A class of nonlinear eigenvalue problems

Nonlinear eigenvalue problems for a class of ordinary differential equations

originates in the works of Mirzov [17] and Elbert [5], who named these equations `half linear’. This theory extends various aspects of oscillation theory, such as the Picone identity [10], Sturm comparison theorem [16, 17], oscillation and nonoscillation criteria of Kneser type [16] and Hille type [14], and other oscillation criteria [3]. One branch of this research is the study of the eigenval...

Solving a Class of Nonlinear Eigenvalue Problems by Newton’s Method

We examine the possibility of using the standard Newton’s method for solving a class of nonlinear eigenvalue problems arising from electronic structure calculation. We show that the Jacobian matrix associated with this nonlinear system has a special structure that can be exploited to reduce the computational complexity of the Newton’s method. Preliminary numerical experiments indicate that the ...

Nonlinear Eigenvalue Problems

Heinrich Voss Hamburg University of Technology 115.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-2 115.2 Analytic matrix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-3 115.3 Variational Characterization of Eigenvalues . . . . . . . . 115-7 115.4 General Rayleigh Functionals . . . . . . . . . . . . . . . . . . . ...

Solving nonlinear eigenvalue problems

A Class of Exactly-Solvable Eigenvalue Problems

The class of differential-equation eigenvalue problems −y′′(x)+x2N+2y(x) = xNEy(x) (N = −1, 0, 1, 2, 3, . . .) on the interval −∞ < x < ∞ can be solved in closed form for all the eigenvalues E and the corresponding eigenfunctions y(x). The eigenvalues are all integers and the eigenfunctions are all confluent hypergeometric functions. The eigenfunctions can be rewritten as products of polynomial...