Related papers: Estimates for eigenvalues of $\mathfrak L$ operato…

Let $x: M\rightarrow \mathbb{R}^{N}$ be an $n$-dimensional compact self-shrinker in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$. In this paper, we study eigenvalues of the operator $\mathcal{L}_r$ on $M$, where $\mathcal{L}_r$ is…

In this paper, we consider an eigenvalue problem of the elliptic operator $$ L_r={\rm div}(T^r\nabla\cdot )$$ on compact submanifolds in arbitrary codimension of space forms $\mathbb{R}^N(c)$ with $c\geq0$. Our estimates on eigenvalues are…

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the…

$\mathfrak{L}_{\nu}$ operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of $\mathfrak{L}^{2}_{\nu}$ operator on a bounded…

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…

Given a Schr\"odinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary…

$\mathfrak{L}_{II}$ operator is introduced by Y.-L. Xin (\emph{Calculus of Variations and Partial Differential Equations. 2015, \textbf{54}(2):1995-2016)}, which is an important extrinsic elliptic differential operator of divergence type…

The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type…

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…

Let $L_0$ be a bounded operator on a Banach space, and consider a perturbation $L=L_0+K$, where $K$ is compact. This work is concerned with obtaining bounds on the number of eigenvalues of $L$ in subsets of the complement of the essential…

In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $\delta$- and $\delta'$-interactions are studied. We show the self-adjointness of these…

The notion of the eigenvalue problem in the Fock space with polynomial eigenfunctions is introduced. This problem is classified by using the finite-dimensional representations of the $\mathfrak{sl}(2)$-algebra in Fock space. In the complex…

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator on locally reducible spacelike submanifold in terms of intrinsic and extrinsic expressions. The limiting-cases are also studied.

We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb{R}^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of…

Estimates for eigenvalues of Schr\"{o}dinger operators on the half-line with complex-valued potentials are established. Schr\"{o}dinger operators with potentials belonging to weak Lebesque's classes are also considered. The results cover…

For an $n$-dimensional compact submanifold $M^n$ in the Euclidean space $\mathbf R^{N}$, we study estimates for eigenvalues of the Paneitz operator on $M^n$. Our estimates for eigenvalues are sharp.

In this paper, we obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we…

In this note, we study the integrodifferential operator $(I-\Delta)^{\log}$ corresponding to the logarithmic symbol $\log(1+|\xi|^2)$, which is a singular integral operator given by $$ (I-\Delta)^{\log}…

We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the…

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general…