Definition & Betydelse | Engelska ordet EIGENFUNCTIONS

Definition av EIGENFUNCTIONS

1. böjningsform av eigenfunction

Exempel på hur man kan använda EIGENFUNCTIONS i en mening

• If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's degree of degeneracy or geometric multiplicity.
• The eigenfunctions corresponding to non-zero eigenvalues are continuous on X and K has the representation.
• The novel Spherical Harmonic involves an imagined universe based on the Hilbert space described by the spherical harmonic eigenfunctions that solve the Laplace Equation, and some prose in the book is written in the shape of the sinusoidal waves found in the spherical harmonics.
• In mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory, which was joint work with Charles François Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, and the fact (also known as Liouville's theorem) that time evolution is measure preserving for a Hamiltonian system.
• While these operators may not be compact, their inverses (when they exist) may be, as in the case of the wave equation, and these inverses have the same eigenfunctions and eigenvalues as the original operator (with the possible exception of zero).
• The statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to the Fourier theorem about the development of any periodic function in a Fourier series.
• In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions.
• Some natural questions are: under what circumstances does this formalism work, and for what operators L are expansions in series of other operators like this possible? Can any function f be expressed in terms of the eigenfunctions (are they a Schauder basis) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite-dimensional spaces and finite-dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and matrix algebra.
• We also noted that one hurdle in its application was the numerical cost of determining the eigenvalues and eigenfunctions of its covariance operator through the Fredholm integral equation of the second kind.
• It has equally found applications to estimate the spectral decompositions of linear operators when the eigenfunctions are parameterized with a linear model, i.
• Crudely speaking, one may define a Laplacian on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings.
• Other contributions in mathematics and mathematical physics include the rigorous foundations of Kenneth Wilson's renormalization group-method, which led to Wilson's Nobel Prize for Physics in 1982, Gibbs measures in ergodic theory, hyperbolic Markov partitions, proof of the existence of Hamiltonian dynamics for infinite particle systems by the idea of "cluster dynamics", description of the discrete Schrödinger operators by the localization of eigenfunctions, Markov partitions for billiards and Lorenz map (with Bunimovich and Chernov), a rigorous treatment of subdiffusions in dynamics, verification of asymptotic Poisson distribution of energy level gaps for a class of integrable dynamical systems, and his version of the Navier–Stokes equations together with Khanin, Mattingly and Li.
• a single Slater determinant composed of orbitals that are the eigenfunctions of the corresponding self-consistent Fock operator.
• The prolate spheroidal wave functions are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated around its long axis, “cigar shape“).
• It also demonstrated the DMD and related methods produce approximations of the Koopman eigenfunctions in addition to the more commonly used eigenvalues and modes.
• These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.
• The eigenfunctions can also be characterized by being rotationally symmetric (thus time-invariant) pure states.
• Borcea, Julius; Bøgvad, Rikard; Shapiro, Boris, Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions.
• The unique quantum ergodicity conjecture of Rudnick and Sarnak asks whether the stronger statement that individual eigenfunctions equidistribure is true.

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