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Nonlinear eigenvalues and analytic hypoellipticity. (English) Zbl 1053.35045
Motivated by the problem of analytic hypoellipticity of sum the of squares of differential operators , the authors are interested in a family of partial differential operators of the type $\mathcal{H}(x,D_{x},\lambda)=-\Delta+\left( P(x)-\lambda\right) ^{2}$ where $$x\mapsto P(x)$$ is a polynomial of degree $$m>1.$$ They prove, under certain conditions on the dimension $$n$$, the polynomial $$P(x)$$ and its degree $$m,$$ that the equation $\mathcal{H}(x,D_{x},\lambda)u=0$ has a solution $$(\lambda,u)$$ with $$u\in\mathcal{S}(\mathbb{R}^{n}),u\neq0.$$
To prove these results, the initial problem is reduced to the spectral analysis of $$(I-2\lambda B+\lambda^{2}A)u=0$$ , where $$A=(-\Delta+P(x))^{-1}$$ and $$B=A^{\frac{1}{2}}PA^{\frac{1}{2}}$$. Then the authors use a standard approach to transform the nonlinear spectral problem to linear ones and apply the Lidskii Theorem concerning the trace of operators to prove the existence of a non trivial solution. As an application, the authors obtain the following result:
Let $$p=2$$ or $$3$$ and let $$P(x)$$ be a positive elliptic polynomial of degree $$m\geq2p$$ in the variables $$(x_{1},\dots,x_{p}).$$ Then the operator $H=\sum\limits_{j=1}^{p}D_{x_{j}}^{2}+\left( P(x)D_{x_{p+1}}-D_{x_{p+2} }\right) ^2$ is not analytic hypoelliptic at the origin.
The theorem is related to the Trèves conjectures [F. Trèves, Symplectic geometry and analytic hypo-ellipticity, Proc. Symp. Pure Math. 65, 201–219 (1999; Zbl 0938.35038)]. The authors also recover known results obtained for the operator $$H$$ in the case of $$p=1$$ .
Reviewer: C. Bouzar (Oran)

MSC:
 35H10 Hypoelliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35S05 Pseudodifferential operators as generalizations of partial differential operators
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References:
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