Pólya’s eigenvalue conjecture

A drumhead fixed along its edge has a sequence of vibration modes. Changing its outline changes their frequencies. Pólya’s conjecture predicts a simple lower limit for each mode using only the drumhead’s area and its position in the ordered list. Here the question concerns flat regions with a fixed edge; the eigenvalue is proportional to the square of the frequency.[1]

A drumhead with its edge fixed has standing-wave modes whose squared frequencies are proportional to its Laplacian eigenvalues. Doubling every length lowers each eigenvalue by a factor of four and multiplies area by four. The question is whether area sets a sharp lower bound for every mode, for every allowed outline, rather than only the high-frequency scale. In a unit square the standing-wave patterns sin(mπx)sin(nπy) have eigenvalues $\pi^2(m^2+n^2)$ for positive integers m and n; the patterns (1,2) and (2,1) have the same eigenvalue and are counted separately.

For a bounded planar Lipschitz domain $\Omega$, let $N_\Omega(t)$ count Dirichlet eigenvalues strictly below t, with multiplicity. The target is $N_\Omega(t)\leq |\Omega|t/(4\pi)$, equivalently $\lambda_k\geq 4\pi k/|\Omega|$. The area term is the leading count in Weyl’s law. That term has a phase-space interpretation: combine positions in the region with vectors describing oscillation direction and spatial frequency. This continuum count does not enumerate the exact standing-wave modes. Scaling $\Omega$ by s multiplies area by $s^2$ and eigenvalues by $s^{-2}$; the proposed comparison therefore depends on shape after size is removed. The strict-below convention differs from the source’s count at a squared threshold and is stated here explicitly.[1] [2]

The prediction works for many shapes, including rectangles, disks and rings. No proof establishes it for every allowed planar shape and every mode. Knowing that very high modes approach the predicted scale is not enough to bound each mode from below.[1]

The 2026 annulus theorem handles the full family of planar annuli. The June 2026 ε-loss result yields k ≤ (1+ε)|Ω|λ_k/(4π) only above a domain- and ε-dependent threshold. Taking smaller ε does not remove the changing threshold or settle the low eigenvalues; a limiting argument cannot supply the exact bound for a fixed k. The all-domain, all-index planar Dirichlet inequality therefore remains the target.[1] [2]

It would reveal a sharp geometric rule shared by very different shapes. Area alone would constrain the entire sequence of fixed-edge vibrations, despite the spectrum’s sensitivity to the outline.[1]

This problem asks how accurately the phase-space volume underlying Weyl’s leading term controls a finite spectral count. Boundary effects and shape can change lower-order behavior without changing that volume. Understanding their sign and size well enough to obtain a universal sharp inequality would clarify the reach of geometric spectral comparison.[1] [2]