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Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1968"


Journal ArticleDOI
TL;DR: In this paper, the eigenvalues C belonging to the manifold of solutions of the Orr-Sommerfeld equation are constructed by application of elementary isoperimetric inequalities, leading to a considerable improvement on the estimate of (αR) regions of linear stability given by Synge.
Abstract: Estimates of the eigenvalues C belonging to the manifold of solutions of the Orr-Sommerfeld equation are constructed by application of elementary isoperimetric inequalities. The inequalities also lead to a considerable improvement on the estimate of (αR) regions of linear stability given by Synge.

75 citations


Journal ArticleDOI
TL;DR: In this article, a general theory of Lusternik-Schnirelman type for nonlinear elliptic boundary value problems of variational type has been presented, and a general theorem on the existence of normalized eigenfunctions for the latter problem has been obtained.
Abstract: with u possibly satisfying additional normalization conditions. I t is our purpose in the present note to describe a way of applying a method of Galerkin type to such problems which works in particular for nonlinear elliptic boundary value problems of variational type. We obtain from it a general theorem on the existence of normalized eigenfunctions for the latter problem, and in the case of T and S odd operators, we obtain also an extremely general form of a theory of Lusternik-Schnirelman type guaranteeing the existence of infinitely many distinct normalized eigenfunctions. We consider first some restrictions that may be placed on the nonlinear operator T. DEFINITION 1. T is said to satisfy condition (S) if for any sequence {uj} in X with u$-+u in X and (T(UJ) — T(U), Uj—u)-^Oi we have Uj—>u in X. DEFINITION 2. T is said to satisfy condition (S)0 if for each sequence {UJ} in X with uj-^u in X, T(u/)—±z in X*, and (T(wy), Uj)—>(z, u), we have Uj-*u in X.

67 citations





Journal ArticleDOI
TL;DR: In this paper, it was shown that for θ ≡ 0 and λ 2 = μ i 2, the bifurcation method can be used to extend these two solutions to the right of μ 2 + δ 0, provided that λ2 = μ 2+δ 0 is not an eigenvalue of the linear operator evaluated at θ = ±ϑ1.
Abstract: We have investigated solutions of equation (3) when λ2 is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for θ ≡ 0 and λ2 = μ i 2 we have exactly two nontrivial solutions which bifurcate to the right of μ i 2 ; these solutions are shown to exist in an interval (μ i 2 , μ i 2 + δ0). The method of Section 3 may then be used to extend these two solutions to the right of μ i 2 +δ0 providing that λ2= μ i 2 +δ0 is not an eigenvalue of the linear operator (13) evaluated at θ= ±ϑ1. Either a solution can be uniquely extended, or there exists a value of λ2where the bifurcation method must be applied again3.

26 citations





Journal ArticleDOI
L. Collatz1
TL;DR: A classification of multiparametric eigenvalue problems in three classes (elliptic, parabolic, hyperbolic problems) is illustrated by examples of matrix and integral equations.

12 citations



Journal ArticleDOI
TL;DR: An eigenvalue problem arising in the treatment of laminar boundary layers nearly described by solutions of the Falkner-Skan equation is handled by application of quasilinearization as discussed by the authors.

Journal ArticleDOI
TL;DR: In this article, the boundary value and eigen value problems for uniformly elliptic equations are discussed. But they are only weak solutions and weak solutions are used to solve weak boundary value problems and weak eigenvalue problems.
Abstract: which is degenerate on the boundary. In this paper we discuss the first boundary value and eigenvalue problems for the elliptic equations of the same form which may degenerate in the interior of the domain. The equations treated in this paper include as their special type uniformly elliptic equations. We treat only weak solutions. However, we weaken the restriction on the coefficients. Our method to solve the problems owes to Sobolev [2], in which we find the boundary value and eigenvalue problems for the Laplace equation. In § 1 we arrange some inequalities to be used in the succeeding sections. Section 2 is devoted to some basic lemmas applied to a variational method. We solve, by a variational method, the first boundary value and eigenvalue problems in § 3 and § 4, respectively. The author wishes to express his sincere thanks to Prof. M. Hukuhara for his helpful suggestions and constant encouragement.



01 Jan 1968
TL;DR: In this article, a systematic procedure for solving the eigenvalue problem for a broad class of Hamiltonian operators containing no terms higher than quadratic in generalized coordinates and th e ir conjugate momenta was developed.
Abstract: A systematic procedure is developed for solving the eigenvalue problem for a broad class of Hamiltonian operators containing no terms higher than quadratic in generalized coordinates and th e ir conjugate momenta. The development is oriented toward practica l applications in the area o f the many-body problem. The procedure accomplishes a canonical reduction of such a Hamiltonian to the form of the Hamiltonian fo r a co llec tion of non in teracting bosons, with the eigenvalues of the Hamiltonian expressed in terms of the solutions to a single secular equation. Application of the resu lts to systems o f in teracting identical bosons is discussed, including a presentation of a useful calcu latlonal technique. The procedures developed are i l lu s tra te d by deta iled treatments o f two specific problems o f in terest in physics. The f i r s t problem considered is an exactly solvable separable potentia l model of p a r t ic le f ie ld theory. This problem consists of the description o f a co llec tio n of l ig h t bosons in teracting with an in f in i t e ly heavy boson v ia a simple separable p o te n tia l . In te res t in th is problem centers on i ts use as a test case for approximation techniques to be used on more complicated systems. The second and more r e a l is t ic problem investigated is the polaron problem of solid s ta te physics. This problem involves the