Asymptotic expansion for nonlinear eigenvalue problems
In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is L ( λ ) = − △ + ( P ( x ) − λ ) 2 in L 2 ( R d ) where P is a positive elliptic polynomial in R d of degree m ⩾ 2 . It is known that for d even, or d =...
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| Published in: | Journal de mathématiques pures et appliquées Vol. 93; no. 2; pp. 149 - 162 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01.02.2010
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| ISSN: | 0021-7824 |
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| Abstract | In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is
L
(
λ
)
=
−
△
+
(
P
(
x
)
−
λ
)
2
in
L
2
(
R
d
)
where
P is a positive elliptic polynomial in
R
d
of degree
m
⩾
2
. It is known that for
d even, or
d
=
1
, or
d
=
3
and
m
⩾
6
, there exist
λ
∈
C
and
u
∈
L
2
(
R
d
)
,
u
≠
0
, such that
L
(
λ
)
u
=
0
. In this paper, we give a method to prove existence of non-trivial solutions for the equation
L
(
λ
)
u
=
0
, valid in every dimension
d
⩾
1
. This is a partial answer to a conjecture in Helffer, Robert and Xue Ping Wang (2004)
[13].
Dans cet article nous considérons un problème aux valeurs propres généralisé pour une famille d'opérateurs dépendant quadratiquement d'un paramètre complexe. Le modèle étudié concerne la famille
L
(
λ
)
=
−
△
+
(
P
(
x
)
−
λ
)
2
dans
L
2
(
R
d
)
où
P un polynôme elliptique dans
R
d
de degré
m
⩾
2
. Si
d est paire ou si
d
=
1
ou
d
=
3
et
m
⩾
6
, on sait alors qu'il existe
λ
∈
C
et
u
∈
L
2
(
R
d
)
,
u
≠
0
, tels que
L
(
λ
)
u
=
0
. L'objet principal de cet article est de donner une méthode pour démontrer l'existence de solutions non triviales pour l'équation
L
(
λ
)
u
=
0
pour toute dimension
d
⩾
1
. On répond ainsi partiellement à une conjecture formulée dans Helffer, Robert et Xue Ping Wang (2004)
[13]. |
|---|---|
| AbstractList | In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is
L
(
λ
)
=
−
△
+
(
P
(
x
)
−
λ
)
2
in
L
2
(
R
d
)
where
P is a positive elliptic polynomial in
R
d
of degree
m
⩾
2
. It is known that for
d even, or
d
=
1
, or
d
=
3
and
m
⩾
6
, there exist
λ
∈
C
and
u
∈
L
2
(
R
d
)
,
u
≠
0
, such that
L
(
λ
)
u
=
0
. In this paper, we give a method to prove existence of non-trivial solutions for the equation
L
(
λ
)
u
=
0
, valid in every dimension
d
⩾
1
. This is a partial answer to a conjecture in Helffer, Robert and Xue Ping Wang (2004)
[13].
Dans cet article nous considérons un problème aux valeurs propres généralisé pour une famille d'opérateurs dépendant quadratiquement d'un paramètre complexe. Le modèle étudié concerne la famille
L
(
λ
)
=
−
△
+
(
P
(
x
)
−
λ
)
2
dans
L
2
(
R
d
)
où
P un polynôme elliptique dans
R
d
de degré
m
⩾
2
. Si
d est paire ou si
d
=
1
ou
d
=
3
et
m
⩾
6
, on sait alors qu'il existe
λ
∈
C
et
u
∈
L
2
(
R
d
)
,
u
≠
0
, tels que
L
(
λ
)
u
=
0
. L'objet principal de cet article est de donner une méthode pour démontrer l'existence de solutions non triviales pour l'équation
L
(
λ
)
u
=
0
pour toute dimension
d
⩾
1
. On répond ainsi partiellement à une conjecture formulée dans Helffer, Robert et Xue Ping Wang (2004)
[13]. In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is $L(\lambda)=-\triangle +(P(x)-\lambda)^2$ in $L^2(\R^d)$ where $P$ is a positive elliptic polynomial in $\R^d$ of degree $m\geq 2$. It is known that for $d$ even, or $d=1$, or $d=3$ and $m\geq 6$, there exist $\lambda\in\C$ and $u\in L^2(\R^d)$, $u\neq 0$, such that $L(\lambda)u=0$. In this paper, we give a method to prove existence of non trivial solutions for the equation $L(\lambda)u=0$, valid in every dimension. This is a partial answer to a conjecture in \cite{herowa}. |
| Author | Robert, Didier Aboud, Fatima |
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| Cites_doi | 10.4171/ZAA/352 10.1070/RM1971v026n04ABEH003985 10.1007/BF01682844 10.1215/S0012-7094-02-11122-3 10.1007/BF02391910 10.1080/03605309408821040 10.1016/S0022-1236(03)00105-8 10.1215/S0012-7094-93-07222-5 10.1016/0022-1236(83)90034-4 |
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| Keywords | Semiclassical analysis Non-selfadjoint operators Nonlinear eigenvalue problems Trace formula Polynomial Asymptotic expansion Nonexistence of solution Mathematics Nonlinear problems Eigenvalue problem Self adjoint operator nonlinear eigenvalue partial differential equation non-seladjoint operator trace formula semiclassical analysis |
| Language | English |
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| References | Dauge, Robert (bib008) 1986; vol. 1256 G. Métivier, Une classe d'opérateurs non hypoelliptiques analytiques, Séminaire Goulaouic–Schwartz, Exposé XII, 1978–1979 Robert (bib019) 1987; vol. 68 Agranovich, Markus (bib002) 1989; 8 Friedman, Shinbrot (bib009) 1968; 121 Markus (bib016) 1988; vol. 71 Robert (bib020) 2004; vol. 26 Boimatov, Kostyuchenko (bib003) 1991; 25 Christ (bib006) 1993 Helffer, Robert, Wang (bib013) 2004; 16 Bouzouina, Robert (bib004) 2002; 111 Aboud (bib001) Krein, Langer (bib015) 1978; 1 Helffer, Robert (bib012) 1983; 53 Lai, Robert (bib018) 1980; 36 Christ (bib007) 1994; 19 Gohberg, Krein (bib010) 1972 Keldysh (bib014) 1971; 26 Chanillo, Helffer, Laptev (bib005) 2004; 209 B. Helffer, Remarques sur des résultats de G. Métivier sur la nonhypoanalyticité, Séminaire de l'Université de Nantes, exposé No. 9, 1978–1979 Seeley (bib021) 1967; vol. 110 Bouzouina (10.1016/j.matpur.2009.08.009_bib004) 2002; 111 Helffer (10.1016/j.matpur.2009.08.009_bib012) 1983; 53 Keldysh (10.1016/j.matpur.2009.08.009_bib014) 1971; 26 Aboud (10.1016/j.matpur.2009.08.009_bib001) Christ (10.1016/j.matpur.2009.08.009_bib006) 1993 10.1016/j.matpur.2009.08.009_bib011 Robert (10.1016/j.matpur.2009.08.009_bib020) 2004; vol. 26 Chanillo (10.1016/j.matpur.2009.08.009_bib005) 2004; 209 Boimatov (10.1016/j.matpur.2009.08.009_bib003) 1991; 25 Robert (10.1016/j.matpur.2009.08.009_bib019) 1987; vol. 68 Agranovich (10.1016/j.matpur.2009.08.009_bib002) 1989; 8 Seeley (10.1016/j.matpur.2009.08.009_bib021) 1967; vol. 110 Krein (10.1016/j.matpur.2009.08.009_bib015) 1978; 1 Markus (10.1016/j.matpur.2009.08.009_bib016) 1988; vol. 71 Christ (10.1016/j.matpur.2009.08.009_bib007) 1994; 19 Helffer (10.1016/j.matpur.2009.08.009_bib013) 2004; 16 Lai (10.1016/j.matpur.2009.08.009_bib018) 1980; 36 Gohberg (10.1016/j.matpur.2009.08.009_bib010) 1972 Friedman (10.1016/j.matpur.2009.08.009_bib009) 1968; 121 Dauge (10.1016/j.matpur.2009.08.009_bib008) 1986; vol. 1256 10.1016/j.matpur.2009.08.009_bib017 |
| References_xml | – volume: 25 start-page: 90 year: 1991 end-page: 93 ident: bib003 article-title: Trace formula for polynomial operators and its applications publication-title: Funktsional. Anal. i Prilozhen. – volume: vol. 1256 start-page: 90 year: 1986 end-page: 122 ident: bib008 article-title: Weyl's formula for a class of pseudodifferential operators with negative order on publication-title: Lecture Notes in Math. – volume: vol. 68 year: 1987 ident: bib019 article-title: Autour de l'approximation semi-classique publication-title: Progress in Mathematics – reference: B. Helffer, Remarques sur des résultats de G. Métivier sur la nonhypoanalyticité, Séminaire de l'Université de Nantes, exposé No. 9, 1978–1979 – volume: 53 start-page: 246 year: 1983 end-page: 268 ident: bib012 article-title: Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles publication-title: J. Funct. Anal. – year: 1972 ident: bib010 article-title: Introduction à la théorie des opérateurs non nécessairement auto-adjoints – start-page: 595 year: 1993 end-page: 639 ident: bib006 article-title: Analytic hypoellipticity, representation of nilpotent groups and non-linear eigenvalue problem publication-title: Duke Math. J. – volume: 209 start-page: 425 year: 2004 end-page: 443 ident: bib005 article-title: Nonlinear eigenvalues and analytic hypoellipticity publication-title: J. Funct. Anal. – volume: 19 start-page: 911 year: 1994 end-page: 941 ident: bib007 article-title: Examples of analytic non-hypoellipticity of publication-title: Comm. 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J. – volume: 26 start-page: 15 year: 1971 end-page: 44 ident: bib014 article-title: On the completeness of the eigenfunctions of some classes of non-self-adjoint linear operators publication-title: Russian Math. Surveys – volume: 36 start-page: 169 year: 1980 end-page: 186 ident: bib018 article-title: Sur un problème aux valeurs propres non linéaire publication-title: J. Math. – reference: G. Métivier, Une classe d'opérateurs non hypoelliptiques analytiques, Séminaire Goulaouic–Schwartz, Exposé XII, 1978–1979 – volume: 8 start-page: 237 year: 1989 end-page: 260 ident: bib002 article-title: On spectral properties of elliptic pseudo-differential operators far from self-adjoint ones publication-title: Z. Anal. Anwendungen – volume: vol. 110 start-page: 288 year: 1967 end-page: 307 ident: bib021 article-title: Complex powers of an elliptic operators publication-title: Singular Integrals – volume: 8 start-page: 237 issue: 3 year: 1989 ident: 10.1016/j.matpur.2009.08.009_bib002 article-title: On spectral properties of elliptic pseudo-differential operators far from self-adjoint ones publication-title: Z. Anal. Anwendungen doi: 10.4171/ZAA/352 – volume: vol. 68 year: 1987 ident: 10.1016/j.matpur.2009.08.009_bib019 article-title: Autour de l'approximation semi-classique – volume: vol. 26 year: 2004 ident: 10.1016/j.matpur.2009.08.009_bib020 article-title: Nonlinear Eigenvalues Problems – ident: 10.1016/j.matpur.2009.08.009_bib001 – volume: vol. 71 year: 1988 ident: 10.1016/j.matpur.2009.08.009_bib016 article-title: Introduction to the Spectral Theory of Polynomial Operator Pencils – year: 1972 ident: 10.1016/j.matpur.2009.08.009_bib010 – volume: 26 start-page: 15 issue: 4 year: 1971 ident: 10.1016/j.matpur.2009.08.009_bib014 article-title: On the completeness of the eigenfunctions of some classes of non-self-adjoint linear operators publication-title: Russian Math. 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PDE doi: 10.1080/03605309408821040 – volume: 25 start-page: 90 issue: 4 year: 1991 ident: 10.1016/j.matpur.2009.08.009_bib003 article-title: Trace formula for polynomial operators and its applications publication-title: Funktsional. Anal. i Prilozhen. – volume: 209 start-page: 425 issue: 2 year: 2004 ident: 10.1016/j.matpur.2009.08.009_bib005 article-title: Nonlinear eigenvalues and analytic hypoellipticity publication-title: J. Funct. Anal. doi: 10.1016/S0022-1236(03)00105-8 – start-page: 595 year: 1993 ident: 10.1016/j.matpur.2009.08.009_bib006 article-title: Analytic hypoellipticity, representation of nilpotent groups and non-linear eigenvalue problem publication-title: Duke Math. J. doi: 10.1215/S0012-7094-93-07222-5 – ident: 10.1016/j.matpur.2009.08.009_bib011 – volume: 53 start-page: 246 issue: 3 year: 1983 ident: 10.1016/j.matpur.2009.08.009_bib012 article-title: Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(83)90034-4 |
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| Snippet | In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is
L
(
λ
)
=... In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is... |
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| StartPage | 149 |
| SubjectTerms | Analysis of PDEs Exact sciences and technology Functional Analysis General mathematics General, history and biography Mathematical Physics Mathematics Non-selfadjoint operators Nonlinear eigenvalue problems Physics Sciences and techniques of general use Semiclassical analysis Spectral Theory Trace formula |
| Title | Asymptotic expansion for nonlinear eigenvalue problems |
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