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Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of this…

We study the eigenvalue problem for the $g-$Laplacian operator in fractional order Orlicz-Sobolev spaces, where $g=G'$ and neither $G$ nor its conjugated function satisfy the $\Delta_2$ condition. Our main result is the existence of a…

Given a Schr\"odinger operator with a real-valued potential on a bounded, convex domain or a bounded interval we prove inequalities between the eigenvalues corresponding to Neumann and Dirichlet boundary conditions, respectively. The…

This document contains additional numerical and graphical evidence to support some of the conjectures mentioned in \cite{GM3}. We give more evidence for $\kappa=3,4$ in dimension 2. As mentioned in that article we still don't quite…

We reconsider the well-known and long-debated problem of the calculation of the eigenvalues of the Herbst Hamiltonian 2\sqrt{p^2 +m^2} - \kappa/r. We give a formulation of the problem that allows, for the first time, a perturbative…

We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n \Delta^{7/5}/\log^{1/5-o(1)}n)$ for any $\Delta$, and by…

We compute the deficiency spaces of operators of the form $H_A{\hat{\otimes}} I + I{\hat{\otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von…

Internal intervals spanned by finite ranges of a conformal coordinate $z$ and terminating at a pair of singularities are a common feature of many string compactifications with broken supersymmetry. The squared masses emerging in…

In this note, we consider locally invertible analytic mappings in two dimensions, with coefficients in a non-archimedean field. Suppose such a map has a Jacobian with eigenvalues $\lambda_1$ and $\lambda_2$ so that $|\lambda_1|>1$ and…

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…

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and…

Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight $\lo$ on a manifold $M$. One can consider a pencil of operators $\hPi(\Delta)=\{\Delta_\l\}$ passing through the operator $\Delta$ such that…

The spectral properties of a singular left-definite Sturm-Liouville operator $JA$ are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart $A$ which is obtained by substituting the…

In this paper we analyze an eigenvalue problem associated to fractional operators of the form \[ L_a^s u(x)=2 \text{p.v.}\int_{\mathbb{R}^n}a(x,y,D^su(x,y))\,\frac{dy}{|x-y|^{n+s}},\] which represents a generalization model for nonlocal,…

The paper is devoted to operators given formally by the expression \begin{equation*} -\partial_x^2+\big(\alpha-\frac14\big)x^{-2}. \end{equation*} This expression is homogeneous of degree minus 2. However, when we try to realize it as a…

Eigenvectors and eigenvalues of discrete graph Laplacians are often used for manifold learning and nonlinear dimensionality reduction. It was previously proved by Belkin and Niyogi that the eigenvectors and eigenvalues of the graph…

Let $P\in \Bbb Q_p[x,y]$, $s\in \Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=\frac{1}{1-p^{-1}}\int_{\Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(\Bbb Z_p)$. We show that if $P$ is homogeneous…

Let $G_n$ be an $n \times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n \times n$ matrix with $\Vert A \Vert \le 1$, and let $\gamma \in (0,1)$. We show that with probability $0.99$, $A + \gamma G_n$ has all of its…

In this paper, we prove the uniform estimates for the resolvent $(\Delta - \alpha)^{-1}$ as a map from $L^q$ to $L^{q'}$ on real hyperbolic space $\mathbb{H}^n$ where $\alpha \in \mathbb{C}\setminus [(n - 1)^2/4, \infty)$ and $2n/(n + 2)…

In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator $ L$, which is introduced by Colding and Minicozzi in [4], on an $n$-dimensional compact self-shrinker in ${R}^{n+p}$. Estimates for…