Related papers: Eigenvalues of the Thurston operator
A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the…
In this paper, we study the principal eigenvalue $\mu(\mathscr{F}_k^-,E)$ of the fully nonlinear operator \[ \mathscr{F}_k^-[u] = \mathcal{P}_k^-(\nabla^2 u) - h |\nabla u| \] on a set $E \Subset \mathbb{R}^n$, where $h \in [0,\infty)$ and…
We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if D is any open set in C^d, and L is a suitable transfer operator acting on Bergman space A^2(D),…
Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical…
In this chapter we are examining several iterative methods for solving nonlinear eigenvalue problems. These arise in variational image-processing, graph partition and classification, nonlinear physics and more. The canonical eigenproblem we…
In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric K\"ahler manifold (i.e. $\lambda_1^\mathbb{T}$, the invariant first eigenvalue) is an unbounded function of the…
The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove…
We show how a Bochner type formula can be used to establish universal inequalities for the eigenvalues of the drifted Cheng-Yau operator on a bounded domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary condition. In the…
Assume that the compact Riemannian spin manifold $(M^n,g)$ admits a $G$-structure with characteristic connection $\nabla$ and parallel characteristic torsion ($\nabla T=0$), and consider the Dirac operator $D^{1/3}$ corresponding to the…
The dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal…
We study the dynamics in C^2 of superattracting fixed point germs and of polynomial maps near infinity. In both cases we show that the asymptotic attraction rate is a quadratic integer, and construct a plurisubharmonic function with the…
The eigenvalues of the Dirac operator on a curved spacetime are diffeomorphism-invariant functions of the geometry. They form an infinite set of ``observables'' for general relativity. Recent work of Chamseddine and Connes suggests that…
In this paper, we investigate the eigenvalue problem for a non-local dispersal operator defined on a bounded spatial domain with Neumann-type boundary conditions. Unlike the classical Laplacian, the non-local operator lacks compactness,…
Within the framework of quantum harmonic analysis, for a locally compact group $G$ with a square-integrable, irreducible unitary representation, we analyze the eigenvalue distributions of convolutions between indicator functions on $G$ and…
In this paper, we introduce cosine Thurston maps. In particular, we construct postsingularly finite topological cosine maps and focus on such maps with strictly preperiodic critical points. We use the techniques of Hubbard, Schleicher, and…
Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite…
We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a…
Recent work in the literature has studied fourth-order elliptic operators on manifolds with boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and…
Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to…
For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue''…