arXiv subjects
Publications and source records attributed to Hironori Kumura.
This paper, focusing on the growth rate of the measure, gives pointwise bounds of solutions of eigenvalue equations of the Laplace-Beltrami operator on noncompact Riemannian manifolds.
The purpose of this paper is to prove the uniqueness theorem of solutions of eigenvalue equations on one end of Riemannian manifolds for drift Laplacians, including the standard Laplacian as a special case; we shall impose "a sort of radiation condition" at infinity on solutions. We shall also provide several Riemannian manifolds whose Laplacians satisfy the absence of embedded eigenvalues and besides the absolutely continuity, although growth orders of their metrics on ends are very complicated.
The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian measure at infinity, in particular, focusing on the limiting absorption principle. As a result, we will obtain the absolute continuity of the Laplace-Beltrami operator.
When a Riemannian manifold $(M,g)$ is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to $ -\frac{1}{r}$, where $r$ stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rotationally symmetric one and derive growth estimates of solutions to the eigenvalue equation, from which the absence of eigenvalues will follows.
This paper discusses the question whether the discrete spectrum of the Laplace-Beltrami operator is infinite or finite. The borderline-behavior of the curvatures for this problem will be completely determined.
The purpose of this paper is to point out that `supremum' in two inequalities of Brooks should be replaced with `infimum'. The results of this paper are already known by Professor Higuci. Hence, I want to delete this paper from this preprint server.
The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curvatures for this problem will be determined: we will assume that the radial curvature $K_{\rm rad.}$ of an end converges to a constant -1 at infinity with the decay order $K_{\rm rad.} + 1 = o(r^{-1})$ and prove the absence of eigenvalues embedded in the essential spectrum. Furthermore, in order to show that this decay order $K_{\rm rad.} + 1 = o(r^{-1})$ is sharp, we will construct a manifold with the radial curvature decay $K_{\rm rad.} + 1 = O(r^{-1})$ and with an eigenvalue $\frac{(n-1)^2}{4} + 1$ embedded in the essential spectrum $[ \frac{(n-1)^2}{4}, \infty)$ of the Laplacian.
Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.
The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schrödinger operator has a finite or infinite number of the discrete pectrum. In this paper, we will give a generalization of this lemma on Euclidean spaces to that on large classes of complete noncompact manifolds. Replacing Euclidean spaces by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some criterions for the above-type question.
We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval $[(n-1)^2K/4, \infty)$ if an $n$-dimensional Riemannian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant $(n-1)K$ converges to zero at infinity.