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Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data

We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues and the normal derivatives of the eigenfunctions on the boundary of a bounded domain. The estimate is of Hölder type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to $O(k^{-a})$ with $a>>1$, then we still have Hölder stability.