## Essential spectrum and heat kernel.(English)Zbl 0634.58031

Let M be a complete Riemannian manifold and $${}^ p\Lambda (M)$$ its vector space of exterior differential p-forms. The Laplacian $$\Delta$$ acts on the space of compactly supported exterior differentiable p-forms. If $$\Delta \phi =\lambda \phi$$, $$\lambda\in {\mathbb{R}}$$, then $$\phi$$ is called an eigen p-form. If $$\lambda =0$$, that means $$\Delta\phi =0$$, then $$\phi$$ is called harmonic p-form. We denote by Sp $$\Delta$$ the spectrum of $$\Delta$$, that means Sp $$\Delta$$ $$=\{\lambda /\Delta\phi= \lambda\phi\}$$. Since $$\Delta$$ is positive semidefinite Sp $$\Delta$$ is a subset of $$[0,\infty)$$. Let $$\rho$$ be the infimum of Sp$$\Delta\setminus 0$$. The essential spectrum of $$\Delta$$ denoted Ess Sp $$\Delta$$, consists of the cluster points in the spectrum of $$\Delta$$ and the eigenvalues with infinite multiplicity. If $$\alpha$$ is the infimum of Ess Sp $$\Delta$$, then we assume that $$\alpha >0$$. It is obvious that Sp $$\Delta\setminus$$ Ess Sp $$\Delta$$ consists of isolated eigenvalues having finite multiplicity. The hypothesis $$\alpha >0$$ therefore implies $$0<\rho \leq \alpha.$$
Let $$u\in L^ 2(\Lambda^ p(M))$$ be a solution of the heat equation $$(\partial /\partial t+\Delta)u=0$$. The aim of the present paper is to study the essential spectrum and heat kernel. Some of the results can be stated as follows:
I. For $$t\geq t_ 0$$, $$\int_{M}| u|^ 2e^{2\sqrt{\beta r}}\leq C_ 3e^{-2\lambda t}+C_ 4e^{-2\gamma t}.$$
II. For $$t\geq t_ 0>0$$, $$\int_{M}| \bar K(t,x,x_ 0)|^ 2 e^{2\sqrt{\beta r}}\leq B_ 1e^{-2\lambda t}+B_ 2e^{-2\gamma t}.$$
The integral runs over the variable x. $$\bar K(t,x,x_ 0)=K(t,x,x_ 0)- H(x,x_ 0|$$, where $$K(t,x,x_ 0)$$ is the fundamental solution of the heat equation. $$H(x,x_ 0)$$ is the kernel for projection onto the space of $$L^ 2(\Lambda^ p(M))$$ harmonic forms and $$x_ 0$$ a basic point on M.
Reviewer: G.Tsagas

### MSC:

 58J35 Heat and other parabolic equation methods for PDEs on manifolds 53C20 Global Riemannian geometry, including pinching 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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### References:

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