Variational principles for real eigenvalues of self-adjoint operator pencils (original) (raw)

Some general local variational principles

Proceedings of the American Mathematical Society, 1992

Local variational min-sup characterizations are presented for the real spectrum of a selfadjoint operator pencil. Instead of minimizing over all subspaces of fixed codimension as in the classical result, the new characterizations minimize over subspaces that are close to extremal subspaces. In this way, the entire real spectrum, including continuous spectrum, can be characterized.

A survey on variational characterizations for nonlinear eigenvalue problems

ETNA - Electronic Transactions on Numerical Analysis, 2021

Variational principles are very powerful tools when studying self-adjoint linear operators on a Hilbert space H. Bounds for eigenvalues, comparison theorems, interlacing results, and monotonicity of eigenvalues can be proved easily with these characterizations, to name just a few. In this paper we consider generalizations of these principles to families of linear, self-adjoint operators depending continuously on a scalar in a real interval.

On the spectrum of linear operator pencils

Matematičnì studìï, 2019

We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of L(.) is the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.

On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations

Proceedings of the American Mathematical Society, 1987

Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equation y(n) + Xp(x)y = 0 for which the general eigenvalues comparison is not true. We use a comparison principle for the zeros of the corresponding Cauchy problem. This paper provides a contribution to the understanding of a problem raised by S. Ahmad and A. C. Lazer [1] in connection with the comparison of the eigenvalues for some multi-point boundary value problems which are not selfadjoint. One is given the equation (1) Lny + Xp(x)y = 0, where p(x) is a continuous function of constant sign on an interval /, A is a parameter, and Lny is a linear differential disconjugate operator of order n, that is, the only solution of Lny = 0 with n zeros on I (counting multiplicity) is y = 0. Let us consider the eigenvalue problem given by equation (1) and the system of boundary conditions ,, Lzy(a)=0, iG{ii,...,ik}, L]V(b)=0, JGiJu.-.Jn-k}, where o, b G I, 1 < k < n-1, Liy, i = 0,..., n-1, are the quasi-derivatives of y(x) (see [7]), and {t'i,..., ¿fc}, {ji, ■ ■ ■ ,jn-k) are two arbitrary sets of indices from the set {0,... ,n-1}. Problems of this type have been studied extensively (cf. [2, 3, 5]). In particular, Elias [5] has shown that if (-l)n_fcp(x) < 0, then the eigenvalues of problems (1) and (2) are real and nonnegative and form a divergence sequence {Am}m£N-Ahmad and Lazer [1] have considered a particular type of boundary condition (2), that is (3) y(a)=y'(a) =-= yik-1\a) = 0, y(b)=y'(b) =-=y(n-k-i\b)=0, and showed that if we set p = Pi, where p¿, i-1,2, are two continuous functions, considering the corresponding sequence of eigenvalues (A¿,m)m6N, i = 1,2, ordered by magnitude, then the condition (4)_ (-l)n-kp2(x) < (-l)"-fepi(x) < 0