Nodal geometry, heat diffusion and Brownian motion

Abstract

Abstract. We use tools from n-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold M . On one hand we extend a theorem of Lieb (see [L]) and prove that any nodal domain Ω λ almost fully contains a ball of radius ∼, which is made precise by Theorem 1.6 below. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains ([Man2]). On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside M (this is Theorem 1.5).

On the Location of Maxima of Solutions of Schrödinger's Equation

“…We can now invoke the argument of Georgiev and Mukherjee [17] (their argument only uses that Brownian motion started in the point of maximum has a quantitatively controlled likelihood of hitting the boundary, which we have just established in the more general context): we present their argument in abbreviated form: if a large part of B were outside of , the likelihood of hitting the boundary would be large, but we just established that it is not.…”

“…Let n 3 and R n be open. Georgiev and Mukherjee [17] used this fact to refine Lieb's theorem. / 1=2 for which jB \ j .1 "/jBj: Little seems to be known about the location of the maximum.…”

“…The second author [32] proved that if u D u with Dirichlet conditions and one starts Brownian motion where juj assumes its maximum, then the likelihood of hitting the boundary within time t D 1 is less than 63.3% (D .e 1/=e). Georgiev and Mukherjee [17] used this fact to refine Lieb's theorem. THEOREM (Georgiev and Mukherjee, [17]).…”

“…Georgiev and Mukherjee [17] used this fact to refine Lieb's theorem. THEOREM (Georgiev and Mukherjee, [17]). The ball with all the properties in Lieb's theorem can be placed around the point where juj assumes its maximum.…”

Faber-Krahn type inequalities and uniqueness of positive solutions on metric measure spaces

“…In an influential work [28] Lieb showed that given any ε ∈ (0, 1) there exists r ε such that for any domain D ⊂ R d , with λ D being its Dirichlet principal eigenvalue in D for the Laplacian, it holds that |D ∩ B(x, r ε λ in several directions. For instance, [35] extends it for Schrödinger operators in R d , [14] establishes this on smooth Riemannian manifolds and [5] obtains an analogous version of this inequality for fractional Laplacian. In a similar direction we also cite [8,9] which studies Faber-Krahn type inequalities for the Schrödinger operators in R d involving singular potentials.…”