, Vol. 194. Springer.CrossRefGoogle Scholar
Benguria, R., Levitin, M., and Parnovski, L. 2009. Fourier transform, null variety, and Laplacian’s eigenvalues. J. Funct. Anal., 257(7), 2088–2123.CrossRefGoogle Scholar
Bordeaux-Montrieux, W. 2010. Estimation de résolvante et construction de quasi-modes près du bord du pseudospectre. Preprint.
Borisov, D. and Krejcirik, D. 2012. The effective Hamiltonian for thin layers with non-hermitian Robin-type boundary. Asymptot. Anal., 76, 49–59.Google Scholar
Boulton, L. S. 2002. Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra. J. Oper. Theory, 47(2), 413–429.Google Scholar
Brézis, H.. 2005. Analyse Fonctionnelle. Editions Masson.Google Scholar
Blanchard, P. and Stubbe, J. 1996. Bound states for Schrödinger Hamiltonians. Phase space methods and applications. Rev. Math. Phys., 35, 504–547.Google Scholar
Cherfils-Clerouin, C., Lafitte, O., and Raviart, P. -A. 2001. Asymptotics results for the linear stage of the Rayleigh–Taylor instability. In Neustupa, J. and Penel, P. (eds.), Mathematical Fluid Mechanics: Recent Results and Open Questions, Advances in Mathematical Fluid Mechanics, pp. 47–71. Birkhäuser.CrossRefGoogle Scholar
Colin de Verdière, Y. 1998. Spectres de graphes, Cours Spécialisé, 4. Société Mathématique de France.Google Scholar
Cycon, H. L., Froese, R., Kirsch, W., and Simon, B. 1987. Schrödinger Operators: With Applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics. Springer.Google Scholar
Courant, R. and Hilbert, D. 1953. Methods of Mathematical Physics. Wiley- Interscience.Google Scholar
Cherfils, C. and Lafitte, O. 2000. Analytic solutions of the Rayleigh equation for linear density profiles. Phys. Rev. E, 62(2), 2967–2970.CrossRefGoogle ScholarPubMed
Combes, J. -M. and Thomas, L. 1973. Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys., 34, 251–270.CrossRefGoogle Scholar
Dauge, M. and Helffer, B. 1993. Eigenvalues variation I, Neumann problem for Sturm–Liouville operators. J. Differ. Equ., 104(2), 243–262.CrossRefGoogle Scholar
Dautray, R. and Lions, J. -L.. 1988–1995. Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Masson.Google Scholar
Davies, E. B. 1996. Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, Vol. 42. Cambridge University Press.Google Scholar
Davies, E. B. 1999. Semi-classical states for non-self-adjoint Schrödinger operators. Commun. Math. Phys., 200, 35–41.CrossRefGoogle Scholar
Davies, E. B. 1999. Pseudospectra, the harmonic oscillator and complex resonances. Proc. R. Soc. Lond. Ser. A, 455, 585–599.CrossRefGoogle Scholar
Davies, E. B. 2000. Wild spectral behaviour of anharmonic oscillators. Bull. Lond. Math. Soc., 32, 432–438.CrossRefGoogle Scholar
Davies, E. B. 2002. Non-self-adjoint differential operators. Bull. Lond. Math. Soc., 34, 513–532.CrossRefGoogle Scholar
Davies, E. B. 2005. Semigroup growth bounds. J. Oper. Theory, 53(2), 225–249.Google Scholar
Davies, E. B. 2007. Linear Operators and Their Spectra, Cambridge Studies in Advanced Mathematics, Vol. 106. Cambridge University Press.CrossRefGoogle Scholar
Dieudonné, J.. 1980. Calcul Infinitésimal. Hermann.Google Scholar
Dimassi, M. and Sjöstrand, J.. 1999. Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, Vol. 268. Cambridge University Press.CrossRefGoogle Scholar
Dencker, N., Sjöstrand, J., and Zworski, M. 2004. Pseudospectra of semi-classical (pseudo)differential operators. Commun. Pure Appl. Math., 57(3), 384–415.CrossRefGoogle Scholar
Deng, W. 2012. Etude du pseudo-spectre d’opérateurs non auto-adjoints liés à la mécanique des fluides. Thèse de doctorat, Université Pierre et Marie Curie.Google Scholar
Eckmann, J. -p., Pillet, C. A., and Rey-Bellet, L. 1999. Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys., 208(2), 275–281.Google Scholar
Engel, K. J. and Nagel, R. 2000. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Vol. 194. Springer.Google Scholar
Engel, K. J. and Nagel, R. 2005. A Short Course on Operator Semi-Groups, Unitext. Springer.Google Scholar
Fournais, S. and Helffer, B. 2010. Spectral Methods in Surface Superconductivity, Progress in Nonlinear Differential Equations and Their Applications, Vol. 77. Birkhäuser.Google Scholar
Gearhart, L. 1978. Spectral theory for contraction semigroups on Hilbert spaces. Trans. Am. Math. Soc., 236, 385–394.CrossRefGoogle Scholar
Gallagher, I., Gallay, T., and Nier, F. 2009. Spectral asymptotics for large skewsymmetric perturbations of the harmonic oscillator. Int. Math. Res. Not., 12(12), 2147–2199.Google Scholar
Gilbarg, D. and Trudinger, N. S. 1998. Elliptic Partial Differential Equations of Second Order. Springer.Google Scholar
Glimm, J. and Jaffe, A. 1987. Quantum Physics: A Functional Integral Point of View, 2nd edition. Springer.CrossRefGoogle Scholar
Grigis, A. and Sjöstrand, J. 1994. Microlocal Analysis for Differential Operators: An Introduction, London Mathematical Society Lecture Note Series, Vol. 196. Cambridge University Press.CrossRefGoogle Scholar
Hager, M. 2006. Instabilité spectrale semi-classique pour des opérateurs non-autoadjoints I: un modèle. Ann. Fac. Sci. Toulouse Math. (6), 15(2), 243–280.CrossRefGoogle Scholar
Hardy, G. H. 1920. Note on a theorem of Hilbert. Math. Z., 6, 314–317.CrossRefGoogle Scholar
Helffer, B. 1984. Théorie Spectrale pour des Opérateurs Globalement Elliptiques, Astérisque, Vol. 112. Société Mathématique de France.Google Scholar
Helffer, B. 1988. Semiclassical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics, Vol. 1336. Springer.CrossRefGoogle Scholar
Helffer, B. 1995. Semiclassical Analysis for Schrödinger Operators, Laplace Integrals and Transfer Operators in Large Dimension: An Introduction, Cours de DEA. Paris Onze Edition.Google Scholar
Helffer, B. 2002. Semiclassical Analysis, Witten Laplacians and Statistical Mechanics, Series on Partial Differential Equations and Applications, Vol. 1. World Scientific.CrossRefGoogle Scholar
Helffer, B. 2011. On pseudo-spectral problems related to a time dependent model in superconductivity with electric current. Confluentes Math., 3(2), 237–251.CrossRefGoogle Scholar
Helffer, B. and Lafitte, O. 2003. Asymptotics methods for the eigenvalues of the Rayleigh equation. Asymptot. Anal., 23(3–4), 189–236.Google Scholar
Helffer, B. and Nier, F. 2004. Hypoelliptic Estimates and Spectral Theory for Fokker– Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, Vol. 1862. Springer.Google Scholar
Helffer, B. and Nourrigat, J. 1985. Hypoellipticité Maximale pour des Opérateurs Polynômes de Champs de Vecteurs, Progress in Mathematics, Vol. 58. Birkhäuser.Google Scholar
Henry, R. 2010. Master’s thesis, UniversitéParis-Sud11.Google Scholar
Helffer, B. and Sjöstrand, J. 1984. Multiple wells in the semiclassical limit I. Commun. Partial Differ. Equ., 9(4), 337–408.CrossRefGoogle Scholar
Helffer, B. and Sjöstrand, J. 2010. From resolvent bounds to semigroup bounds. Preprint, .
Herbst, I. 1979. Dilation analyticity in constant electric field I. The two body problem. Commun. Math. Phys., 64, 279–298.CrossRefGoogle Scholar
Hérau, F. and Nier, F. 2004. Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential. Arch. Ration. Mech. Anal., 171(2), 151–218.CrossRefGoogle Scholar
Hérau, F., Sjöstrand, J., and Stolk, C. 2005. Semi-classical analysis for the Kramers–Fokker–Planck equation. Commun. Partial Differ Equ., 30(5–6), 689–760.CrossRefGoogle Scholar
Huang, F. L. 1985. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ., 1, 43–56.Google Scholar
Hérau, F., Hitrik, M., and Sjöstrand, J. 2008. Tunnel effect for Kramers–Fokker–Planck type operators. Ann. Henri Poincaré, 9(2), 209–274.CrossRefGoogle Scholar
Hérau, F., Hitrik, M., and Sjöstrand, J. 2008. Kramers–Fokker–Planck type operators: return to equilibrium and applications. Int. Math. Res. Not., Article ID rnn057, 48 pp.
Hislop, P. D. and Sigal, I. M. 1995. Introduction to Spectral Theory: With Applications to Schrödinger Operators, Applied Mathematical Sciences, Vol. 113. Springer.Google Scholar
Hörmander, L. 1967. Hypoelliptic second order differential equations. Acta Math, 119, 147–171.CrossRefGoogle Scholar
Hörmander, L. 1985. The Analysis of Linear Partial Differential Operators. Springer.Google Scholar
Huet, D. 1976. Décomposition Spectrale et Opérateurs. Presses universitaires de France.Google Scholar
Kato, T. 1966. Perturbation Theory for Linear Operators. Springer.Google Scholar
Lafitte, O. 2001. Sur la phase linéaire de l’instabilité de Rayleigh-Taylor. Séminaire EDP de l’Ecole Polytechnique. .
Langer, H. and Tretter, C. 1997. Spectral properties of the Orr–Sommerfeld problem. Proc. R. Soc. Edinb. Sect. A, 127, 1245–1261.CrossRefGoogle Scholar
Laptev, A. 1997. Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J. Funct. Anal., 151(2), 531–545.CrossRefGoogle Scholar
Lévy-Bruhl, P. 2003. Introduction à la Théorie Spectrale. Editions Dunod.Google Scholar
Lieb, E. and Loss, M. 1996. Analysis, Graduate Studies in Mathematics, Vol. 14. American Mathematical Society.Google Scholar
Lions, J. -L. and Magenes, E. 1968. Problèmes aux Limites Non-homogènes. Tome 1. Editions Dunod.Google Scholar
Lions, J. -L. 1957. Lecture on Elliptic Partial Differential Equations. Tata Institute of Fundamental Research, Bombay.Google Scholar
Lions, J. -L. 1962. Problèmes aux limites dans les EDP. Séminaire de Mathématiques supérieures de l’université de Montréal.
Li, P. and Yau, S. T. 1983. On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys., 88(3), 309–318.CrossRefGoogle Scholar
Martinet, J. 2009. Sur les propriétés spectrales d’opérateurs non-autoadjoints provenant de la mécanique des fluides. Thèse de doctorat, Université Paris-Sud 11.Google Scholar
Pazy, A. 1983. Semigroups of Linear Operators and Applications to Partial Differential Operators, Applied Mathematical Sciences, Vol. 44. Springer.CrossRefGoogle Scholar
Prüss, J. 1984. On the spectrum of C0-semigroups. Trans. Am. Math. Soc., 284, 847– 857.CrossRefGoogle Scholar
Pravda-Starov, K. 2006. A complete study of the pseudo-spectrum for the rotated harmonic oscillator. J. Lond. Math. Soc., 73(3), 745–761.CrossRefGoogle Scholar
Risken, H. 1989. The Fokker–Planck Equation: Methods of Solution and Applications, 2nd edition. Springer.CrossRefGoogle Scholar
Robert, D. 1987. Autour de l’Approximation Semi-classique, Progress in Mathematics, Vol. 68. Birkhäuser.Google Scholar
Roch, S. and Silbermann, B. 1996. C*-algebras techniques in numerical analysis. J. Oper. Theory, 35, 241–280.Google Scholar
Reed, M. and Simon, B. 1972. Methods of Modern Mathematical Physics, Vol. I: Functional Analysis. Academic Press.Google Scholar
Reed, M. and Simon, B. 1975. Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness. Academic Press.Google Scholar
Reed, M. and Simon, B. 1976. Methods of Modern Mathematical Physics, Vol. III: Scattering Theory. Academic Press.Google Scholar
Reed, M. and Simon, B. 1978. Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators. Academic Press.Google Scholar
Rubinstein, J., Sternberg, P., and Zumbrun, K. 2010. The resistive state in a superconducting wire: bifurcation from the normal state. Arch. ation. Mech. Anal R., 195(1), 117–158.CrossRefGoogle Scholar
Rudin, W. 1974. Real and Complex Analysis. McGraw-Hill.Google Scholar
Rudin, W. 1997. Analyse Fonctionnelle. Ediscience International.Google Scholar
Simon, B. 1979. Functional Integration and Quantum Physics, Pure and Applied Mathematics, Vol. 86. Academic Press.Google Scholar
Simon, B. 2005. Trace Ideals and Their Applications, 2nd edition, Mathematical Surveys and Monographs, Vol. 120. American Mathematical Society.Google Scholar
Sibuya, Y. 1975. Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient. North-Holland.Google Scholar
Simader, C. G. 1978. Essential self-adjointness of Schrödinger operators bounded from below. Math. Z., 159, 47–50.CrossRefGoogle Scholar
Sjöstrand, J. 2003. Pseudospectrum for differential operators. Séminaire à l’Ecole Polytechnique, Exp. No. XVI, Séminaire Equations aux Dérivées Partielles, Ecole Polytechnique, Palaiseau.Google Scholar
Sjöstrand, J. 2009. Spectral properties for non self-adjoint differential operators. In Proceedings ofColloque sur les équations aux Dérivées Partielles. évian.Google Scholar
Sjöstrand, J. 2010. Resolvent estimates for non-selfadjoint operators via semigroups. In Laptev, A. (ed.), Around the Research of Vladimir Maz’ya III, International Mathematical Series, Vol. 13, pp. 359–384. Springer/Tamara Rozhkovskaya.CrossRefGoogle Scholar
Sjöstrand, J. and Zworski, M. 2007. Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier, 57(7), 2095–2141.CrossRefGoogle Scholar
Staffans, O. 2005. Well-Posed Linear Systems. Cambridge University Press.CrossRefGoogle Scholar
Trefethen, L. N. 1997. Pseudospectra of linear operators. SIAM Rev., 39, 383–400.CrossRefGoogle Scholar
Trefethen, L. N. 2000. Spectral Methods in MATLAB. SIAM.CrossRefGoogle Scholar
Trefethen, L. N. and Embree, M. 2005. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Villani, C. 2009. Hypocoercivity, Memoirs of the AMS, Vol. 202, No. 950. AmericanMathematical Society.Google Scholar
Yosida, K. 1980. Functional Analysis, Grundlehren der mathematischen Wissenschaften, Vol. 123. Springer.Google Scholar
Zuily, C. 2000. Eléments de Distributions et d’équations aux Dérivées Partielles, Collection Sciences Sup. Editions Dunod.Google Scholar
Zworski, M. 2001. A remark on a paper by E. B. Davies. Proc. Am. Math. Soc., 129, 2955–2957.CrossRefGoogle Scholar
Zworski, M. 2012. Semiclassical Analysis, Graduate Studies in Mathematics, Vol. 138. AmericanMathematical Society.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Bibliography
  • Bernard Helffer
  • Book: Spectral Theory and its Applications
  • Online publication: 05 March 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781139505727.017
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Bibliography
  • Bernard Helffer
  • Book: Spectral Theory and its Applications
  • Online publication: 05 March 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781139505727.017
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Bibliography
  • Bernard Helffer
  • Book: Spectral Theory and its Applications
  • Online publication: 05 March 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781139505727.017
Available formats
×