The method of empirical eigenfunctions is developed in a general framework. In particular it is shown that the method of snapshots leads to the determination of the empirical eigenfunctions in any number of dimensions in terms of an equivalent one-dimensional problem. The methodology is discussed within the framework of some turbulent simulations and it is shown how this facilitates the flow an...

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 ...

The linear canonical transform (LCT) with a, b, c, d parameter plays an important role in quantum mechanics, optics, and signal processing. The eigenfunctions of the LCT are also important because they describe the self-imaging phenomenon in optical systems. However, the existing solutions for the eigenfunctions of the LCT are divided into many cases and they lack a systematic way to solve thes...

Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost and it is difficult to design nonstationary GP priors in practice. In this paper, we propose a sparse Gaussian process model, EigenGP, based on the Karhunen-Loève (KL) expansion of a GP prior. We use the Nyström approximation to obtain data dependent...

Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-pote...

Abstract In this work, we consider a boundary value problem for q -Dirac equation. We prove orthogonality of the eigenfunctions, realness eigenvalues, and study asymptotic formulas eigenfunctions. show that eigenfunctions form complete system, obtain expansion formula with respect to derive Parseval’s equality. construct Weyl solution function. uniqueness theorem inverse

Contents 1. Introduction 1 2. Some preliminaries 5 3. The point spectrum of U p 7 4. A structure theorem for eigenfunctions 14 5. A decomposition into finite dimensional eigenspaces 21 6. Simultaneous eigenfunctions 23 7. A first application: tensor products of Hecke operatorsand the Riemann zeta function 27 8. A second application: completely multiplicative coefficients 30 9. Appendix: Explici...

We consider an eigenvalue problem for a divergence form elliptic operator Aε with high contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of ‘order one’ size. The local perturbation of coefficients for such operator could result in emergence of localised waves eigenfunctions with corresponding eigenva...

We consider an eigenvalue problem for a divergence form elliptic operator Aε with high contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of ‘order one’ size. The local perturbation of coefficients for such operator could result in emergence of localized waves eigenfunctions with corresponding eigenva...