Related papers: Dirichlet eigenvalues of Schwarz triangle (3/2 3/2…

Among all triangles of given diameter, the equilateral triangle is shown to minimize the sum of the first $n$ eigenvalues of the Dirichlet Laplacian, for each $n \geq 1$. In addition, the first, second and third eigenvalues are each proved…

The Dirichlet eigenvalues of the Laplacian on a triangle that collapses into a line segment diverge to infinity. In this paper, to track the behavior of the eigenvalues during the collapsing process of a triangle, we establish a…

We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a scale-invariant functional of the first Dirichlet…

We compute Dirichlet eigenvalues and eigenfunctions explicitly for spherical lunes and the spherical triangles which are half the lunes, and show that the fundamental gap goes to infinity when the angle of the lune goes to zero. Then we…

This paper solves the open problem of the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a complete solution to Conjecture 6.47 posed by R. Laugesen and B. Siudeja in A. Henrot's book ``Shape…

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling…

In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with…

We study the first Dirichlet eigenfunction of a class of Schr\"odinger operators with a convex potential V on a domain $\Omega$. We find two length scales $L_1$ and $L_2$, and an orientation of the domain $\Omega$, which determine the shape…

Schwarz triangle functions play a fundamental role in the solutions of the generalised Chazy equation. Chazy has shown that for the parameters $k=2$ and $3$, the equations can be linearised. We determine the Schwarz triangle functions that…

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first $k$ Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of…

We study the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in $\mathbb{R}^{n+2}$ with $n\geq 1$, imposing the Steklov condition on the outer boundary sphere, denoted by $\Gamma_S$, and the Dirichlet condition on the inner…

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want…

In this article we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator $\sum_{i=1}^{d}(-\partial_i^2)^{s}, \, s\in (1/2,1]$ on an open and bounded subdomain $\Omega \subset \mathbb{R}^d$ and predict…

Despite the moduli space of triangles being three dimensional, we prove the existence of two triangles which are not isometric to each other for which the first, second and fourth Dirichlet eigenvalues coincide, establishing a numerical…

In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter $D$ of sphere $\mathbb S^n$ is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$. We…

We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The…

We calculate accurate eigenvalues and eigenfunctions of the Schr\"odinger equation for a two-dimensional quantum dipole. This model proved useful for the study of elastic effects of a single edge dislocation. We show that the Rayleigh-Ritz…

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/\lambda_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower…

We address the question of determining the eigenvalues $\lambda\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$…

We calculate the spectrum and a basis of eigenvectors for the Spin Dirac operator over the standard 3-sphere. For the spectrum, we use the method of Hitchin which we transfer to quaternions and explain in more detail. The eigenbasis (in…