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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Institute of Physics Publishing (IOP)
Guillemin, Victor and Zuoqin Wang. "The generalized Legendre transform and its applications to inverse spectral problems." Inverse Problems 32:1 (December 2015), 015001.