Guillemin–Wang inverse spectral conjecture
For the relevant semiclassical magnetic Schrödinger inverse problems, the spectral data should determine the coefficients in the following symmetry-restricted settings: (i) an even one-dimensional potential should be determined by the spectrum of the semiclassical Schrödinger operator ; and (ii) a radial magnetic field in two dimensions should be determined by the spectrum of the corresponding purely magnetic Schrödinger operator.
References
Primary source
Additional references
- Inverse spectral results for the magnetic Schrödinger operator in low dimensions — arXiv — Devon Farrell, Anna Holloway, Nikhil Savale
Progress summary
A new preprint claims progress in two symmetry-restricted settings, while the general inverse problem remains open.
The Guillemin–Wang conjecture concerns whether spectral data determine the relevant potentials and magnetic fields. Earlier work established important special cases; a new preprint claims the two-dimensional radial-magnetic and even one-dimensional cases.
Known results
- Guillemin and Wang (2009): in one dimension, suitable single-well and asymmetry assumptions give determination of the potential up to reflection.
- Guillemin and Wang (2009): radially symmetric electric potential and magnetic field in two dimensions are spectrally determined.
- Guillemin and Uribe (2005): low-lying eigenvalues determine Taylor data near a non-degenerate minimum under symmetry hypotheses.
October 2026 preprint
Devon Farrell, Anna Holloway, and Nikhil Savale claim new inverse spectral results for even one-dimensional potentials and two-dimensional radial magnetic fields. This advances the symmetry-restricted theory but does not settle general magnetic Schrödinger operators; the claim is unverified.
Current status (as of October 2026): Two symmetry-restricted cases have a new unverified preprint claim, while the general conjecture remains open.
Solutions 0
No solutions have been posted yet.