## The generalized Legendre transform and its applications to inverse spectral problems

##### Author(s)

Guillemin, Victor W; Wang, Zuoqin
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Let M be a Riemannian manifold, τ : G x M --> M an isometric action on M of an n-torus G and V : M --> R a bounded G-invariant smooth function. By G-invariance the Schrödinger operator, P = -h[superscript 2][Delta]M + V, restricts to a self-adjoint operator on L[superscript 2](M)[subscript alpha over h], alpha being a weight of G and 1[over h] a large positive integer. Let [c[subscript alpha], [infinity]] be the asymptotic support of the spectrum of this operator. We will show that c[subscript alpha] extend to a function, W : g*-->R 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 dW onto the graph of dV.

##### Date issued

2015-12##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Inverse Problems

##### Publisher

Institute of Physics Publishing (IOP)

##### Citation

Guillemin, Victor and Zuoqin Wang. "The generalized Legendre transform and its applications to inverse spectral problems." Inverse Problems 32:1 (December 2015), 015001.

Version: Original manuscript

##### ISSN

0266-5611

1361-6420