Multiparameter Eigenvalue Problems and Expansion Theorems

Abstract

PREFACESince the pUblication of Atkinson's book "Multiparameter Eigenvalue Problems, Vol. I" in 1972, multiparameter spectral theory has become a subject of growing interest. Several authors have made important contributions to the theory; see the references at the end of this book. There are also two monographs of Sleeman (1978a) and McGhee and Picard (1988) It is not assumed that the reader knows already some multiparameter spectral theory but it is supposed that the reader is familiar with the usual one-parameter eigenvalue problems for compact Hermitian operators and their inverses. Brouwer's degree of maps will be used in the first two chapters. Basic properties of the tensor product of linear spaces will be needed in Chapters 4,5,6.

Spectral collocation solutions to multiparameter Mathieu’s system

“…Volkmer analyses in his monograph [1] the above four systems in detail. For these right definite MEP, he provides results concerning the existence and countability of eigenvalues, numbers of zeros of eigenfunctions and the completeness of even and odd sets of eigenmodes.…”

“…For each of the four systems we know from Volkmer [1] that for every pair of nonnegative indices (i, j) there exists a pair (a ij , q ij ) with nonzero functions F ij and G ij such that G ij has exactly i zeros on (0, π 2 ) and F ij has exactly j zeros on (0, ξ 0 ). This is one way how we can index the solutions.…”

Root vectors for geometrically simple two-parameter eigenvalues

“…We consider a pair of two-parameter equations of the form The spectral theory of such systems has been developed from various viewpoints; see [1,8,22,24,26] for books on the subject. In most of the literature, the A ij are selfadjoint and satisfy definiteness conditions guaranteeing that the eigenvalues are simple in certain senses.…”

Right-Definite Multiparameter Sturm–liouville Problems With Eigenparameter-Dependent Boundary Conditions

“…We refer to Ince [14] for the oscillation theorem and to [17] for oscillation results under weaker conditions on the coefficients and under alternative definiteness conditions.…”