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We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $-4{d}^{2}/d{s}^{2}+{\kappa}^{2}\left(s\right)$ with potential given by the curvature of a closed curve.

This article is concerned with estimations from below for the remainder term in Weyl’s law for the spectral counting function of certain rational (2ℓ + 1)-dimensional Heisenberg manifolds. Concentrating on the case of odd ℓ, it continues the work done in part I [21] which dealt with even ℓ.

We control the gap between the mean value of a function on a submanifold (or a point), and its mean value on any tube around the submanifold (in fact, we give the exact value of the second derivative of the gap). We apply this formula to obtain comparison theorems between eigenvalues of the Laplace-Beltrami operator, and then to compute the first three terms of the asymptotic time-expansion of a heat diffusion process on convex polyhedrons in euclidean spaces of arbitrary dimension. We also write...

The aim of this paper is to demonstrate how a fairly simple nilpotent Lie algebra can be used as a tool to study differential operators on ${\mathbb{R}}^{n}$ with polynomial coefficients, especially when the property studied depends only on the degree of the polynomials involved and/or the number of variables.