A Homotopy Method for Finding All Solutions of a Multiparameter Eigenvalue Problem

2016 ◽  
Vol 37 (2) ◽  
pp. 550-571 ◽  
Author(s):  
Bo Dong ◽  
Bo Yu ◽  
Yan Yu
Keyword(s):  
Eigenvalue Problem ◽  
Homotopy Method ◽  
All Solutions ◽  
Multiparameter Eigenvalue Problem

scholarly journals The spinless relativistic kink-like problem

2015 ◽  
Vol 30 (12) ◽  
pp. 1550062 ◽  
Author(s):  
Wolfgang Lucha ◽  
Franz F. Schöberl
Keyword(s):  
Quantum Theory ◽  
Eigenvalue Problem ◽  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Schrodinger Equation ◽  
Bound State ◽  
State Equation ◽  
Hyperbolic Tangent ◽  
Central Potential ◽  
All Solutions

We constrain the possible bound-state solutions of the spinless Salpeter equation (the most obvious semirelativistic generalization of the nonrelativistic Schrödinger equation) with an interaction between the bound-state constituents given by the kink-like potential (a central potential of hyperbolic-tangent form) by formulating a bunch of very elementary boundary conditions to be satisfied by all solutions of the eigenvalue problem posed by a bound-state equation of this type, only to learn that all results produced by a procedure very much liked by some quantum-theory practitioners prove to be in severe conflict with our expectations.

scholarly journals An application of the homotopy method to the generalised symmetric eigenvalue problem

1988 ◽  
Vol 30 (2) ◽  
pp. 230-249
Author(s):  
Wen-Wei Lin ◽  
Gerhard Lutzer
Keyword(s):  
Eigenvalue Problem ◽  
Positive Definite ◽  
Homotopy Method ◽  
Smooth Curves ◽  
Symmetric Eigenvalue Problem ◽  
Tridiagonal Matrices ◽  
Almost All ◽  
Curve Tracing

AbstractThe homotopy method is used to find all eigenpairs of a generalised symmetric eigenvalue problem Ax = λBx with positive definite B. The determination of n eigenpairs (x, λ) is reduced to curve tracing of n disjoint smooth curves in Rn × R × [0, 1]. The method might be attractive if A and B are sparse symmetric. In this paper it is shown that the method will work for almost all symmetric tridiagonal matrices A and B.

11.— A Left Definite Multiparameter Eigenvalue Problem in Ordinary Differential Equations

1976 ◽  
Vol 74 ◽  
pp. 145-155 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Ellipticity Condition ◽  
Spectral Parameters ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville ◽  
Image Position

SynopsisThe main result of this paper is to establish the completeness of the eigenfunctions for the multiparameter eigenvalue problem defined by the system of ordinary differential equations0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditionsr = 1, …, k. In addition it is assumed that the coefficients ars of the spectral parameters λs, satisfy the ellipticity condition , s = 1, …, k, for all xrɛ[0, 1], r = 1, …, k, and some real k-tuple μ1, …, μk and where is the co-factor of asr in the determinant . The theory developed here contrasts with the results known when …k is assumed non-vanishing for all xrɛ[0,1].

Variational Methods for One and Several Parameter Non-Linear Eigenvalue Problems

1981 ◽  
Vol 33 (1) ◽  
pp. 210-228 ◽  
Author(s):  
Paul Binding
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Linear Operators ◽  
Main Thrust ◽  
Non Linear ◽  
Multiparameter Eigenvalue Problem ◽  
The One ◽  
Image Position ◽  
Parameter Problem

We shall consider a multiparameter eigenvalue problem of the form(1.1)where λ ∈ Rk while Tn and Vn(λ) are self-adjoint linear operators on a Hilbert space Hn. If λ = (λ1, …, λk) ∈ Rk and satisfy (1.1) then we call λ an eigenvalue, x an eigenvector and (λ, x) an eigenpair. While our main thrust is towTards the general case of several parameters λn, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation.

scholarly journals Asymptotic spectrum of multiparameter eigenvalue problems

1996 ◽  
Vol 39 (1) ◽  
pp. 119-132 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Hilbert Space ◽  
Maximum Principle ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Asymptotic Spectrum ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

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