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Simon's L^2 Conjecture

Simon's L^2 conjecture is the assertion that a real potential V on R^d, d>=2, satisfying

I_(d,0)(V)=int_(R^d)||x||^(1-d)|V(x)|^2dx<infty

implies that a self-adjoint realization of the Schrödinger operator -Delta+V has an absolutely continuous spectral component of infinite multiplicity at almost every positive energy (Simon 2000, Becker and Yang 2026). It is problem 8 in Simon's problems.

Becker and Yang (2026) report a counterexample for every d>=2 and 0<=sigma<1/3. Their smooth bounded nonnegative V can have arbitrarily small ||V||_infty+I_(d,0)(V)+I_(d,sigma)(V), where I_(d,sigma)(V)=int_(R^d)||x||^(1-d+sigma)|V(x)|^2dx, while -Delta+V on L^2(R^d) has operator spectrum [0,infty) and purely point spectral measures. Its eigenfunctions form a complete orthonormal basis. Purely point spectral measures do not mean that every positive energy is an eigenvalue. The complete claimed refutation has not been independently verified here. Becker and Yang (2026) report substantive AI contributions to the potential construction and proof.

See also

Operator Spectrum, Schrödinger Operator, Simon's Problems

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References

Becker, S. and Yang, M. "Lenses Make Many Points: Pure Point Spectrum Disproves Simon's L^2-Conjecture." 22 Oct 2026. https://arxiv.org/abs/2610.31637.Simon, B. "Schrödinger Operators in the Twenty-First Century." 17 Feb 2000. https://www.ma.utexas.edu/mp_arc-bin/mpa?yn=00-78.

Cite this as:

Weisstein, Eric W. "Simon's L^2 Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SimonsL2Conjecture.html

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