Characterization of eigenfunctions of the Laplace-Beltrami operator through heat propagation in small time
Let \(X\) be a Riemann manifold and \(\Delta\) the positive Laplace-Beltrami operator on \(X\). \(e^{-t\Delta}\) denotes the heat propagator. \(\Delta f=cf\) always implies \(e^{-t\Delta}f=e^{-tc}f \forall t>0 \). In this note the authors endeavour is to explore the converse of this. They show: For \(n\geq 2\) and \( p > \frac{2n}{n-1}\), if \(f \in L^p (\mathbb R^n)\) satisfies \( e^{-t\Delta_{\mathbb R^n}} f = e^{-tc} f \) for some \(c > 0, t > 0\), then \(\Delta_\mathbb R^n f = cf \). If \(f \in L^\infty (\mathbb R^n)\) satisfies \( e^{-t\Delta_{\mathbb R^n}} f = e^{-tc} f \) for some \(c \geq 0, t > 0\), then \(\Delta_\mathbb R^n f = cf \). For a rank one Riemannian symmetric space of noncompact type \(X\) the situation is different. The authors show that the answer to such a question is affirmative only if \(t\) is within a sharp range \(0 < t < T \), where the critical time \(T \) depends on the proposed eigenvalue \(c\) and the integrability of the eigenfunction (or temperedness of the eigendistribution). In fact, \(f \in L^p(X)\) in the above discussion can be replaced by appropriately tempered distributions. It appears that the tempered distributions are ideal objects in this setup as they are less restrictive.
eigenfunction of the Laplace-Beltrami operator
scientific article; zbMATH DE number 7220396
scientific article