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Results for “EIGENFUNCTION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

On Eigenfunctions of the Fourier Transform

A nontrivial example of an eigenfunction in the sense of the theory of distributions for the planar Fourier transform was described by the authors in their previous work. In this paper, a method for obtaining other eigenfunctions is proposed. Positive homogeneous distributions in ℝ{sup n} of order −n/2 are considered, and it is shown that F(ω)|x|{sup −n/2}, |ω| = 1, is an eigenfunction in the sense of the theory of distributions of the Fourier transform if and only if F(ω) is an eigenfunction of a certain singular integral operator on the unit sphere of ℝ{sup n}. Since Y{sub m,n}{sup (k)}(ω)|x|{sup −n/2}, where Y{sub m,n}{sup (k)} denote the spherical functions of order m in ℝ{sup n}, are eigenfunctions of the Fourier transform, it follows that Y{sub m,n}{sup (k)} are eigenfunctions of the above-mentioned singular integral operator. In the planar case, all eigenfunctions of the Fourier transform of the form F(ω)|x|{sup −1} are described by means of the Fourier coefficients of F(ω).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers

The application of quantum algorithms to the study of many-particle quantum systems requires the ability to prepare wave functions that are relevant in the behavior of the system under study. Hamiltonian symmetries are important instruments used to classify relevant many-particle wave functions and to improve the efficiency of numerical simulations. In this work, quantum circuits for the exact and approximate preparation of total spin eigenfunctions on quantum computers are presented. Two different strategies are discussed and compared: exact recursive construction of total spin eigenfunctions based on the addition theorem of angular momentum, and heuristic approximation of total spin eigenfunctions based on the variational optimization of a suitable cost function. The construction of these quantum circuits is illustrated in detail, and the preparation of total spin eigenfunctions is demonstrated on IBM quantum devices, focusing on three- and five-spin systems on graphs with triangle connectivity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Torus geometry eigenfunctions of an interacting multi-Landau-level Hamiltonian

A short-ranged, rotationally symmetric multi-Landau-level model Hamiltonian for strongly interacting electrons in a magnetic field was proposed [A. Anand et al., Phys. Rev. Lett. 126, 136601 (2021)] with the key feature that it allows exact many-body eigenfunctions on the disk not just for quasiholes but for all charged and neutral excitations of the entire Jain sequence filling fractions. We extend this to geometries without full rotational symmetry, namely, the torus and cylinder geometries, and present their spectra. Exact diagonalization of the interaction on the torus produces the low-energy spectra at filling fraction v = n/(2⁢pn + 1) that is identical, up to a topological (2⁢pn + 1)-fold multiplicity, to that of the integer quantum Hall spectra at v = n, for the incompressible state as well as all excitations. While the ansatz eigenfunctions in the disk geometry cannot be generalized to closed geometries such as torus or sphere, we show how to extend them to cylinder geometry. Meanwhile, we show eigenfunctions for charged excitations at filling fractions between 1/3 and 2/5 can be written on the torus and the spherical geometries.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1972

The calculation of the eigenvalues and eigenfunctions of Mathieu's equation

The eigenfunctions of Mathieu's equation are expanded in trigonometric series, and the resulting eigenvalue problem is cast in matrix form. This matrix is found to be a symmetric, triagonal matrix, and the eigenvalues are computed using the bisection method. The eigenfunction expansion coefficients are obtained by the standard recursion method. This computational technique for the eigenvalues and eigenfunctions of Mathieu's equation is both rapid and accurate.

Hodge, D. B.↗
NASA NTRS1988

Eigenfunctions of the magnetospheric radial-diffusion operator

The applicability of eigenfunction methods to the theory of magnetospheric radial diffusion is investigated analytically, with a focus on the operator (Lambda) for radiation-belt and ring-current particles. It is shown that the eigenfunctions of Lambda can be expressed in terms of a linear combination of ordinary Bessel functions of the first and second kinds, and that the solution for the drift-averaged phase-space density can be expressed as the sum of a quasi-static solution and a time-dependent superposition of the eigenfunctions of Lambda. Results for illustrative and practical sample problems are presented in extensive tables and graphs and discussed in detail.

Schulz, Michael↗
NASA NTRS1991

Dynamical eigenfunction decomposition of turbulent channel flow

The results of an analysis of low-Reynolds-number turbulent channel flow based on the Karhunen-Loeve (K-L) expansion are presented. The turbulent flow field is generated by a direct numerical simulation of the Navier-Stokes equations at a Reynolds number Re(tau) = 80 (based on the wall shear velocity and channel half-width). The K-L procedure is then applied to determine the eigenvalues and eigenfunctions for this flow. The random coefficients of the K-L expansion are subsequently found by projecting the numerical flow field onto these eigenfunctions. The resulting expansion captures 90 percent of the turbulent energy with significantly fewer modes than the original trigonometric expansion. The eigenfunctions, which appear either as rolls or shearing motions, possess viscous boundary layers at the walls and are much richer in harmonics than the original basis functions.

Ball, K. S.↗
DOE OSTI2019

On Justification of the Asymptotics of Eigenfunctions of the Absolutely Continuous Spectrum in the Problem of Three One-Dimensional Short-Range Quantum Particles with Repulsion

The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrödinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrödinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1985

The exact eigenfunctions and eigenvalues of a two-dimensional rigid rotor obtained using Gaussian wave packet dynamics

Exact eigenfunctions for a two-dimensional rigid rotor are obtained using Gaussian wave packet dynamics. The wave functions are obtained by propagating, without approximation, an infinite set of Gaussian wave packets that collectively have the correct periodicity, being coherent states appropriate to this rotational problem. This result leads to a numerical method for the semiclassical calculation of rovibrational, molecular eigenstates. Also, a simple, almost classical, approximation to full wave packet dynamics is shown to give exact results: this leads to an a posteriori justification of the De Leon-Heller spectral quantization method.

Reimers, J. R.↗
NASA NTRS1986

Eigenfunction methods in magnetospheric radial-diffusion theory

Complete sets of orthonormal basis functions constructed according to a generalization of the quantum-mechanical WKB approximation can be used to generate a nearly-diagonal matrix representation of the radial-transport operator for ring-current ions in the presence of radial diffusion and charge exchange. The resulting eigenfunctions (constructed by weighting the basis functions in proportion to the respective components of the eigenvectors of the matrix representation) and eigenvalues provide a spatial and temporal description of the evolving phase-space density during and following a magnetospheric disturbance (e.g., a magnetic storm). A linear superposition of the basis functions can also be used to eliminate any discrepancy between the steady-state solution of the transport equation and the appropriate WKB approximation of this steady-state solution.

Schulz, Michael↗