Abel–Lidskii bases in non-selfadjoint inverse boundary problem

Ya. V. Kurylev   ; M. Lassas

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 27, Tome 250 (1998), pp. 161-190

Résumé

Let $M$ be a manifold with bondary $\partial M\ne\varnothing$. Let $A$ be a 2-nd order elliptic PDO on $M$. Denote by $R_\lambda(x,y)$, $x$, $y\in M$, $\lambda\in\mathbb C\setminus\sigma(A)$ the Schwartz kernel of $(A-\lambda I)^{-1}$. We consider the Gel'fand inverse boundary problem of the reconstruction of $(M,A)$ via given $R_\lambda(x,y)$, $x$, $y\in\partial M$, $\lambda\in\mathbb C$. We prove that if the main symbol of $A$ satisfies some geometrical condition (Bardos–Lebeau–Rauch condition) then these data determine $M$ uniquely and $A$ to within the group of the generalized gauge transformations on $M$. The above mentioned geometric condition means, roughly speaking, that any geodesics (in the metric generated by $A$) leaves $M$.