Gelf'and Inverse Problem for a Quadratic Operator Pencil

✍ Scribed by Yaroslav Kurylev; Matti Lassas

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 198 KB
• Volume: 176
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3615

✦ Synopsis

In this paper we consider the inverse boundary value problem for the operator pencil A(*)=a(x, D)&i*b 0 (x)&* 2 where a(x, D) is an elliptic second-order operator on a differentiable manifold M with boundary. The manifold M can be interpreted as a Riemannian manifold (M, g) where g is the metric generated by a(x, D). We assume that the Gel'fand data on the boundary is given; i.e., we know the boundary M and the boundary values of the fundamental solution of A(*), namely, R * (x, y), x, y # M, * # C.

We show that if (M, g) satisfies some geometric condition then the Gel'fand data determine the manifold M, the metric g, the coefficient b 0 (x) uniquely and also the equivalence class of a(x, D) with respect to the group of generalized gauge transformations.

2000 Academic Press a(x, D)=&g &1Â2 ( j +b j ) g 1Â2 g jl ( l +b l )+q,