Related papers: Accurate eigenvalues of bounded oscillators
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…
We establish uniform lower and upper bounds for the eigenvalues of the Hodge Laplacian acting on differential forms on closed Riemannian manifolds with a lower Ricci curvature bound, a positive lower bound on the injectivity radius, and an…
In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and…
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 analyze the distribution of the eigenvalues of the quantum-mechanical rotating harmonic oscillator by means of the Frobenius method. A suitable ansatz leads to a three-term recurrence relation for the expansion coefficients. Truncation…
The generalized pseudospectral method is employed for the accurate calculation of eigenvalues, densities and expectation values for the spiked harmonic oscillators. This allows \emph{nonuniform} and \emph{optimal} spatial discretization of…
If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given…
We introduce an efficient numerical method for second order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory…
In this paper, using the Lewis-Riesenfeld method, we determine the explicit form of the wavefunctions of one- and three-dimensional harmonic oscillators with time-dependent mass and frequency within the framework of the Dunkl derivative,…
The vertical modes of linearized equations of motion are widely used by the oceanographic community in numerous theoretical and observational contexts. However, the standard approach for solving the generalized eigenvalue problem using…
We prove that the Dirichlet eigenvalues of the Laplace-Beltrami operator on a compact Riemannian manifold with cylindrical boundary can be approximated by the spectrum of truncated graph Laplacians constructed from…
In this paper, we study the adaptive planewave discretization for a cluster of eigenvalues of second-order elliptic partial differential equations. We first design an a posteriori error estimator and prove both the upper and lower bounds.…
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenb\"ock techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor.…
In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and…
In this paper, we compute universal estimates of eigenvalues for a class of coupled systems of elliptic differential equations in divergence form on a bounded domain in Euclidean space, which includes the well-known Lam\'e and the Laplacian…
We obtain highly accurate solutions to the Thomas-Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Pad\'e-Hankel method, numerical integration, power series with Pad\'e and Hermite-Pad\'e approximants and…
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…
This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for isolated…
We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we…
We study the rate of convergence for (variational) eigenvalues of several non-linear problems involving oscillating weights and subject to different kinds of boundary conditions in bounded domains.