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About 5,464 papers for “Eigenfunction”

Eigenfunction

March-Russell, John · AccessScience · 2015

AccessScience is an authoritative and dynamic online resource that contains incisively written, high-quality educational material covering all major scientific disciplines. An acclaimed gateway to scientific knowledge, AccessScience is continually expanding the ways it can demonstrate and explain core, trustworthy scientific information that inspires and guides users to deeper knowledge.

Cited by 1

Eigenfunction rules

L. Dobrzyński, Housni Al Wahsh, Abdellatif Akjouj et al. · Greater South Information System · 2021

It is well known that eigenfunctions have to be continuous at each space point of the structure in which they are localized. This is not the only rule they have to fulfill. Rules involving eigenfunction derivatives are also used in the bulk and at the boundary of all physical systems. The aim of this section is to try and reach a general formulation of the eigenfunction rules. At each space point of this network, it is found that the eigenfunctions are continuous and the sum of their outgoing first derivatives is equal to zero. These rules fix the values of the discrete eigenvalues.

Cited by 0Open access

Generalized Eigenfunction Expansions

Emin Karayel · Figshare · 2014

Generalized eigenfunction expansions based on commutator relations. Improving on the embeddings obtained by Hilbert-Schmidt-Riggings.

Cited by 0Open access

Generalized Eigenfunction Expansions

Karayel, Emin · Figshare · 2014

Generalized eigenfunction expansions based on commutator relations. Improving on the embeddings obtained by Hilbert-Schmidt-Riggings.

Cited by 0Open access

A nonhomogeneous eigenfunction expansion

Allan M. Krall · Transactions of the American Mathematical Society · 1965

Non-selfadjoint expressions of the form ly --y" + q(x)y have begun to be studied extensively in the past few years, and in two instances Naimark [4] (on the interval [0, oo)) and Kemp [3] (on the interval (-co, oo)) have achieved expansions in terms of eigenfunctions of /.By applying a suitable boundary condition, this paper generates an ordinary expansion and also a nonhomogeneous expansion, that is, an expansion involving solutions of -v" + (q(x) -X)y = K(x), in [0, oo), thus extending the work of Naimark [4].Nonhomogeneous expansions originated with Hilb [2] and Betschler [1].1.The operator L. We consider a differential expression of the form ly = -y" + q(x)y, 0 í£ x < oo, where q(x) is an arbitrary measurable complex function satisfying J""| q(x) | dx < oo.We denote by D0 those functions / defined on [0, oo) and satisfying l./isinL2(0, oo), 2. /' exists and is absolutely continuous on every finite subinterval of [0, oo), 3. Z/isinL2(0, oo).Let K(x) be an arbitrary complex-valued function in L2(0, oo), and let a and ß be arbitrary complex numbers.We denote by D those functions/satisfying 1./is in ö0, 2. Sl?K(x)f(x)dx -ßf(0) + af(0) = 0.We define the operator L by Lf = // for all / in D.Theorem 1.1.1/ | a \2 + \ ß \2 ¿ 0, D is dense in L2(0, oo).Let U be those functions g in D0 vanishing with their derivatives for large x and in a neighborhood of the origin and satisfying $çK(x)g(x)dx = 0. Since U is dense in Rx (see [4, pp.105-108]) and U c D, we see that D is dense in Rx.If D is not dense in L2 (0, oo), there is a nonzero vector orthogonal to D and hence to Rx.This vector must be a scalar multiple of R.But then » K(x)f(x)dx = 0 for all/in D, and hence ßf(0)-af'(0) -0 for all/in D. Clearly this is impossible.2. Solutions of ly -Xy = 0. Since we are working with the interval [0, oo), it is convenient to use the same solutions of ly -Xy -0 as [4].

Cited by 29