Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1301.6202

Heat Equation on the Cone and the Spectrum of the Spherical Laplacian

B S Balakrishna
View original source ↗Read source document ↗Export metadata ↓

Abstract

Spectrum of the Laplacian on spherical domains is analyzed from the point of view of the heat equation on the cone. The series solution to the heat equation on the cone is known to lead to a study of the Laplacian eigenvalue problem on domains on the sphere in higher dimensions. It is found that the solution leads naturally to a spectral function, a `generating function' for the eigenvalues and multiplicities of the Laplacian, expressible in closed form for certain domains on the sphere. Analytical properties of the spectral function suggest a simple scaling procedure for estimating the eigenvalues. Comparison of the first eigenvalue estimate with the available theoretical and numerical results for some specific domains shows remarkable agreement.

Explore related subjects

math.SPmath-phmath.MPmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

B S Balakrishna. 2013-03-25. Heat Equation on the Cone and the Spectrum of the Spherical Laplacian. https://arxiv.org/abs/1301.6202

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes

This article is concerned with mixed Steklov--Robin eigenvalues on bounded domains in $\mathbb{R}^{n}, n \geq 2$, with Lipschitz boundary. Specifically, we consider domains with symmetry of order $4$ containing a spherical hole. We obtain isoperimetric inequalities for the $k$-th Steklov-Robin eigenvalues for each $k \in \{2, 3, \dots, n+1\}$. We provide examples to emphasize the fact that the symmetry assumptions, on the family of domains considered, are crucial.

math.SP↗
arXiv2026

Steklov isospectrality of conformal metrics

The Steklov spectrum of a smooth compact Riemannian manifold $(M,g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map) This article is devoted to the Steklov spectral inverse problem of recovering the metric $g$, up to natural gauge invariance, from its Steklov spectrum. Positive results are established in dimension $n\geq 3$ for conformal metrics under the assumption that the geodesic flow on the boundary is Anosov with simple length spectrum. The paper combines wave trace formula techniques with the injectivity of the geodesic X-ray transform for functions on closed Anosov manifolds. It is shown that knowledge of the Steklov spectrum determines the jet at the boundary of the underlying Riemannian metric within its conformal class. In this particular context, this parallels the well-known results of the Calderón problem, where we are given the entire Dirichlet-to-Neumann map instead. As a simple corollary, assuming real-analyticity of the conformal factor, Steklov isospectral metrics must coincide. Using similar arguments, we are also able to prove under the same assumption of hyperbolicity of the geodesic flow on the boundary, that generically any smooth potential $q$ can be recovered from the Steklov spectrum, in the sense that its jet at the boundary is determined by the spectrum of the DN map for the Schrödinger operator with potential $q$. Consequently, in this case, two analytic Steklov isospectral potentials must be equal.

math.SP↗
arXiv2026

Revisiting the Weak Coupling Phenomenon for Two-Dimensional Schrödinger Operators

We study the existence of negative eigenvalues for two-dimensional Schrödinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to describe the unique negative eigenvalue emerging from the threshold of the essential spectrum of one- and two-dimensional Schrödinger operators. The aim of this paper is to extend Simon's results in two dimensions to a broader class of potentials, allowing for both stronger singularities and slower decay at infinity, at the cost of losing uniqueness of weakly coupled eigenvalues.

math.SP↗