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We show that a bi-invariant metric on a compact connected Lie group is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric on there is a positive integer such that, within a neighborhood of in the class of left-invariant metrics of at most the same volume, is uniquely determined by the first distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where is simple, can be chosen to be two....
In this paper, we compute the spectrum of the Lichnerowicz laplacian on the symmetric forms of degree 2 on the sphere Sn and the real projective space RPn. This is obtained by generalizing to forms the calculations of the spectrum of the laplacian on fonctions done via restriction of harmonic polynomials on euclidean space.
What is called the “Semi-classical trace formula” is a formula expressing the smoothed density of states of the Laplace operator on a compact Riemannian manifold in terms of the periodic geodesics. Mathematical derivation of such formulas were provided in the seventies by several authors. The main goal of this paper is to state the formula and to give a self-contained proof independent of the difficult use of the global calculus of Fourier Integral Operators. This proof is close in the spirit of...
We consider the laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the curvature radii of the Neumann boundary to the Dirichlet one...
We consider the Laplacian in a domain squeezed
between two parallel curves in the plane,
subject to Dirichlet boundary conditions on one of the curves
and Neumann boundary conditions on the other.
We derive two-term asymptotics for eigenvalues
in the limit when the distance between the curves tends to zero.
The asymptotics are uniform and local in the sense that
the coefficients depend only on the extremal points where
the ratio of the curvature radii of the Neumann boundary
to the Dirichlet one...
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the area of the Neumann...
Let (M,g) be a compact Riemannian manifold without boundary, with dim M ≥ 3, and f: ℝ → ℝ a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem
, σ ∈ M, ω ∈ H₁²(M),
is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the function K̃. Here, stands for the Laplace-Beltrami operator on (M,g), and α, K̃ are smooth positive functions. These multiplicity...
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