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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)

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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