Spectral Theory and its Applications

1. Introduction 2. Unbounded operators 3. Representation theorems 4. Semibounded operators 5. Compact operators 6. Spectral theory for bounded operators 7. Applications in physics and PDE 8. Spectrum for self-adjoint operators 9. Essentially self-adjoint operators 10. Discrete spectrum, essential spectrum 11. The max-min principle 12. An application to fluid mechanics 13. Pseudospectra 14. Applications for 1D-models 15. Applications in kinetic theory 16. Problems References Index.

[1]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[2]  J. L McGregor.  Solvability criteria for certain N-dimensional moment problems☆ , 1980 .

[3]  B. Helffer,et al.  Superconductivity Near the Normal State Under the Action of Electric Currents and Induced Magnetic Fields in $${\mathbb{R}^2}$$ , 2010 .

[4]  F. Hérau,et al.  Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential , 2004 .

[5]  O. Staffans.  Well-Posed Linear Systems , 2005 .

[6]  Israel Michael Sigal,et al.  Introduction to Spectral Theory: With Applications to Schrödinger Operators , 1995 .

[7]  Steen Pedersen,et al.  Moment problems and subnormality , 1990 .

[8]  F. Nier,et al.  Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator , 2008, 0809.0574.

[9]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[10]  Konrad Schmüdgen,et al.  Unbounded Operator Algebras and Representation Theory , 1990 .

[11]  M. Zworski.  A remark on a paper of E. B. Davies , 2001 .

[12]  Bernard Helffer,et al.  Multiple wells in the semi-classical limit I , 1984 .

[13]  E. Davies,et al.  Non‐Self‐Adjoint Differential Operators , 2002 .

[14]  R. Benguria,et al.  Fourier transform, null variety, and Laplacian's eigenvalues , 2008, 0801.1617.

[15]  P. Masani.  Dilations as Propagators of Hilbertian Varieties , 1978 .

[16]  B. Simon,et al.  Schrödinger operators with magnetic fields , 1981 .

[17]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[18]  L. Thomas,et al.  Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators , 1973 .

[19]  Hamburger and Stieltjes moment problems in several variables , 2002 .

[20]  Bernard Helffer,et al.  Spectral Methods in Surface Superconductivity , 2010 .

[22]  E. B. Davies,et al.  Spectral instability for some Schrödinger operators , 2000, Numerische Mathematik.

[23]  D. Huet.  Décomposition spectrale et opérateurs , 1976 .

[24]  F. Hérau,et al.  Tunnel Effect for Kramers–Fokker–Planck Type Operators , 2007, math/0703684.

[25]  A. Devinatz.  Two parameter moment problems , 1957 .

[26]  Moments and Positivity , 2000 .

[27]  B. Simon.  Functional integration and quantum physics , 1979 .

[28]  B. Sz.-Nagy.  A moment problem for self-adjoint operators , 1952 .

[29]  J. S. MacNerney.  Hermitian moment sequences , 1962 .

[30]  J. Sjoestrand.  Resolvent Estimates for Non-Selfadjoint Operators via Semigroups , 2009, 0906.0094.

[31]  A Complete Study of the Pseudo‐Spectrum for the Rotated Harmonic Oscillator , 2006 .

[32]  W. Stinespring.  Positive functions on *-algebras , 1955 .

[33]  Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra , 1999, math/9909179.

[34]  R. Dautray,et al.  Analyse mathématique et calcul numérique pour les sciences et les techniques , 1984 .

[35]  Alain Grigis,et al.  Microlocal Analysis for Differential Operators: An Introduction , 1994 .

[36]  J. Stochel.  Decomposition and disintegration of positive definite kernels on convex *-semigroups , 1992 .

[37]  J. Rubinstein,et al.  The Resistive State in a Superconducting Wire: Bifurcation from the Normal State , 2007, 0712.3531.

[38]  Larry Gearhart,et al.  Spectral theory for contraction semigroups on Hilbert space , 1978 .

[39]  M. Berger,et al.  Le Spectre d'une Variete Riemannienne , 1971 .

[40]  J. Nourrigat,et al.  Hypoellipticit'e maximale pour des op'erateurs polyn?omes de champs de vecteurs , 1980 .

[41]  Shing-Tung Yau,et al.  On the Schrödinger equation and the eigenvalue problem , 1983 .

[42]  J. Friedrich.  Operator Moment Problems , 1991 .

[43]  C. Berg,et al.  Harmonic Analysis on Semigroups , 1984 .

[44]  Jan Prüss,et al.  On the spectrum of ₀-semigroups , 1984 .

[45]  R. Curto,et al.  Flat Extensions of Positive Moment Matrices: Recursively Generated Relations , 1998 .

[46]  Christiane Tretter,et al.  Spectral properties of the Orr-Sommerfeld problem , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[47]  F. H. Szafraniec,et al.  The complex moment problem and subnormality : a polar decomposition approach , 1998 .

[48]  ON THE COMMUTATIVE FAMILY OF SUBNORMAL OPERATORS , 1958 .

[49]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .

[50]  Ameer Athavale.  Holomorphic kernels and commuting operators , 1987 .

[51]  Mildred Hager.  Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints I: un modèle , 2006 .

[52]  I. Herbst.  Dilation analyticity in constant electric field , 1979 .

[53]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[54]  A. Atzmon.  A moment problem for positive measures on the unit disc. , 1975 .

[55]  Moment Problems for Multi-sequences of Operators , 1998 .

[56]  C. Simader.  Essential self-adjointness of Schrödinger operators bounded from below , 1978 .

[57]  E. Davies,et al.  Linear Operators and their Spectra , 2007 .

[58]  A. Lubin.  Weighted shifts and commuting normal extension , 1979, Journal of the Australian Mathematical Society.

[59]  B. Helffer.  Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics , 2002 .

[60]  Mouez Dimassi,et al.  Spectral asymptotics in the semi-classical limit , 1999 .

[61]  Christiaan C. Stolk,et al.  Semiclassical Analysis for the Kramers–Fokker–Planck Equation , 2004, math/0406275.

[62]  B. Helffer,et al.  Eigenvalues Variation. I.: Neumann Problem for Sturm-Liouville Operators , 1993 .

[63]  B. Helffer.  Théorie spectrale pour des opérateurs globalement elliptiques , 1984 .

[64]  Cherfils,et al.  Analytic solutions of the rayleigh equation for linear density profiles , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[65]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[66]  J. Agler.  The Arveson Extension Theorem and coanalytic models , 1982 .

[67]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[68]  Andrew J. Bernoff,et al.  Onset of superconductivity in decreasing fields for general domains , 1998 .

[69]  D. Robert.  Autour de l'approximation semi-classique , 1987 .

[70]  B. Simon.  Trace ideals and their applications , 1979 .

[71]  B. Helffer.  ON PSEUDO-SPECTRAL PROBLEMS RELATED TO A TIME-DEPENDENT MODEL IN SUPERCONDUCTIVITY WITH ELECTRIC CURRENT , 2011 .

[72]  E. Davies.  Semi-Classical States for Non-Self-Adjoint Schrödinger Operators , 1998, math/9803129.

[73]  F. Vasilescu.  Quaternionic Cayley Transform , 1999 .

[74]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[75]  Christian Berg,et al.  Rotation invariant moment problems , 1991 .

[76]  D. Handelman.  Representing polynomials by positive linear functions on compact convex polyhedra. , 1988 .

[77]  Ari Laptev,et al.  Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces , 1997 .

[78]  E. Davies.  Wild Spectral Behaviour of Anharmonic Oscillators , 2000 .

[79]  P. Masani.  Quasi-isometric measures and their applications , 1970 .

[80]  K. Schmüdgen.  TheK-moment problem for compact semi-algebraic sets , 1991 .

[81]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[82]  渋谷 泰隆.  Global theory of a second order linear ordinary differential equation with a polynomial coefficient , 1975 .

[83]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.

[84]  E. Haviland,et al.  On the Momentum Problem for Distribution Functions in More Than One Dimension. II , 1935 .

[85]  M. Zworski,et al.  Pseudospectra of semiclassical (pseudo‐) differential operators , 2004 .

[86]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[87]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[88]  W. Rudin.  Real and complex analysis , 1968 .

[89]  B. Simon,et al.  Schrödinger operators with magnetic fields. I. general interactions , 1978 .

[90]  Hans L. Cycon,et al.  Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry , 1987 .

[91]  F. Vasilescu.  Existence of Unitary Dilations as a Moment Problem , 2003 .

[92]  E. Davies,et al.  Pseudo–spectra, the harmonic oscillator and complex resonances , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[93]  J. Glimm,et al.  Quantum Physics: A Functional Integral Point of View , 1981 .

[94]  P. Raviart,et al.  Asymptotic Results for the Linear Stage of the Rayleigh—Taylor Instability , 2001 .

[95]  David Krejcirík,et al.  The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions , 2012, Asymptot. Anal..

[96]  G. Hardy,et al.  Note on a theorem of Hilbert , 1920 .

[97]  Y. C. Verdière,et al.  Spectres de graphes , 1998 .

[98]  W. Arendt.  Vector-valued laplace transforms and cauchy problems , 2002 .

[99]  J. Prüss.  On the Spectrum of C 0 -Semigroups , 1984 .

[100]  V. Georgescu,et al.  On the Virial Theorem in Quantum Mechanics , 1999 .

[101]  F. Nier,et al.  Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians , 2005 .

[102]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[103]  P. Blanchard,et al.  Bound states for Schrodinger Hamiltonians: Phase space methods and applications , 1996 .

[104]  L. Hörmander.  Hypoelliptic second order differential equations , 1967 .

[105]  И.М. Гельфанд,et al.  On the imbedding of normed rings into the ring of operators in Hilbert space , 1943 .

[106]  Pseudospectra of semi-classical (pseudo)differential operators , 2003, math/0301242.

[107]  J. Sjoestrand,et al.  Elementary linear algebra for advanced spectral problems , 2003, math/0312166.

[108]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[109]  J. Eckmann,et al.  Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures , 1998, chao-dyn/9804001.

[110]  Y. ALMOG,et al.  The Stability of the Normal State of Superconductors in the Presence of Electric Currents , 2008, SIAM J. Math. Anal..