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Topic: Eigenfunction

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	Eigenfunction (Site not responding. Last check: 2007-10-11)
		In mathematics an eigenfunction f of a linear operator A on a function space is an eigenvector of the linear operator; it satisfies
		Eigenfunctions play an important role in quantum mechanics where the Schrödinger equation
		Due to the of the hamiltonian operator $\mathcal H its$ eigenfunctions are orthogonal functions.
	www.freeglossary.com /Eigenfunction (120 words)

	Eigenfunction Expansions, Operator Algebras and Riemannian Symmetric Spaces
		This Research Note pays particular attention to studying the convergence of the expansion and to the case where D is a family of partial differential operators.
		The elements of the range of these eigenprojections are the eigenfunctions, which solve the appropriate eigenvalue equations by duality.
		The spectral measure is abstractly defined, but its absolute continuity with respect to Hausdorf measure on the joint spectrum is shown to occur when the eigenfunctions are very well-behaved.
	www.ramex.com /title.asp?id=5247 (437 words)

	And while I'm at it: eigenfunction of position
		I've yet to see a decent argument as to why the eigenfunctions of the position operator are delta functions.
		The way I've understood it is that the components of the matrix that is used to represent the position operator in the position basis are the "values" of the Dirac delta functional: anddelta;(x,x').
		eigenfunctions corresponding to a continuous spectrum of eigenvalues can't be normalized like those of a discrete spectrum can be.
	www.physicsforums.com /showthread.php?t=34156 (1042 words)

	2. Some Basic Ideas about Quantum Mechanics
		This is known as the collapse of the wavefunction and the probability of the wavefunction collapsing into a particular eigenfunction depends on how much that eigenfunction contributed to the original superposition.
		More precisely, the probability that a given eigenfunction will be chosen is proportional to the square of the coefficient of that eigenfunction in the superposition, normalised so that the overall probability of collapse is unity (i.e.
		These special solutions of the time-dependent Schrodinger equation turn out to be the energy eigenfunctions, and can be written as a time-independent factor multiplied by a sinusoidal time-dependent factor related to the energy (in fact the frequency of the sine wave is given by the relation E=h*frequency).
	newton.ex.ac.uk /research/qsystems/people/jenkins/mbody/mbody2.html (2015 words)

	Tetrahedron's Home Page
		The tetrahedral eigenfunctions are visualized for some quantal levels belonging to the symmetry class possessing Dirichlet boundary conditions on the surface of the tetrahedron.
		A sum over the eigenfunctions which reveals the contribution of the rotation elements as well as the boost of the shortest periodic orbit.
		However the periodic orbit shown as a blue tube is not clearly revealed due to the large contribution of the rotation elements.
	www.uni-ulm.de /~raurich/Orbifold/tetraeder.html (453 words)

	An Investigation of Eigenfunctions over the Equilateral Triangle and Square
		For this paper the eigenvalues and eigenfunctions of Laplace's equation with homogeneous Neumann and Dirichlet boundary conditions over the square and equilateral triangle are investigated.
		This is an application of the study of eigenvalues and eigenfunctions of a heat flow problem posed on an equilateral triangle.
		This consisted of evaluating the eigenfunction at the center of the shape and assigning it a corresponding color.
	cauchy.math.colostate.edu /Projects/Sjoberg/paper/paper.html (3145 words)

	Particle in a Box
		In those degenerate cases there are n-1 degrees of freedom for writing the eigenfunctions where n is the degeneracy.
		This is because the eigenfunctions of one operator are automatically eigenfunctions of the second operator.
		If the system is in a pure state described by one of these eigenfunctions the application of the operators will return sharp, well-defined eigenvalues in both cases, i.e.
	itl.chem.ufl.edu /4412_aa/schroed.html (785 words)

	Normal Derivatives on the Sierpinski Gasket
		We were very interested in learning more about normal derivatives of eigenfunctions at the boundary points, especially in relation to the heat kernel.
		Wherein j represents a certain sequence of spectral decimation which determines lambda, lambda is the eigenvalue of u, u is a Dirichlet eigenfunction across the gasket, x and y are two interior points, and t is time.
		The upper limit indicates that the normal derivative of the eigenfunction is on the order of the square root of the eigenvalue, which is exactly the case for the unit interval.
	www.math.cornell.edu /~Bengal/page2b.html (1247 words)

	Amazon.com: eigenfunction (Site not responding. Last check: 2007-10-11)
		Eigenfunction Expansions, Operator Algebras and Riemannian Symmetric Spaces (Pitman Research Notes in Mathematics Series, 355) by Robert M Kauffman (Hardcover - Sep 25, 1996)
		Eigenfunction branches of nonlinear operators, and their bifurcations (Lecture notes in mathematics, 104) by George H Pimbley (Unknown Binding - 1969)
		Representations for eigenfunctions of expected value operators in the Wishart distribution (Institute of Statistics mimeo series) by Donald St. P Richards (Unknown Binding - 1982)
	www.amazon.com /s?ie=UTF8&keywords=eigenfunction&tag=robbsbiologybook&index=blended&link_code=qs&page=1 (462 words)

	Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential, ...
		Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential, Wendell H. Fleming, Shuenn-Jyi Sheu
		Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential
		The asymptotic behaviors of the principal eigenvalue and the corresponding normalized eigenfunction of the operator $G^\varepsilon f = (\varepsilon/2)\triangle f + g \triangledown f +(l/\varepsilon)f$ for small $\varepsilon$ are studied.
	projecteuclid.org /getRecord?id=euclid.aop/1023481117 (746 words)

	The Method of Eigenfunction Expansion - Science Articles
		Rather than trying to explain the techniques in words or with difficult to understand abstract theorems, letâs demonstrate how to use the method with two concrete examples.
		While that elementary option is available in this case, we will use the method of eigenfunction expansion so that we can demonstrate the method using a simple equation.
		We now have two series expansions using the same eigenfunctions that are equal, therefore each individual term in the series on the left must equal the corresponding term on the right.
	www.physicspost.com /articles.php?articleId=212 (770 words)

	(1) Show that the 2px orbital is an eigenfunction of H and L2 but not an eigenfunction of Lz (Site not responding. Last check: 2007-10-11)
		orbital is an eigenfunction of H and L
		is an eigenfunction of H with eigenvalue E
		> = 0 because the orbitals are eigenfunctions of a Hermitian operator, H (or L
	www.rci.rutgers.edu /~kroghjes/KK-J421521/HW6Sol.htm (606 words)

	[No title]
		Any eigenfunction of a linear operator can be multiplied by a constant and still be an eigenfunction of the operator.
		This means that if f(x) is an eigenfunction of A with eigenvalue k, then cf(x) is also an eigenfunction of A with eigenvalue k.
		[(-h2/2m) d2/dx2 + V(x)] ((x) = E ((x), ((x) is the eigenfunction, E is the eigenvalue, & the Hamiltonian operator is (-h2/2m) d2/dx2 + V(x) The Hamiltonian function was originally defined in classical mechanics for systems where the total energy was conserved.
	www.njit.edu /v2/Faculty/Venanzi/notes3.doc (1976 words)

	[No title]
		But by doing this, we increase the uncertainty in momentum because all of the plane waves we mix together have different values of k (and therefore different momenta) and furthermore each may contribute to a different extent depending on the coefficient Ck used in the expansion of the final wavepacket describing our more localized particle.
		In particular, for the position/momentum uncertainty principle EMBED Equation.3 we see that the commutator is not 0 and therefore there is no wavefunction for a particle that is an eigenfunction both the position and the momentum operators.
		In other words, there is not at the same time both an exact (or sharp) momentum eigenvalue and position eigenvalue for a particle When two operators commute, they can have a common set of eigenfunctions.
	faculty.une.edu /cas/cnash/CH111/FreeParticlenew.doc (1014 words)

	Fourier Series: Eigenfunction Approach (Site not responding. Last check: 2007-10-11)
		To be able to view the math in this document, use the PDF version, or please consider using another browser, such as Mozilla, Netscape 7 or above or Microsoft Internet Explorer 6 or above (MathPlayer required for IE).
		Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems, calculating the output of an LTI system ℋ given ⅇ
		equation 2 essentially breaks down f(t) into pieces, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system).
	cnx.org /content/m10496/latest (1049 words)

	SSRN-Black's Model of Interest Rates as Options, Eigenfunction Expansions and Japanese Interest Rates by Viatcheslav ...
		When the shadow rate is modeled as a diffusion, we interpret the zero-coupon bond as a Laplace transform of the area functional of the underlying shadow rate diffusion (evaluated at the unit value of the transform parameter).
		Using the method of eigenfunction expansions, we derive analytical solutions for zero-coupon bonds and bond options under the Vasicek and shifted CIR processes for the shadow rate.
		The current implied value of the instantaneous shadow rate in Japan is negative.
	papers.ssrn.com /sol3/papers.cfm?abstract_id=513751 (368 words)

	Hartree-Fock Theory - (2) the method (Site not responding. Last check: 2007-10-11)
		In the simplest form, we are also assuming the electrons are non-interacting.
		This is equivalent to claiming the electrons must satisfy the Pauli exclusion principle.
		We minimize the energy with respect to changes in the Slater determinant eigenfunction through a variational derivative with respect to the spin-orbitals using a lagrange multiplier which turns out to be the energy eigenvalue for the spin-orbital.
	hermes.phys.uwm.edu /projects/elecstruct/hermsk/HF/HF.Theory2.html (500 words)

	Tension matrix in a scaling eigenfunction basis
		The unscaled eigenfunctions are taken to have unit normalization in the billiard.
		The scaling eigenfunctions no longer go to zero on the original billiard boundary
		is the scaling eigenfunction representation of the tension quadratic form, and we will see that it has wonderful properties.
	www.math.dartmouth.edu /~ahb/thesis_html/node73.html (293 words)

	SSRN-The Eigenfunction Expansion Method in Multi-factor Quadratic Term Structure Models by Nina Boyarchenko, Sergei ... (Site not responding. Last check: 2007-10-11)
		The eigenvalues, eigenfunctions and adjoint functions are calculated using elements of the representation theory of Lie algebras not only in the self-adjoint case but in non-selfadjoint case as well; the eigenfunctions and adjoint functions are expressed in terms of the Hermite polynomials.
		We also consider the subordination of the same class of models, and show that in the framework of the eigenfunction expansion approach, the subordinated models are (almost) as simple as pure Gaussian models.
		Boyarchenko, Nina and Levendorskii, Sergei Z., "The Eigenfunction Expansion Method in Multi-factor Quadratic Term Structure Models" (January 9, 2006).
	papers.ssrn.com /Sol3/papers.cfm?abstract_id=874839 (367 words)

	[No title]
		Let’s assume that the eigenfunction is separable into a product of a radial function and a function involving the two angles.
		A lemma in quantum mechanics states that if two operators commute, then an eigenfunction of one operator is also an eigenfunction of another operator.
		The vector EMBED Equation.2 is a new eigenfunction of EMBED Equation.2 with an eigenvalue that is one EMBED Equation.2 less than the value of the z component of angular momentum.
	www.cbu.edu /~jvarrian/447/Out460_4.doc (2389 words)

	Eigenfunction Expansions (Site not responding. Last check: 2007-10-11)
		In this special case we are considering, p(x)=1 and we have two sets of eigenfunctions, un and vn.
		We will use a special eigenfunction expansion, called the FOURIER EXPANSION or FOURIER SERIES, that is a combination of the two eigenfunction expansions.
		The eigenfunction u0=1 is used but v0=0 is not.
	www.ma.iup.edu /projects/CalcDEMma/slfourier/slfourier2.html (140 words)

	[No title]
		Commutation, you'll recall, involves the fact that sometimes the order in which you 'operate' on a wavefunction matters to the final result.
		In the case we looked at above (the momentum of the particle in the ground state of the particle-in-a-box) we did not get the same wavefunction back when we operated on it.
		So the particle in a box wavrefunction we looked at was not an eigenfunction of the momentum operator.
	faculty.une.edu /cas/cnash/Part2FreeParticle.doc (1286 words)

	Eigenfunction expansion method in PDE solutions
		In case you ask: the linear differential operator d^2/dx^2 acts on the eigenfunction sin(nx), and yields the eigenfunction again scaled by an eigenvalue (-n^2).
		Because the space is a function space although functions can be part of a vector space because a vector space is the most general space in linear algebra?
		To work out by separation of variables is not possible, so we have to use eigenfunction expansion.
	www.physicsforums.com /showthread.php?threadid=134608 (1810 words)

	CJM - Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary ...
		Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions
		Form domains are characterized for regular $2n$-th order differential equations subject to general self-adjoint boundary conditions depending affinely on the eigenparameter.
		Corresponding modes of convergence for eigenfunction expansions are studied, including uniform convergence of the first $n-1$ derivatives.
	journals.cms.math.ca /cgi-bin/vault/view/binding1001 (76 words)

	[No title]
		Why is this so different from the eigenfunctions you have seen so far?
		Include a sketch of the resulting eigenfunction in your book.
		Question 4: Comment on the behaviour of the eigenfunction inside the barrier and also to the right of the barrier.
	bison.ph.bham.ac.uk /~wjc/Teaching/SCHROD04.doc (448 words)

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