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## ‣ First Principles Semiclassical Calculations of Vibrational Eigenfunctions

Ceotto, Michele; Valleau, Stéphanie; Tantardini, Gian Franco; Aspuru-Guzik, Alán
Fonte: American Institute of Physics Publicador: American Institute of Physics
Tipo: Artigo de Revista Científica
Português
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Vibrational eigenfunctions are calculated on-the-fly using semiclassical methods in conjunction with ab initio density functional theory classical trajectories. Various semiclassical approximations based on the time-dependent representation of the eigenfunctions are tested on an analytical potential describing the chemisorption of CO on Cu(100). Then, first principles semiclassical vibrational eigenfunctions are calculated for the (CO_2) molecule and its accuracy evaluated. The multiple coherent states initial value representations semiclassical method recently developed by us has shown with only six ab initio trajectories to evaluate eigenvalues and eigenfunctions at the accuracy level of thousands trajectory semiclassical initial value representation simulations.; Chemistry and Chemical Biology

## ‣ Tables of eigenvalues and eigenfunctions of the Orr-Sommerfeld equation for plane Poiseuille flows

Clark, W. H.; Gawain, Theodore Henry
Fonte: Monterey, California. Naval Postgraduate School Publicador: Monterey, California. Naval Postgraduate School
Tipo: Relatório
Português
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In the report the authors present a numerical technique for computing the eigenvalues and eigenfunctions of the Orr-Sommerfeld equation for infinitesimal disturbances in plane Poiseuille flows. For the case alpha = 1.0, Rsube = 6667 the eigenvalues, beta sub Nm, (n = 1,2,3,4 m = 1,2,...199) and eigenfunctions, phi sub nm(y), (n = 1,2,3,4 m = 1,2,...8) are presented in tabular and graphical form. In addition the function, chi sub nm(y), which is orthogonal to phi sub nm(y), over the interval -1 or - y or - plus or minus 1 is tabulated. In a previous report (Gawain and Clark)1971) it was shown that these eigenfunctions can be extremely useful in describing certain aspects of the nonlinear mechanics of wave disturbances in plane Poiseuille flows. It is hoped that the present report will serve both as a complement to the previously mentioned report and as a useful reference for similar future investigations. (Author)

## ‣ Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds

Braun, Volker; Brelidze, Tamaz; Douglas, Michael R.; Ovrut, Burt A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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A numerical algorithm for explicitly computing the spectrum of the Laplace-Beltrami operator on Calabi-Yau threefolds is presented. The requisite Ricci-flat metrics are calculated using a method introduced in previous papers. To illustrate our algorithm, the eigenvalues and eigenfunctions of the Laplacian are computed numerically on two different quintic hypersurfaces, some Z_5 x Z_5 quotients of quintics, and the Calabi-Yau threefold with Z_3 x Z_3 fundamental group of the heterotic standard model. The multiplicities of the eigenvalues are explained in detail in terms of the irreducible representations of the finite isometry groups of the threefolds.; Comment: 67 pages, 16 figures, 9 tables. v2: References added

## ‣ Eigenvalues and eigenfunctions of the anharmonic oscillator $V(x,y)=x^{2}y^{2}$

Fernández, Francisco M.; Garcia, Javier
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/10/2013 Português
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We obtain sufficiently accurate eigenvalues and eigenfunctions for the anharmonic oscillator with potential $V(x,y)=x^{2}y^{2}$ by means of three different methods. Our results strongly suggest that the spectrum of this oscillator is discrete in agreement with early rigorous mathematical proofs and against a recent statement that cast doubts about it.

## ‣ Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

Barbatis, G.; Burenkov, V. I.; Lamberti, P. D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi$ defined on a fixed reference domain $\Omega$. Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.; Comment: 34 pages. Minor changes in the introduction and the refercenes. Published in: Around the research of Vladimir Maz'ya II, pp23--60, Int. Math. Ser. (N.Y.), vol. 12, Springer, New York 2010

## ‣ On Eigenvalues and Eigenfunctions Absent in the Actual Solid State Theory

Pereyra, Pedro
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/09/2000 Português
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In this letter new, closed and compact analytic expressions for the evaluation of resonant energies, resonant bound-states, eigenvalues and eigenfunctions for both scattering and bounded $n$-cell systems are reported. It is shown that for (scattering and bounded) 1-D systems the eigenfunctions $\Psi_{\mu ,\nu}(z)$ are simple and well defined functions of the Chebyshev polynomials of the second kind $U_{n}$, and the energy eigenvalues $E_{\mu ,\nu }$ (in the $\mu$-th band) are determined by the zeros of these polynomials. New insights on the energy gap and the localization effect induced by phase coherence are shown.; Comment: 10 pages, 5 figures

## ‣ Eigenvalues and Eigenfunctions of Double Layer Potentials

Miyanishi, Yoshihisa; Suzuki, Takashi
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/01/2015 Português
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For bounded domains, eigenvalues and eigenfunctions of double layer potentials are considered. The aim of this paper is to establish some relationships between eigenvalues, eigenfunctions and the geometry of domain boundaries.; Comment: 19 pages. Preliminary version, comments appreciated

## ‣ Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions

Berti, Emanuele; Cardoso, Vitor; Casals, Marc
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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Spin-weighted spheroidal harmonics are useful in a variety of physical situations, including light scattering, nuclear modeling, signal processing, electromagnetic wave propagation, black hole perturbation theory in four and higher dimensions, quantum field theory in curved space-time and studies of D-branes. We first review analytic and numerical calculations of their eigenvalues and eigenfunctions in four dimensions, filling gaps in the existing literature when necessary. Then we compute the angular dependence of the spin-weighted spheroidal harmonics corresponding to slowly-damped quasinormal mode frequencies of the Kerr black hole, providing numerical tables and approximate formulas for their scalar products. Finally we present an exhaustive analytic and numerical study of scalar spheroidal harmonics in (n+4) dimensions.; Comment: 26 pages, 10 figures. Corrected typos in Eqs. (2.16f) and (2.16g)

## ‣ Eigenvalues and Eigenfunctions of Two Coupled Normal Metal Nano-rings

Fang, Lei; Schmeltzer, David
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/07/2015 Português
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A general scheme is developed to deal with 1D lattice systems that could be topologically complicated. It is aimed to give a complete study of two coupled normal metal rings. Our method starts with an investigation of the local expressions of the eigenfunctions. By connecting different parts of the system, all the eigenvalues and eigenfunctions can be obtained. It is found that there is a possibility for the existence of localized states, which is beyond previous expectations.; Comment: 20 pages, 7 figures

## ‣ Eigenvalues and Eigenfunctions of Woods Saxon Potential in PT Symmetric Quantum Mechanics

Berkdemir, Ayse; Berkdemir, Cuneyt; Sever, Ramazan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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Using the Nikiforov Uvarov method, we obtained the eigenvalues and eigenfunctions of the Woods Saxon potential with the negative energy levels based on the mathematical approach. According to the PT Symmetric quantum mechanics, we exactly solved the time independent Shcrodinger equation for the same potential. Results are obtained for the s states.; Comment: 12 pages. submitted to Physics Letters A

## ‣ Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift

Biezuner, Rodney Josué; Ercole, Grey; Giacchini, Breno Loureiro; Martins, Eder Marinho
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains $\Omega\subset R^{N}$. This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function $u\in L^{2}(\Omega)$.; Comment: In this version the numerical tests in Section 6 were considerably improved and the Section 5 entitled "Normalization at each step" was introduced. Moreover, minor adjustments in the Section 1 (Introduction) and in the Section 7 (Fi nal Comments) were made. Breno Loureiro Giacchini was added as coauthor

## ‣ A Finite Element Algorithm for High-Lying Eigenvalues and Eigenfunctions with Homogeneous Neumann and Dirichlet Boundary Conditions

Baez, G.; Leyvraz, F.; Mendez-Sanchez, R. A.; Seligman, T. H.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 26/05/2000 Português
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We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions or combinations of either for different parts of the boundary. In order to solve the generalized eigenvalue problem, we use an inverse power plus Gauss-Seidel algorithm. For Neumann boundary conditions the method is much more efficient than the equivalent finite difference algorithm. We have cheked the algorithm comparing the cumulative level density of the espectrum obtained numerically, with the theoretical prediction given by the Weyl formula. A systematic deviation was found. This deviation is due to the discretisation and not to the algorithm. As an application we calculate the statistical properties of the eigenvalues of the acoustic Bunimovich stadium and compare them with the theoretical results given by random matrix theory.; Comment: 8 pages, 8 figures

## ‣ Asymptotic expressions of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the boundary conditon

Şen, Erdoğan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/02/2013 Português
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The aim of this study is to find asymptotic expressions of eigenvalues and eigenfunctions of a discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition. Applications of differential equations with retarded argument can be encountered in the theory of selfoscillatory systems, in the study of problems connected with combustion in rocket engines, in a number of problems in economics, biophysics, and many other fields. The problems in these areas can be solved reducing differential equations with retarded argument. In this study discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition were investigated and asymptotic formulas were obtained for eigenvalues and eigenfunctions for using areas which mentioned above; Comment: Master thesis, 35 pages

## ‣ Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function

Şen, Erdoğan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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In this paper, by using the similar methods of [O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694] we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with discontinuous weight at two interior points together with spectral parameter-dependent boundary conditions. We give an operator-theoretic formulation for the considered problem and obtain asymptotic formulas for the eigenvalues and eigenfunctions.; Comment: 11 pages

## ‣ Exact Eigenvalues and Eigenfunctions of the Hulthen Potential in the PT-Symmetry for Any Angular Momentum

Ikhdair, Sameer M.; Sever, Ramazan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/08/2005 Português
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The Schrodinger equation with the PT-symmetric Hulthen potential is solved exactly by taking into account effect of the centrifugal barrier for any l-state. Eigenfunctions are obtained in terms of the Jacobi polynomials. The Nikiforov-Uvarov method is used in the computations. Our numerical results are in good agreement with the ones obtained before. Keywords: Energy Eigenvalues and Eigenfunctions; Hulthen potential; PT-symmetry; Nikiforov-Uvarov Method.; Comment: 24 pages

## ‣ Asymptotic formulations of the eigenvalues and eigenfunctions for a boundary value problem

Şen, Erdoğan; Bayramov, Azad
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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In this work a discontinuous boundary-value problem with retarded argument which contains spectral parameter in the transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions.; Comment: 12 pages, accepted for publication in Mathematical Methods in the Applied Sciences

## ‣ Asymptotic formulas for eigenvalues and eigenfunctions of a new boundary-value-transmission problem

Mukhtarov, O. Sh.; Aydemir, K.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/03/2013 Português
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In this paper we are concerned with a new class of BVP' s consisting of eigendependent boundary conditions and two supplementary transmission conditions at one interior point. By modifying some techniques of classical Sturm-Liouville theory and suggesting own approaches we find asymptotic formulas for the eigenvalues and eigenfunction.

## ‣ Gravity Modes in ZZ Ceti Stars. II. Eigenvalues and Eigenfunctions

Wu, Yanqin; Goldreich, Peter
Fonte: American Astronomical Society Publicador: American Astronomical Society
Tipo: Article; PeerReviewed Formato: application/pdf
Publicado em 10/07/1999 Português
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We report on numerical calculations of nonadiabatic eigenvalues and eigenfunctions for g-modes in ZZ Ceti variables. The spectrum of overstable l = 1 modes delineates the instability strip. Its blue edge occurs where ωτ_c ≈ 1 for the n = 1 mode. Here ω is radian frequency and τ_c is about 4 times the thermal time at the bottom of the surface convection zone. As a ZZ Ceti cools, its convection zone deepens, longer period modes become overstable, but the critical value of ωτ_c separating overstable and damped modes rises. The latter is a consequence of enhanced radiative damping for modes that propagate immediately below the convection zone. The critical value of ωτ_c is of observational significance, because modes with the smallest value of ωτ_c are most observable photometrically. Maximum periods for overstable modes predicted for our cooler model envelopes are about a factor of 2 longer than the observational upper limit of 1200 s. We assess a number of plausible resolutions for this discrepancy among which convective overshoot and nonlinear saturation look promising. The nonadiabatic eigenfunctions enable us to predict relative amplitudes and phases of photospheric variations of flux and velocity, quantities made accessible by recent observations. We also present asymptotic formula for damping rates of high-order modes...

## ‣ Stability of pole solutions for planar propagating flames: I. Exact eigenvalues and eigenfunctions

Vaynblat, Dimitri; Matalon, Moshe
Fonte: Instituto de Tecnologia da Califórnia Publicador: Instituto de Tecnologia da Califórnia
Tipo: Article; PeerReviewed Formato: application/pdf
Publicado em 02/02/2000 Português
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It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states-the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.

## ‣ Reliable Tracking Algorithms for Principal and Minor Eigenvector Computations

Baumann, Markus; Helmke, Uwe; Manton, Jonathan
Fonte: Institute of Electrical and Electronics Engineers (IEEE Inc) Publicador: Institute of Electrical and Electronics Engineers (IEEE Inc)
Tipo: Conference paper
Português
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Many problems in control and signal processing require the tracking of certain eigenvectors of a time-varying matrix; the eigenvectors associated with the largest eigenvalues are called the principal eigenvectors and those with the smallest eigenvalues the minor eigenvectors. This paper presents a novel algorithm for tracking minor eigenvectors. One interesting feature, inherited from a recently proposed minor eigenvector flow upon which part of this work is based, is that the algorithm can be used also for tracking principal eigenvectors simply by changing the sign of the matrix whose eigenvectors are being tracked. The other key feature is that the algorithm has a guaranteed accuracy. Indeed, the algorithm is based on a flow which can be interpreted as the combination of a homotopy method and a Newton method, the purpose of the latter to compensate for discretisation errors.