Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.

[1]  Didier Robert.  Non linear eigenvalue problems , 2004 .

[2]  M. S. Baouendi,et al.  Nonanalytic-hypoellipticity for some degenerate elliptic operators , 1972 .

[3]  Failure of analytic hypoellipticity in a class of differential operators , 2003 .

[4]  O. Oleinik,et al.  ON THE ANALYTICITY OF SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1973 .

[5]  G. Métivier.  Une classe d'opérateurs non hypoelliptiques analytiques , 1979 .

[6]  M. A. Kaashoek,et al.  Review: A. S. Markus, Introduction to the spectral theory of polynomial operator pencils , 1989 .

[7]  D. Robert,et al.  Sur un probleme aux valeurs propres non line aire , 1980 .

[8]  M. Solomjak,et al.  Spectral Theory of Self-Adjoint Operators in Hilbert Space , 1987 .

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

[10]  F. Trèves.  Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the -neumann problem , 1978 .

[11]  B. Helffer,et al.  Proprietes asymptotiques du spectre dioperateurs pseuwdifferentiels sur IRn , 1982 .

[12]  Singular solutions for sums of squares of vector fields , 1991 .

[13]  D. Robert.  Non linear eigenvalues problems with a small parameter , 1984 .

[14]  Certain sums of Squares of Vector Fields Fail to be Analytic Hypoelliptic , 1991 .

[15]  J. Sjöstrand.  Analytic wavefront sets and operators with multiple characteristics , 1983 .

[16]  Hypoellipticity: Geometrization and speculation , 1998, math/9801142.

[17]  G. Métivier.  Hypoellipticite analytique sur des groupes nilpotents de rang $2$ , 1980 .

[18]  Avner Friedman,et al.  Nonlinear eigenvalue problems , 1968 .

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

[20]  A Necessary Condition For Analytic Hypoellipticity , 1994 .

[21]  Singular solutions for a class of Grusin type operators , 1996 .

[22]  Some non-analytic-hypoelliptic sums of squares of vector fields , 1992, math/9201260.

[23]  Non-analytic hypoellipticity in the presence of symplecticity , 1998 .

[24]  Analytic regularity for an operator with Treves curves , 2004 .

[25]  B. Helffer.  Partial differential equations on nilpotent groups , 1984 .

[26]  D. S. Tartakoff,et al.  The local real analyticity of solutions to □b and the $$\bar \partial $$ -Neumann problem , 1980 .

[27]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[28]  Bernard Helffer,et al.  Caracterisation des operateurs hypoelliptiques homogenes , 1979 .

[29]  V. Grusin.  ON A CLASS OF HYPOELLIPTIC OPERATORS , 1970 .

[30]  Kirillov theory, Treves strata, Schrodinger equations and analytic hypoellipticity of sums of squares , 2001, math/0107106.