The generalized Legendre transform and its applications to inverse spectral problems. (3rd December 2015)

• Record Type: Journal Article
• Authors: Guillemin, Victor; Wang, Zuoqin
• Is Part Of: Inverse problems. Volume 32:Number 1(2016:Jan.)
• Publication Date: 2015-12-03
• Subjects: inverse spectral problem -- Schrödinger operator -- equivariant spectrum -- Legendre transform; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/1/015001
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: Let M be a Riemannian manifold, an isometric action on M of an n -torus G and a bounded G -invariant smooth function. By G -invariance the Schrödinger operator, restricts to a self-adjoint operator on α being a weight of G and a large positive integer. Let be the asymptotic support of the spectrum of this operator. We will show that c α extend to a function, and that, modulo assumptions on τ and V one can recover V from W, i.e. prove that V is spectrally determined. The main ingredient in the proof of this result is the existence of a 'generalized Legendre transform' mapping the graph of d W onto the graph of d V .