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Pólya Eigenvalue Conjecture

The Pólya eigenvalue conjecture asserts that the Dirichlet and Neumann eigenvalues of the negative Laplacian on a bounded domain Omega subset= R^d satisfy

mu_k<=4pi^2(k/(omega_d|Omega|))^(2/d)<=lambda_k
(1)

for k>=1, where omega_d is the volume of the unit ball, |Omega| is the volume of the domain, 0=mu_0<=mu_1<=... are the eigenvalues for Neumann boundary conditions, and lambda_1<=lambda_2<=... are the eigenvalues for Dirichlet boundary conditions, counted with multiplicity (Dai and Sun 2026).

Dai and Sun (2026) proved the Neumann inequality for every circular annulus A_(r,R)={x in R^2:r<||x||<R} with 0<r<R,

mu_k<=(4k)/(R^2-r^2)
(2)

for all k>=1. Their proof combines analytic estimates with computer-assisted interval certificates covering the remaining parameter ranges.

Tang and Wang (2026a) reported the strict Dirichlet inequality

|Omega|lambda_j(Omega)>4pij
(3)

for every nonempty bounded, simply connected planar domain Omega and every j>=1. The authors supply an accompanying formalization.

Tang and Wang (2026b) separately reported the strict Neumann inequality

|Omega|mu_j(Omega)<4pij
(4)

for every nonempty bounded, simply connected planar domain Omega with Lipschitz boundary and every j>=1. The indexing begins with mu_0=0. The general conjecture remains open.

Tang and Wang (2026ab) acknowledge ChatGPT assistance with technical arguments and Aristotle and Codex assistance with Lean formalizations. The registered Neumann development passed machine checks, but independent specialist review of the complete argument had not been reported as of Oct. 15, 2026.

For a round spherical shell A_(r,R)^d={x in R^d:r<||x||<R} with 0<r<R, Li (2026a) proved the strict Dirichlet bound

N_D(A_(r,R)^d,Lambda)<(R^d-r^d)/(2^dGamma(d/2+1)^2)Lambda^(d/2)
(5)

for every d>=2 and every Lambda>0. Here N_D counts eigenvalues at most Lambda, with multiplicity, and Gamma is the gamma function. Li (2026a) also proved the corresponding strict inequality for the Aharonov-Bohm operator on planar annuli at arbitrary magnetic flux. These are Dirichlet results and do not assert a Neumann inequality.

Li (2026b) subsequently proved the strict Neumann inequality for the same spherical shells:

N_N^<(A_(r,R)^d,Lambda)>(R^d-r^d)/(2^dGamma(d/2+1)^2)Lambda^(d/2)
(6)

for every d>=2, 0<r<R, and Lambda>0. Here N_N^< counts eigenvalues strictly below Lambda, with multiplicity, including the zero Neumann eigenvalue. Li (2026b) also proved the corresponding strict bound for balls.

See also

Dirichlet Boundary Conditions, Eigenvalue, Laplacian, Neumann Boundary Conditions

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References

Dai, G. and Sun, Y. "Pólya's Conjecture for Neumann Eigenvalues of Annuli." 4 Oct 2026. https://arxiv.org/abs/2610.05146.Li, Y. "Pólya's Conjecture for Dirichlet Eigenvalues of Spherical Shells and Aharonov-Bohm Annuli." 20 Oct 2026a. https://arxiv.org/abs/2610.27707.Li, Y. "Pólya's Conjecture for Neumann Eigenvalues of Spherical Shells." 22 Oct 2026b. https://arxiv.org/abs/2610.31464.Tang, Q. and Wang, Z. "Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains." 9 Oct 2026a. https://arxiv.org/abs/2610.13139.Tang, Q. and Wang, Z. "The Neumann Pólya Inequality on Simply Connected Planar Domains." 2026b. https://github.com/QuanyuTang/polya-neumann-simply-connected-formalization-Palomar-version-v2/tree/f07f1c7051862ba146429db6eda5dc4e8b020f74.

Cite this as:

Weisstein, Eric W. "Pólya Eigenvalue Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolyaEigenvalueConjecture.html

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