[permalink] [id link]
The eigenfunction expansions to certain linear operators defined over a given domain, will often yield a countably infinite set of orthogonal functions which will span that domain.
from
Wikipedia
Some Related Sentences
eigenfunction and expansions
The twin subjects of eigenfunction expansions and functional decomposition, both briefly alluded to in this Wikipedia article, are not completely independent.
eigenfunction and certain
# to be an eigenfunction of certain Casimir operators on G ; and
eigenfunction and linear
where the last right-hand term of the previous equation is easily obtained by remembering that the Fourier transform of a Dirac pulse is a complex exponential function, an eigenfunction of every linear system.
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor.
* The usage of generalized eigenfunction differs from this ; it is part of the theory of rigged Hilbert spaces, so that for a linear operator on a function space this may be something different.
eigenfunction and operators
In this way, it has been deduced that an eigenfunction of the Hamiltonian operator of a system with discrete translational symmetry such as a crystal is always an eigenfunction of the discrete symmetrical translation operators with eigenvalue.
eigenfunction and over
where the sum over has been replaced by an integral over and the matrix and vector have been replaced by the ' kernel ' and the eigenfunction.
eigenfunction and given
This eigenfunction gives the probability of the occurrence of a given integer in a continued fraction expansion, and is known as the Gauss – Kuzmin distribution.
eigenfunction and often
It is perhaps worthwhile to note that both the eigenfunction and eigenvector solutions to these two equations respectively, often yield an orthogonal set of functions / vectors which span ( i. e., form a basis set for ) the function / vector spaces under consideration.
These approximate methods were ( and are ) often used together with the central field approximation, to impose that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction.
eigenfunction and functions
* Each energy eigenfunction is assumed to be describable by a single Slater determinant, an antisymmetrized product of one-electron wave functions ( i. e., orbitals ).
Finding the Hartree – Fock one-electron wave functions is now equivalent to solving the eigenfunction equation:
eigenfunction and which
Specifically, in the study of signals and systems, the eigenfunction of a system is the signal which when input into the system, produces a response with the complex constant.
The plane wave spectrum arises naturally as the eigenfunction or " natural mode " solution to the homogeneous electromagnetic wave equation in rectangular coordinates ( see also Electromagnetic radiation, which derives the wave equation from Maxwell's equations in source-free media, or Scott ).
In MP theory the zeroth-order wave function is an exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator.
This proof of the Hellmann – Feynman theorem requires that the wavefunction be an eigenfunction of the Hamiltonian under consideration ; however, one can also prove more generally that the theorem holds for non-eigenfunction wavefunctions which are stationary ( partial derivative is zero ) for all relevant variables ( such as orbital rotations ).
eigenfunction and .
The reason for this is that the complex exponential is the eigenfunction of differentiation.
This solution of the vibrations of a circular drum | vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.
From the Schrödinger equation, the phase of an eigenfunction with energy E goes as.
In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero ( i. e., the forcing function / forcing vector is zero ), the equation may still admit a non-trivial solution, known in applied mathematics as an eigenfunction solution, in physics as a " natural mode " solution and in electrical circuit theory as the " zero-input response.
Each propagation mode of the waveguide is known as an eigenfunction solution ( or eigenmode solution ) to Maxwell's equations in the waveguide.
In this case, λ is the eigenvalue of the opposite of the differentiation operator with N ( t ) as the corresponding eigenfunction.
In this way, each eigenfunction of the Hamiltonian can be an eigenfunction of the translation operator.
on the real line R, but isn't square-integrable for the usual Borel measure on R. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory.
This was supplied by the apparatus of Schwartz distributions, and a generalized eigenfunction theory was developed in the years after 1950.
By convention, n corresponds to the number of nodes of the radial eigenfunction, l indicates the total number of nodal lines on spheres, and m tells how many of these nodal lines cross the equator.
If the distribution has support only at the point, then the integral equation reduces to a differential eigenfunction equation.
expansions and certain
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
The concept of nature as a whole, the physical universe, is one of several expansions of the original notion ; it began with certain core applications of the word φύσις by pre-Socratic philosophers, and has steadily gained currency ever since.
The devas are expansions of Brahman into various forms, each with a certain quality.
Asymptotic expansions typically arise in the approximation of certain integrals ( Laplace's method, saddle-point method, method of steepest descent ) or in the approximation of probability distributions ( Edgeworth series ).
In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions ; indeed, of compositions with such functions.
The game is a third-person shooter in the same style as Everything or Nothing, with expansions in the story and certain details changed ( such as trading SPECTRE for OCTOPUS, due to legal problems ).
expansions and linear
Therefore the problem of computing derivatives, antiderivatives, integrals, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition.
More formally, this linear approximation is given by using two 1st order Taylor expansions, namely:
For exactly isotropic materials, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient.
expansions and operators
Some natural questions are: under what circumstances does this formalism work, and for what operators are expansions in series of other operators like this possible?
expansions and defined
Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the " gaps " between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions.
Peano's ground-breaking paper contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator.
More recent editions of the Lexicon Minus have corrections and expansions ; also, in later editions all words are defined in both English, French, and German, making it of greater international importance than Lewis and Short.
expansions and over
The Zhou initially moved their capital west to an area near modern Xi ' an, on the Wei River, a tributary of the Yellow River, but they would preside over a series of expansions into the Yangtze River valley.
In addition to the main rule book ( now in its fourth edition ) there have been over 100 supplemental books published with adventures and expansions to both the rules and the game settings.
Many additions and expansions to the town were made over subsequent decades, adding several streets and more than 300 lots.
On February 22, 2008, Fantasy Flight Games announced that they would be taking over the license for Talisman, and continuing to produce both the 4th edition and its expansions.
In the 19th century, the story had undergone expansions and transformations by many notable authors and had been translated into numerous languages, totaling over 100 various editions.
The expansions of the Kings of the Vojislavljević dynasty led to the control over the other Slavic lands, including Zahumlje, Bosnia and Rascia.
The park has expanded several times over the decades to its present size ; one of the most recent expansions occurred in 1978, when grassroots efforts, spearheaded by the Sierra Club, fought off attempts by the Walt Disney Corporation to purchase a high-alpine former mining site south of the park for use as a ski resort.
Japan's underground networks, while individually smaller, are the most extensive overall with an estimated 76 underground shopping streets totaling over 900, 000 square metres of floor space in 1996, with many expansions since then.
I therefore decided to split up the fall in pressure of the steam into small fractional expansions over a large number of turbines in series, so that the velocity of the steam nowhere should be great ... I was also anxious to avoid the well-known cutting action on metal of steam at high velocity.
As Carpenter and Prichard write in The Oxford Companion to Children's Literature ( Oxford, 1995 ), " with all the expansions and contractions over the past two centuries ( this includes a long history of abridgments, condensations, Christianizing, and Disney products ), Wyss's original narrative has long since been obscured.
After four expansions, the line currently has 20 stations over a total track length of 18. 828 km, of which 16. 654 km are passenger track.
The largest Innovia Metro system today is part of the Vancouver SkyTrain network, which has seen several major expansions over its lifetime, with several more being planned.
By February 2009, the title ( expansions included ) had sold over 4 million units, making it one of the most successful PC video games ever.
Circumscription and delimitation of the tribe has experienced expansions and contractions over the decades as genera and groups of genera have been moved in and out,
It is planned these expansions will cater for more than 40, 000 people over the next 10 to 20
At its opening it had a total capacity of 73, 000 and expansions over the past decade have since increased that to 76, 867.
By 1715, however, the French built Fort Michilimackinac to re-establish a presence along the Straits of Mackinac, with several modifications and expansions to the palisade walls over the decades.
Extensions and expansions over the centuries led to its formal consecration in 1559 when the first bishop Nicolaas van Nieuwland was appointed.
It originally seated 50, 611 for baseball, but several expansions over the years brought it to 58, 729.
The stadium originally sat 35, 000, but many expansions took place over the years, raising the capacity to over 46, 500.
There exist many such expansions of a stochastic process: if the process is indexed over b, any orthonormal basis of L < sup > 2 </ sup >( b ) yields an expansion thereof in that form.
This hospital has undergone several expansions over the last twenty years.
The bauxite mine is the world's largest with planned expansions increasing the margin over other mines in 2010.
0.950 seconds.