Singular principal normality in the Cauchy problem (Q2565391)

In an earlier paper the first two authors studied the family of Cauchy problems \[ \begin{aligned} P_\alpha (t,y,D_t,D_y)u & =\bigl(D^2_t- D^2_{y_1}-D^2_{y_2} +i(D^2_{y_1} +2tD_{y_1} D_{y_2}+ (1+\alpha) t^2D^2_{y_2})\bigr) u+au=0 \\ u(0,y_1,y_2) & =u_t (0,y_1,y_2)=0. \end{aligned} \] They proved by using methods of geometrical optics that to each \(\alpha\in [0,1/3)\) there exist a coefficient \(a\) and a solution \(u\) belonging to \(C^\infty\) in a neighbourhood of the origin such that \(0\in\text{supp} u\subseteq \{t\geq 0\}\). If \(\alpha>1/3\), then \(P_\alpha\) has the compact uniqueness property for \(C^\infty\)-solutions with respect to \(\{t=0\}\) at the origin. One has to take into consideration that \(P_\alpha\) does not possess the property of principal normality, but is of principal type. The hyperplane \(\Sigma= \{t=0\}\) is noncharacteristic and \(P_\alpha\) has simple characteristic roots with respect to \(\Sigma\) at the origin. In the present paper the authors generalize these very precise results for \(P_\alpha\) to linear operators \(P\) of higher order with simple characteristic roots. For these operators they introduce the notion of singular principal normality (extension of the usual principal normality). 1. Under both assumptions (simple characteristic roots, singular principal normality) the compact uniqueness property is shown for \(P\) proving a Carleman estimate (compare with \(\alpha> 1/3\) in the above example). 2. Moreover, a non-uniqueness result is proved for classes of operators \(P\) with violated singular principal normality (compare with \(\alpha\in [0,1/3)\) in the above example). The nonuniqueness result is stable even for perturbations by operators of order 0.