Related papers: On minimal eigenvalues of Schrodinger operators on…

Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $\Omega\subset M$ be a suitable domain, and let $\lambda(\Omega)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We prove several…

Let C be a smooth closed curve of length 2 Pi in R^3, and let k(s) be its curvature, regarded as a function of arc length. We associate with this curve the one-dimensional Schrodinger operator H_C = -d^2/ds^2 + k^2 acting on the space of…

We are concerned with the non-normal Schr\"odinger operator $$ H=-\Delta+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this…

This paper investigates the asymptotic behavior of the principal eigenvalue $\lambda(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -\Delta_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c…

We study the smallest positive eigenvalue $\lambda_1(M)$ of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold $M$ which fibers over the circle, with fiber a closed surface of genus $g\geq 2$. We show the existence of a…

In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $\mu_1(\Omega)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $\Omega$ in $\R^n$. Our estimate does not require any convexity assumption on…

We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the…

In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kahler foliation. The limiting case is characterized by the existence of quaternion-Kahler Killing spinors. We end this paper by…

For Schr\"odinger operators on an interval with either convex or symmetric single-well potentials, and Robin or Neumann boundary conditions, the gap between the two lowest eigenvalues is minimised when the potential is constant. We also…

As an extension to the paper by Breuer, Grinshpon, and White \cite{B}, we study the linear statistics for the eigenvalues of the Schr\"odinger operator with random decaying potential with order ${\cal O}(x^{-\alpha})$ ($\alpha>0$) at…

The purpose of this paper is to study spectral properties of non-self-adjoint Schr\"odinger operators $-\Delta-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type…

We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and the corresponding eigenfunctions of a…

Let $ \mathcal{L} = -\Delta + V $ be a Schr\"odinger operator acting on $ L^2(\mathbb{R}^n) $, where the nonnegative potential $ V $ belongs to the reverse H\"older class $ RH_q $ for some $ q \geq n/2 $. This article is primarily concerned…

We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest…

For a bounded open set Omega in a complete oriented Riemannian n-manifold and a positive finite Borel measure mu with support contained in Omega, we define an associated Krein-Feller operators (or Laplacian) Delta_mu by assuming the…

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains $\Omega$ of a Riemannian manifold $(\mathcal{M},g)$. More precisely, we…

In this paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schr\"odinger operator $-4\Delta+R$ consists of discrete eigenvalues with finite multiplicities, if the scalar curvature $R$ satisfies…

In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold $M$ isometrically immersed into another Riemannian manifold $\bar M$ for arbitrary codimension. We first assume the pull back Weitzenb\"{o}ck…

We establish inequalities for the eigenvalues of Schr\"odinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related…

We investigate the Schr\"{o}dinger operators $H_\varepsilon=-\Delta +W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $\gamma$. The operators $H_\varepsilon$…