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Cramér’s formula for Heisenberg manifolds. (English) Zbl 1090.58018

The authors consider the difference \(R(t)\) between the counting function \(N(t)\), representing the number of eigenvalues of the Laplace Beltrami operator on a \(3\)-dimensional Heisenberg manifold which are less or equal to \(t\), and the first term of the asymptotic expansion of \(N(t)\) as \(t\) tends to \(\infty\), and prove a formula for the asymptotics of \(\int_{1}^{T}| R(t)| ^{2}\,dt\) for \(T\) large. Then a similar result is proved for Heisenberg manifolds of dimension \(2n+1\) with \(n>1\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
31C12 Potential theory on Riemannian manifolds and other spaces
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