Inverse Spectral Problem for Schrödinger Operators

Details Inverse Spectral Problem for Schrödinger Operators Inverse Spectral Problem for Schrödinger Operators

2008-01-21

arxiv.org/abs/0801.3283v2

Mathematics

Spectral Theory

22 pages

Scientific paper

10.1007/s00220-008-0718-8

In this article we improve some of the inverse spectral results proved by Guillemin and Uribe in \cite{GU}. They proved that under some symmetry assumptions on the potential $V(x)$, the Taylor expansion of $V(x)$ near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schr\"odinger operator in $\mathbb R^n$. We prove some similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of $V(x)$. We establish our results by finding some explicit formulas for wave invariants at the bottom of the well.

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