By Wolfgang Arendt, Wolfgang P. Schleich

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3 Weyl’s Law with Remainder Term. 7. 3 Weyl’s Law with Remainder Term. 1 The Laplacian on the Flat Torus T2 In special cases it is possible to derive exact formulae for the counting function N(λ) which contain in addition to the Weyl term (and possible higher order terms) an explicit expression for a remainder function. 4. To illustrate the method in a simple case, we consider the eigenvalue problem –ΔT2 u = λu, where ΔT2 denotes the Laplacian on a flat torus T2 := S1L ~ S1L = R2 /(LZ ~ LZ) characterized by a length scale L > 0.

330 . . 324 . . 318 . . (Iwaniec and Mozzochi 1988 [57]). 315 . . is still far away from 1/4! ) Since P(x) is a wildly fluctuating function, it might be that some very rare spikes exceeding the conjectured x1/4 -behavior make it extremely difficult to improve the best existing bound. In order to “tame” these spikes, one can consider moments of P(x) and hope that the spikes are being washed out. 9 making use of the trace formula for ν(x) which we shall now derive. Note added in proof: in a recent unpublished paper [59] it is claimed to present a proof of Hardy’s conjecture.

3) is equivalent to the asymptotic behavior N(λ) = |Ω| λ + o(λ) 4π (λ → ∞ ) . 4) These results are now called Weyl’s law. 5) 3 4 1 Weyl’s Law and the extension to the vector Helmholtz wave equation describing the vibrations of the electric field E in an empty cavity Ω with perfectly reflecting walls ∂Ω. 8, it is exactly this electrodynamic cavity problem, studied extensively in those years by theoretical physicists, which was one of the open problems that provoked Weyl to start his seminal work. 2).

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